{On the representations of $\Spp(p,q)$ and $\SO^\ast(2n)$ unitarily induced from derived functor modules}
{Hisayosi Matumoto}\thanks{Supported in part by NSF-grant DMS-9100367 and Grant-in-aid for Scientific Research (No.\ 50272597) .}
Graduate School of Mathematical Sciences
3-8-1 Komaba
Tokyo
153-8914
JAPAN
We obtain a decomposition formula of a representation of $\Spp(p,q)$ or $\SO^\ast(2n)$ unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of $SO^\ast(2n)$ and $\Spp(p,q)$ induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.\footnote{Keywords: unitary representation, parabolic induction, derived functor module \\ AMS Mathematical Subject Classification: 22E46, 22E47}
math0203107.tex

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\title{{\sf On the representations of $\Spp(p,q)$ and $\SO^\ast(2n)$ unitarily induced from derived functor modules}}
\author{{\sf Hisayosi Matumoto}\thanks{Supported in part by NSF-grant
DMS-9100367 and Grant-in-aid for Scientific Research (No.\ 50272597) .}
\\Graduate School of Mathematical Sciences\\ University of Tokyo\\ 3-8-1
Komaba, Tokyo\\ 153-8914, JAPAN\\ e-mail: hisayosi@ms.u-tokyo.ac.jp}
\date{}
\maketitle
\begin{abstract}
We obtain a decomposition formula of a representation of  $\Spp(p,q)$ or
 $\SO^\ast(2n)$ unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules.
In particular, we show such an induced representation is decomposed into
 a direct sum of irreducible unitarily induced modules from derived
 functor modules under some regularity condition on the parameters.
In particular, representations of $SO^\ast(2n)$ and $\Spp(p,q)$ induced from
 one-dimensional unitary representations of their parabolic subgroups
 are irreducible.\footnote{Keywords: 
 unitary representation, parabolic induction,
 derived functor module \\ AMS Mathematical Subject Classification: 22E46, 22E47}
\end{abstract}

\setcounter{section}{0}
\section*{\S\,\, 0.\,\,\,\, Introduction}
\setcounter{subsection}{0}

Our object of study is the decomposition of unitarily induced modules of
a real reductive Lie groups from derived functor modules.
In [22], the case of $U(m,n)$ is treated.
In this article, we study the case of $\Spp(p,q)$ and $\SO^\ast(2n)$.
Reducibilities of the representations of $U(m,n)$ unitarily
induced from derived functor modules is coming from the reducibility of
particular degenerate principal series of $\U(n,n)$ found by Kashiwara-Vergne
[9].
In the case of $\Spp(p,q)$ and $\SO^\ast(2n)$ the situation is quite similar, at least for regular
values of the parameter.
The reducibilities are also reduced to the
Kashiwara-Vergne decomposition.

We are going into more details.
Put $G=\Spp(p,q)$ \,\, $(p\geqslant q)$ or $G=\SO^\ast(2n)$. 
We fix a Cartan involution $\theta$ as usual.
Let $\kappa=(k_1,...,k_s)$ be a finite sequence of positive integers
such that 
\[
 k_1+\cdots+k_s\leqslant \left\{ \begin{array}{ll} q & \,\,\,\, \mbox{if $G=\Spp(p,q)$},
\\
\frac{n}{2} & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}.
\end{array}
\right. .
\]
If $G=\Spp(p,q)$, put  $p^\prime=p-k_1-\cdots-k_s$ and
$q^\prime=q-k_1-\cdots-k_s$.
If $G=\SO^\ast(2n)$, put $r=n-2(k_1-\cdots-k_s)$.
Then, there is a parabolic subgroup $P_\kappa$ of $G$, whose Levi
subgroup $M_\kappa$ is written as 
\[ M_\kappa\cong 
\left\{\begin{array}{ll} 
\GL(k_1,\qrt)\times\cdots\times\GL(k_s,\qrt) \times\Spp(p^\prime,q^\prime) & \,\,\,\, \mbox{if $G=\Spp(p,q)$}\\
\GL(k_1,\qrt)\times\cdots\times\GL(k_s,\qrt)\times \SO^\ast(2r) & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}
\end{array}\right. .\]
Here, formally, we denote by $\Spp(0,0)$ and $\SO^\ast(0)$ the trivial group $\{1\}$.
Any parabolic subgroup of $G$ is $G$-conjugate to some $P_\kappa$.
$\GL(k,\qrt)$ has some particular irreducible unitary representation
so-called quaternionic Speh representations defined as follows.
We consider $\GL(k,\cpx)$ as a subgroup of $\GL(k,\qrt)$.
For $\ell\in\itg$ and $t\in\sqrt{-1}\rel$, we define a one-dimensional unitary representation
$\xi_{\ell,t}$ of $\GL(k,\cpx)$ as follows.
\[
 \xi_{\ell, t}(g)=\left(\frac{\det(g)}{|\det(g)|}\right)^\ell|\det(g)|^{t}.
\]
$\GL(k,\cpx)$ is the centralizer in $\GL(k,\qrt)$ of the group
consisting of complex scalar
matrices with the eigenvalue of the absolute value one.
So, there is a $\theta$-stable  parabolic subalgebra
$\qqq(k)$ with a Levi
subgroup $\GL(k,\cpx)$.
We choose the nilradical $\nnn(k)$ so that $\xi_{\ell,t}$ is good with respect
to $\qqq(k)$ for sufficient large $\ell$.
Derived functor modules with respect to $\qqq(k)$ are called quaternionic Speh representations.
For $t\in \sqrt{-1}\rel$, there is a one-dimensional unitary representation
$\tilde{\xi}_t$ of $\GL(k,\qrt)$ whose restriction to $\GL(k,\cpx)$ is
$\xi_{0,t}$.
We put  
\[
 A_k(\ell,t)=({}^u\rca_{\qqq(k),\Oo(k)}^{\gl(k,\qrt)\otimes_\rel\cpx,
\Spp(k)})^{k(k+1)}(\xi_{\ell+2k,t})   \,\,\,\,\,  \,\, (\ell \in\itg).
\]
Here, ``${}^u\rca$'' means the cohomological induction ([14], also see 1.3).
We also put
\[
 A_k(-\infty,t)=\tilde{\xi}_t.
\]
For $\ell\in\itg$, $A_k(\ell,t)$ is derived functor module in the good (resp.\ weakly fair) range
in the sense of [35] if and only if $\ell\geqslant 0$ (resp.\
$\ell\geqslant -k$).
It is more or less known by [34] that any derived functor module
of $\GL(k,\qrt)$ is a unitary parabolic induction from one-dimensional
representations or quaternionic Speh representations.
So, it suffices to  consider the following induced representation.
\begin{equation*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z). \tag{$\circledast$}
\end{equation*}
Here, $Z$ is a derived functor module of $\Spp(p^\prime,q^\prime)$ or
$\SO^\ast(2r)$ in the weakly fair range.
Moreover,  $\ell_i\in \{\ell\in\itg\mid \ell\geqslant
-k_i\}\cup\{-\infty\}$, and $t_i\in\sqrt{-1}\rel$ for $1\leqslant
i\leqslant s$.
If we apply Harish-Chandra's result that the equivalence class of a
representation parabolically induced from a unitary representation
$(\pi_M, M)$ depends only on the conjugacy class of $(\pi_M,M)$, we see
that permuting the $A_{k_i}(\ell_i,t_i)$s does not change the induced representation.
 We assume that $\ell_i+1\in 2\itg$ and $t_i=0$ for some $1\leqslant
i\leqslant s$.
Thus, we may assume $\ell_s+1\in 2\itg$ and $t_s=0$.
Let $\kappa^\prime=(k_1,...,k_{s-1})$.
Then from the induction-by-stages, we have
\begin{multline*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z)
\\
\cong
\Ind_{P_\kappa^\prime}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes A_{k_{s-1}}(\ell_{s-1},t_{s-1})
 \boxtimes\Ind_{P_{(k_s)}}^{M_{\kappa^\prime}^\circ}(A_{k_s}(\ell_s,0)\boxtimes Z)).
\end{multline*}
Here, $M^\circ_{\kappa^\prime}$ is $\Spp(p^\prime+k_s,q^\prime+k_s)$ or
$\SO^\ast(2(r+2k_s))$.

Our reducibility result is:

{\bf Theorem A} \,\,\, (Theorem 3.6.5)

{\it $\Ind_{P_{(k_s)}}^{M_{\kappa^\prime}}(A_{k_s}(\ell_s,0)\boxtimes Z)$
is decomposed into a direct sum of derived functor modules of
$M_{\kappa^\prime}^\circ$ in weakly fair range.
We obtain an explicit decomposition formula.}

Whenever there is  $1\leqslant
i\leqslant s$ such that $\ell_i+1\in 2\itg$ and $t_i=0$, we can apply
the above procedure.
Assuming that we understand the reducibility of derived functor modules,
we can reduce the irreducible
decomposition of the above induced module to the following.
\begin{equation*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)\boxtimes
A_{k_{h+1}}(\ell_{h+1},t_{h+1})\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z). \tag{$\lozenge$}
\end{equation*}
Here, $\ell_i$ is not odd integer if $1\leqslant i\leqslant h$,
$\sqrt{-1}t_i>0$ if $h<i\leqslant s$, and $Z$ is an irreducible
representation of  $M_\kappa^\circ$ whose infinitesimal character plus
the half sum of positive roots can be
realized as a weight of a finite dimensional representation of $G$.
Put $\tau=(k_1,...,k_h)$ and $\tau^\prime=(k_{h+1},...,k_s)$.
Also put $a=k_1+\cdots+k_h$ and  $b=k_{h+1}+\cdots+k_s$.
In this setting we have:

{\bf Theorem B} \,\,\, (Theorem 4.2.2)
{\it The following are equivalent.

(1) \,\,\,\,
The induced representation $\lozenge$ is irreducible.

(2) \,\,\,\, The following induced module is irreducible.
\begin{equation*} 
\Ind_{P_{\tau}}^{\SO^\ast(4a)}(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)). 
\end{equation*}
}

Under an appropriate regularity condition on $\ell_1,...,\ell_h$, we may
show the irreducibility of the induced module in the above (2).

On the induced representation in Theorem B(2), we have a partial answer:

{\bf Lemma C} \,\,\, (Theorem 5.1.1)

{\it If $\ell_1,...,\ell_h$ are all $-\infty$ (namely,
if $A_{k_1}(\ell_1,0)$,..., $A_{k_h}(\ell_h,0)$ are trivial representations,)
$\Ind_{P_{\tau}}^{\SO^\ast(4a)}(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0))$
is irreducible.
}

As a corollary of this result, we have:

{\bf Corollary D} \,\,\, (Corollary 5.1.2)
{\it
Representations of $SO^\ast(2n)$ and $\Spp(p,q)$ induced from
 one-dimensional unitary representations of their parabolic subgroups
 are irreducible.
}

For some special parabolic subgroups, the irreducibility of the above
kind of induced representations has been known.
If the parabolic subgroup is minimal, the irreducibility of the induced representation is a special case of a general result in [16] (also see [5]).
Studies of Johnson, Sahi, and Howe-Tan 
([7], [23], [6]) also include
the irreducibility of the induced modules from a unitary one-dimensional representations of some maximal parabolic subgroups. 


The remaining problems on the reducibility of the representations of
$\Spp(p,q)$ and $\SO^\ast(2n)$ unitarily induced from derived functor
modules in the weakly fair region are:

(1) \,\,\, Vanishing and irreducibilities  of derived functor modules of
$\Spp(p,q)$ and $\SO^\ast(2n)$ in the weakly fair range. 

(2)\,\,\,  Irreducibilities of the induced representation of the form:
\[
 \Ind_{P_\tau}^{SO^\ast(4a)}(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)).
\]
(Here, $\ell_i$ \,\, $(1\leqslant i\leqslant h)$) are even
integers or $-\infty$.)

Regrettably, I do not have a complete answer on the above problem.
For a type A group $U(m,n)$, general theories on translation principle
are applicable to the above problem on  irreducibilities.
Together with Trapa's result [28], we have a complete answer.
Unfortunately, neither $\Spp(p,q)$ nor  $\SO^\ast(2n)$ are of type A.
So, situation is more difficult than the case of $U(m,n)$.
In fact, irreducibility of a derived functor module of
$\Spp(p,q)$ fails in some singular parameter (Vogan).
If the degeneration of the parameter is not so bad, Vogan ([35])
found an idea to control irreducibilities. 
Using the idea, he proved irreducibility of discrete
series of semisimple symmetric spaces.
This idea works in this case.
In fact, using Vogan's idea Kobayashi studied irreducibilities  of derived functor modules of
$\Spp(p,q)$ in [15].
In subsequent article, I would like to take up this problem.

One of the main ingredient of this article is the change-of-polarization
formula (Theorem 2.2.3).
It means we may exchange, under some positivity condition, the order of cohomological induction and
parabolic induction in the Grothendieck group of the category of the
Harish-Chandra modules.
The change of polarization for a standard module was originated by Vogan
([31]) and completed by Hecht, Mili\v{c}i\'{c}, Schmid, and Wolf
(cf.\ [24]).   Also see [14] Theorem 11.87.
For the degenerate setting, some case is observed for  $\GL(n)$ in
[34].
We apply this idea in [21].
In Theorem 2.2.3, we gave a formulation of the change-of-polarization
in the general setting.

The other ingredient of this article is comparison of the Hecke algebra
module structures. 
In fact, the irreducible decomposition of standard representation is
determined only by the Hecke algebra module structure via so-called
Kazdhan-Lusztig
algorithm ([1]).
This deep result enable us to compare irreducibilities of induced
representations of different groups with the same Hecke algebra module structures.
Using this idea, we show Theorem B.

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setcounter{section}{1}
\section*{\S\,\,1.\,\,\,\, Preliminaries}
\setcounter{subsection}{0}
\subsection{ General notations}

In this article, we use the following notations.

As usual we denote the Hamilton quaternionic field, the complex number field, the real number field, the rational number field, the ring of integers, and the set of non-negative integers by $\qrt$,  $\cpx$, $\rel$, $\rat$, $\itg$, and $\nat$ respectively.

We denote by $\emptyset$ the empty set
and denote by $A-B$ the set theoretical difference of $A$ from $B$.
For each set $A$, we denote by $\card A$ the cardinality of $A$.
For a complex number $a$ (resp.\ a matrix $X$ over $\cpx$), we denote by
$\bar{a}$ (resp.\ $\overline{X}$) the complex conjugation. For the
Hamilton quaternionic field,
we also use a similar notation.
If $p>q$, we put $\sum_{i=p}^q =0$.

Let $R$ be a ring and let $M$ be a left $R$-module.  We denote by $\Ann_R(M)$ the annihilator of $M$ in $R$.
In this article, a character of a Lie group $G$ means a (not necessarily unitary) continuous homomorphism of $G$ to $\cpx^\times$.
For a matrix $X=(a_{ij})$, we denote by ${}^tX$, $\tr X$, and $\det X$ the transpose $(a_{ji})$ of $X$, the trace of $X$, and the determinant of $X$ respectively.
For a positive integer $k$, we denote by $I_k$ (resp.\ $0_k$) the $k\times k$-identity (resp.\ zero) matrix.

Let $n, n_1,...,n_\ell$ be positive integers such that $n=n_1+\cdots+n_\ell$.
For $n_i\times n_i$-matrices $X_i$ \, $(1\leqs i\leqs \ell)$ , we put
\[ \diag(X_1,...,X_\ell)=\begin{pmatrix} X_1 &0& \hdotsfor{2} &0 \\
0& X_2 & 0 & \dots &0 \\
\hdotsfor{5} \\
 0 & \hdotsfor{2} &0 & X_\ell
\end{pmatrix}
\]


We denote by ${\mathfrak S}_\ell$ the $\ell$-th symmetric group.

For a complex Lie algebra $\gggg$, we denote by $U(\gggg)$ its universal
enveloping algebra.
We denote by $Z(\gggg)$ the center of $U(\gggg)$.
For a Harish-Chandra module $V$, we denote by $[V]$ the corresponding distribution character.
In this article, an irreducible Harish-Chandra module should be
non-zero.


\subsection{{Notations for root systems}}

Let $G$ be a connected real reductive linear group, and let $G_\cpx$ be its complexification.
We fix a maximal compact subgroup $K$ of $G$ and denote by $\theta$ the corresponding Cartan involution.
We denote by $\gggg_0$ (resp.\ $\kkk_0$) the Lie algebra of $G$ (resp.\
$K$).
Let $H$ be a $\theta$-stable Cartan subgroup of $G$ and let $\hhh_0$ be
its Lie algebra.
We denote by $\gggg$ , $\kkk$, and $\hhh$  the complexifications of
$\gggg_0$ , $\kkk_0$, and $\hhh_0$, respectively.
We denote by $\hhh^\ast$ the complex dual of $\hhh$.
We denote the induced involution from $\theta$ on $\gggg$, $\hhh$,
$\hhh^\ast$ by the same letter $\theta$.
We denote by $\sigma$ he complex conjugation on $\gggg$ with respect to $\gggg_0$.
We denote by $W(\gggg,\hhh)$ (resp.\ $\Delta(\gggg,\hhh)$) the Weyl group (resp.\ the root system) with respect to the pair ($\gggg$, $\hhh$).
Let $\langle \,\,.\,\,\rangle$ be a $W(\gggg,\hhh)$-invariant bilinear
form on $\hhh^\ast$ induced from an invariant non-degenerate bilinear
form of $\gggg$.

A root $\alpha\in \Delta(\gggg,\hhh)$ is called imaginary (resp.\ real) if $\theta(\alpha)=\alpha$ (resp.\ $\theta(\alpha)=-\alpha$).
A root $\alpha\in \Delta(\gggg,\hhh)$ is called complex if $\alpha$ is neither real nor imaginary.
A imaginary root $\alpha\in \Delta(\gggg,\hhh)$ is called compact (resp.\ noncompact) if the root space for $\alpha$ is contained (resp.\ not contained) in $\kkk$.

We denote by $\pca(\hhh)$ the integral weight lattice in $\hhh^\ast$. 
Namely, we put
\[\pca(\hhh)=\left\{\lambda\in\hhh^\ast \left| 2\frac{\langle
\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\itg \,\,\,
(\alpha\in\Delta(\gggg,\hhh))\right\}\right. .\]
We also put
\[\pca_G(\hhh)=\left\{\lambda\in\hhh^\ast | \mbox{$\lambda$ is a weight of some finite dimensional representation of $G$.} \right\}.\]
We denote by $\qca(\hhh)$ the root lattice, namely the set of integral linear combination of elements of $\Delta(\gggg,\hhh)$.
We have $\qca(\hhh)\subseteq\pca_G(\hhh)\subseteq\pca(\hhh)\subseteq\hhh^\ast$.

For $\lambda\in\hhh^\ast$, we denote by $\chi_\lambda$ the corresponding Harish-Chandra homomorphism $\chi_\lambda : Z(\gggg)\rightarrow \cpx$.

We fix a $\theta$-stable maximally split Cartan subgroup $\sH$ of $G$ and denote by $\shh$ its complexified Lie algebra.
For simplicity, we write $\Delta$, $W$, $\pca$, $\pca_G$, $\qca$ for $\Delta(\gggg,\shh)$, $W(\gggg,\shh)$, $\pca(\shh)$, $\pca_G(\shh)$, $\qca(\shh)$, respectively.

We choose regular weights $\lambda\in\hhh^\ast$ and $\spl\lambda\in\shh^\ast$ such that $\chi_{\lambda}=\chi_{\spl\lambda}$.
Then, there is a unique isomorphism $\bi_{\spl\lambda,\lambda} : \shh^\ast\rightarrow\hhh^\ast$ induced from an inner automorphism of $G$
 such that $\bi_{\spl\lambda,\lambda}(\spl\lambda)=\lambda$. 
We denote by the same letter $\bi_{\spl\lambda,\lambda}$ the corresponding isomorphism of $W$ onto $W(\gggg,\hhh)$.


\subsection{{Cohomological inductions}}

We fix the notations on the Vogan-Zuckerman cohomological inductions of Harish-Chandra modules. 
Here, we adapt the definition found in {\sf [14]}.
Let $G$ be a real reductive linear Lie group which is contained in the
complexification $G_\cpx$.
We assume $G_\cpx$ is a connected complex reductive linear group.


\begin{dfn}
Assume that a parabolic subalgebra $\qqq$ has a Levi decomposition $\qqq=\llll+\uuu$ such that $\llll$ is stable under $\theta$ and $\sigma$. 
Such a Levi decomposition is called an orderly Levi decomposition.
\end{dfn}

A $\theta$-stable or $\sigma$-stable parabolic subalgebra has a unique
 orderly Levi decomposition.
In fact, if $\qqq$ is $\theta$ (resp.\  $\sigma$)-stable, then
 $\llll=\qqq\cap\sigma(\qqq)$ (resp.\ $\llll=\qqq\cap\theta(\qqq)$  ).

Let $\qqq$ be a parabolic subalgebra of $\gggg$ with an orderly Levi decomposition $\qqq=\llll+\uuu$.
We fix a $\theta$ and $\sigma$-stable Cartan subalgebra $\hhh$ of $\llll$ and a Weyl group invariant non-degenerate bilinear form $\langle\,\,,\,\,\rangle$.
Let $L$ be the corresponding Levi subgroup in $G$ to $\llll$.

We denote by ${}^u\rca^{\gggg, K}_{\qqq, L\cap K}$ the right adjoint functor of the forgetful functor of the category of $(\gggg, K)$-modules to the category of $(\qqq, L\cap K)$-modules. 
Introducing trivial $\uuu$-action, we regard an $(\llll, L\cap K)$-module as a $(\qqq, L\cap K)$-module.
So, we also regard ${}^u\rca^{\gggg, K}_{\qqq, L\cap K}$ as a functor of the category of $(\llll, L\cap K)$-modules  to the category of $(\gggg, K)$-modules.
We denote by $({}^u\rca^{\gggg, K}_{\qqq, L\cap K})^i$ the $i$-th right derived functor.
(See [14] p671)

We review a normalized version.
We define a one-dimensional representation
$\delta(\uuu)$ of $\llll$ by $\delta(\uuu)(X)=\frac{1}{2}\tr (\ad(X)|_{\uuu})$.
Following [14] p720, we define a one-dimensional representation
$\cpx_{2\delta(\uuu)^\prime}$ of $L$ as follows.
(For later use, we introduce slightly more general setting.)
Let $V$ be a finite dimensional semisimple $\llll$-module.
We define a one-dimensional representation $\delta(V)$ of $\llll$ by $\delta(V)(X)=\frac{1}{2}\tr (X|_V)$.
Let $V=V_1\oplus V_2\oplus\cdots\oplus V_k$ be the decomposition of $V$ into irreducible $\llll$-modules.
We distinguish between those $V_i$ that are self-conjugate with respect to $\sigma$ and those are not.
We define a one dimensional representation $\xi_{2\delta(V)^\prime}$  of  $L$ on a space $\cpx_{{2\delta(V)^\prime}}$ by
\[\xi_{2\delta(V)^\prime}(\ell)=\left(\prod_{\mbox{$i$ with $V_i$ self-conjugate}}|\det(\ell|_{V=i})|\right)
\left(\prod_{\mbox{$i$ with $V_i$ not self-conjugate}}\det(\ell|_{V=i})\right).\]

Let $L^\sim$ be the metaplectic double cover of $L$ with respect to $\cpx_{{2\delta(V)^\prime}}$.
Namely,
\[ L^\sim=\{ (\ell,z)\in L\times \cpx^\times\mid \xi_{2\delta(V)^\prime}(\ell)=z^2\}.\]
We define the one-dimensional $L^\sim$-module
$\cpx_{{\delta(V)^\prime}}$ by the projection to the second factor of $L^\sim\subseteq L\times \cpx^\times$. 
Of course, the definition of $L^\sim$ depends on $V$.
Hereafter, we consider the case of $V=\uuu$ (the adjoint action of $L$ on $\uuu$).
Let $(K\cap L)^\sim$ be the maximal compact subgroup of $L$ corresponding to $K\cap L$.

Let $Z$ be a Harish-Chandra $(\llll,(K\cap L)^\sim)$-module such that
 $Z\otimes\cpx_{\delta(\uuu)}$ is a Harish-Chandra $(\llll, K\cap L)$-module.
We put
\[({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^i(Z)=({}^u\rca^{\gggg, K}_{\qqq, L\cap K})^i(Z\otimes\cpx_{\delta(\uuu)^\prime}).\]
Let $\lambda$ be the infinitesimal character of $Z$ with respect to $\hhh$.
(It is well-defined up to the Weyl group action of $\llll$.)
Then $({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^i(Z)$ is a Harish-Chandra $(\gggg, K)$-module of an infinitesimal character $\lambda$.

We consider three particular cases.

(1) \,\, {\it Hyperbolic case} \,\,\,\,  If $\qqq$ is $\sigma$-stable,
then there is a parabolic subgroup $Q=LU$ whose complexified Lie algebra is $\qqq$ and whose nilradical is $U$.
In this case, we have 
$({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^i(Z)=0$
for all $i>0$.
In fact, $ ({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^0(Z)$ is nothing but the parabolic induction $\Ind_Q^G(Z)$.

We clarify the definition of the parabolic induction.
First, we remark that $L^\sim$ is just a direct product $L\times \{\pm 1\}$ in this case and $\cpx_{\delta(\uuu)}$
 can be reduced to a representation of $L$ (say $(\xi_{\delta(\uuu)}, \cpx_{\delta(\uuu)})$ ) .

$\Ind_Q^G(Z)$ (or we also write $\Ind(Q\uparrow G;Z)$) is the $K$-finite part of
\[\{f\in C^\infty(G)\otimes H\mid f(g\ell n)=\pi(\ell^{-1})f(g) \,\,\,\,\, (g\in G, \ell\in L, n\in U)\}.\]
Here, $(\pi, H)$ is a Hilbert globalization of $Z\otimes\cpx_{\delta(\uuu)}$.
If $Z$ is unitarizable, so is $\Ind(Q\uparrow G;Z)$ (unitary induction).
We also consider the unnormalized parabolic induction as follows.
\[{}^u\Ind(Q\uparrow G;Z)=\Ind(Q\uparrow G;Z\otimes\cpx_{\delta(\buu)}).\]


(2) \,\, {\it Elliptic case} \,\,\,\, Assume $\qqq$ is $\theta$-stable and put $S=\dim(\uuu\cap\kkk)$.
We call $Z$ weakly good (or $\lambda$ is in the weakly good range), if
$\Real\langle\lambda,\alpha\rangle\geqslant 0$ holds for each root $\alpha$ of $\hhh$ in $\uuu$.
We call $Z$ integrally good (resp.\ weakly integrally good ), if $\langle\lambda,\alpha\rangle>0$ (resp.\
$\langle\lambda,\alpha\rangle\geqslant 0$) holds for each root
$\alpha$ of $\hhh$ in $\uuu$ such that $2\frac{\langle \lambda,
\alpha\rangle}{\langle \alpha, \alpha\rangle}\in \itg$.

\begin{thm}
$\mbox{([33]  Theorem 2.6)}$

(a) \,\, If $Z$ is weakly integrally good, then
 $({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^i=0$ for $i\neq S$.

(b) \,\, If $Z$ is  irreducible and weakly integrally good,
 $({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^S(Z)$ is irreducible or zero.

(c) \,\, If $Z$ is  irreducible and integrally good, $({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^S(Z)$ is irreducible.

(d) \,\, If $Z$ is  unitarizable and weakly good, $({}^n\rca^{\gggg, K}_{\qqq,
L\cap K})^S(Z)$ is unitarizable.
\end{thm}

\bigskip

(3) \,\, {\it Standard modules} \,\,\,\,\,
A regular character $(H,\Gamma, \lambda)$ is a triple satisfying the
following conditions (R1)-(R6).
(cf.\ [30])

(R1) \,\,\,\, $H$ is a $\theta$-stable Cartan subgroup of $G$.

(R2) \,\,\,\, $\Gamma$ is a (non-unitary) character of $H$.

(R3) \,\,\,\, $\lambda$ is in $\hhh^\ast$. (Here, $\hhh$ is the complexified Lie algebra of $H$.)

In order to write down remaining conditions, we introduce some notations.
Let $\ttt$ (resp.\ $\aaa$) the $+1$ (resp.\ $-1$) eigenspace in $\hhh$ with respect to $\theta$.
We denote by $\mmm$ the centralizer of $\aaa$ in $\gggg$.
Then $\Delta(\mmm,\hhh)$ is the set of imaginary roots in $\Delta(\gggg,\hhh)$.

(R4) \,\,\,\, $\lambda$ is regular with respect to $\Delta(\mmm,\hhh)$.

(R5) \,\,\,\, $\langle \lambda,\alpha\rangle$ is real for any $\alpha\in\Delta(\mmm,\hhh)$.

Under the above conditions (R4) and (R5), there is a unique positive system
$\Delta^+_\lambda(\mmm,\hhh)$ of $\Delta(\mmm,\hhh)$ such that
$\langle\alpha,\lambda\rangle>0$ for all $\alpha\in\Delta^+_\lambda(\mmm,\hhh)$.
We denote by $\rho_\lambda(\mmm,\hhh)$ (resp.\
$\rho_\lambda^c(\mmm,\hhh)$) the half sum of positive imaginary roots
(resp. positive compact imaginary roots) with respect to $\Delta^+_\lambda(\mmm,\hhh)$.
We put $\mu_\lambda=\lambda+\rho_\lambda(\mmm,\hhh)-2\rho_\lambda^c(\mmm,\hhh)$.

(R6) \,\,\,\, $\mu_\lambda$ is the differential of $\Gamma$.

\medskip

We fix a regular character $\gamma=(H,\Gamma,\lambda)$. 
We denote by $M$ the centralizer of $\aaa$ in $G$.
The above conditions (R1)-(R5) assure that there is a unique relative
discrete series representation $\sigma$ with infinitesimal character
$\lambda$ such that the Blattner parameter of $\sigma$ is $\Gamma$.
Here, a relative discrete series means a representation whose restriction to semisimple part is in discrete series.  
We do not require the unitarizability of $\sigma$ itself.
We fix a parabolic subgroup $P$ of $G$ such that $M$ is a Levi part of $P$.
We define the standard module $\pi_G(\gamma)$ (We simply write $\pi(\gamma)$, if there is no confusion.) for  a regular character $\gamma=(H,\Gamma,\lambda)$ by $\pi_G(\gamma)=\Ind_P^G(\sigma)$.
The distribution character $[\pi_G(\gamma)]$ is independent of the
choice of $P$.

We may describe $\pi_G(\gamma)$ in terms of the cohomological induction
as follows.
First, let $\bbb_0$ be the Borel subalgebra of $\mmm$ corresponding to
$(\hhh, \Delta^+_\lambda(\mmm,\hhh))$ and let $\uuu_1$ be its
nilradical.
Then $\bbb_1$ is $\theta$-stable and $\sigma\cong ({}^u\rca^{\mmm,
M\cap K}_{\bbb_1, H})^{\dim
\uuu_1\cap\kkk}(\Gamma\otimes\cpx_{2\delta(\uuu_1\cap\kkk)^\prime})$.
Let $\nnn$ be the nilradical of the complexified Lie algebra of $P$.
We put $\bbb=\bbb_1+\nnn$ and $\uuu=\uuu_1+\nnn$.
$\bbb$ is a Borel subalgebra of $\gggg$ and $\uuu$ is the nilradical of $\bbb$.
Using the induction-by-stage formula ([14]  Corollary
11.86), we have
\[
 \pi_G(\gamma)\cong  ({}^u\rca^{\gggg,
K}_{\bbb, H})^{\dim
\uuu\cap\kkk}(\Gamma\otimes\cpx_{2\delta(\uuu_1\cap\kkk)^\prime}\otimes\cpx_{\delta(n)})
\]
There are various presentations of the standard representation as a
cohomological induction from a character on a Borel subalgebra (cf.\
[24], [14] XI).

For a regular character $\gamma=(H,\Gamma, \lambda)$ and $k\in K$, we
put $k\cdot\gamma=(\Ad(k)H, \Gamma\circ\Ad(k^{-1}), \lambda\circ\Ad(k^{-1}))$.
Then, $k\cdot\gamma$ is also a regular character.
For two regular characters $\gamma_1$ and $\gamma_2$, $[\pi_G(\gamma_1)]=[\pi_G(\gamma_2)]$ if and only if $k\cdot\gamma_1=\gamma_2$ for some $k\in K$.

Let $\gamma=(H,\Gamma, \lambda)$ be a regular character and assume
$\lambda$ is regular with respect to $\Delta(\gggg,\hhh)$.
Then,  a standard module $\pi_G(\gamma)$ has a unique irreducible subquotient (Langlands subquotient) $\bar{\pi}_G(\gamma)$ such that all the minimal $K$-types of $\pi_G(\gamma)$ is contained in $\bar{\pi}_G(\gamma)$.
$\bar{\pi}_G(\gamma)$ is independent of the choice of $P$.
Each irreducible Harish-Chandra $(\gggg, K)$-module with a regular infinitesimal character is isomorphic to some $\bar{\pi}_G(\gamma)$, and
for two regular characters $\gamma_1$ and $\gamma_2$, $\bar{\pi}_G(\gamma_1)\cong\bar{\pi}_G(\gamma_2)$ if and only if $k\cdot\gamma_1=\gamma_2$ for some $k\in K$ (Langlands classification).




For a $\theta$-stable Cartan subgroup $H$ of $G$ and a regular weight $\eta\in\shh^\ast$, 
we denote by $R_G(H,\eta)$ the set of the regular characters $(H,\Gamma,\lambda)$ such that $\chi_\lambda=\chi_\eta$.
For a regular weight $\eta\in\shh^\ast$, we denote by $R_G(\eta)$, the set of all the regular character $\gamma$ such that $\pi(\gamma)$ has an infinitesimal character $\eta$.
$R_G(\eta)$ is the union of $R_G(H,\eta)$'s.
We call a $\theta$-stable Cartan subgroup $H$ of $G$ $\eta$-integral if $R_G(H,\eta)\neq \emptyset$.

A root $\alpha\in \Delta$ is called real, complex, compact imaginary,
noncompact imaginary with respect to $\gamma=(H,\Gamma, \lambda)\in R_G(\eta)$, if $\bi_{\eta,\lambda}(\alpha)$ is real, complex, compact imaginary, noncompact imaginary, respectively.
For $\gamma=(H,\Gamma, \lambda)\in R_G(\eta)$, we put $\theta_\gamma=\bi_{\eta,\lambda}^{-1}\circ\theta\circ\bi_{\eta,\lambda}$.   
$\theta_\gamma$ acts on $\Delta$.
Obviously, $\theta_\gamma$ only depends on the $K$-conjugacy class of
$\gamma$.


\setcounter{section}{2}
\section*{\S\,\,2.\,\,\,\, Change of polarization}
\setcounter{subsection}{0}


\subsection{$\sigma\theta$ pair}

\mbox{}

We consider here the following setting.

Let $G$ be a real reductive linear Lie group which is contained in the complexification $G_\cpx$.
We fix a maximal compact subgroup $K$ of $G$ and let $\theta$ be the corresponding Cartan involution.
We denote by $\gggg_0$ (resp.\ $\kkk_0$ )the Lie algebra of $G$ (resp.\ $K$) and denote by $\gggg$ (resp.\ $\kkk$) its complexification.
We denote also by the same letter $\theta$ the  complexified Cartan involution on $\gggg$.
We denote by $\sigma$ the complex conjugation on $\gggg$ with respect to $\gggg_0$.

\begin{dfn}

A pair $(\ppp,\qqq )$ is called a $\sigma\theta$ pair, if it satisfies the following conditions (S1) and (S2).

(S1) \,\,\,\, 
$\qqq$ (resp.\ $\ppp$) is a $\theta$-stable (resp.\ $\sigma$-stable) parabolic subalgebra of $\gggg$.

(S2) \,\,\,\,
There exists a $\theta$ and $\sigma$-stable Cartan subalgebra $\hhh$  of $\gggg$ such that $\hhh\subseteq \ppp\cap\qqq$.

\end{dfn}

Hereafter, we fix a $\sigma\theta$ pair $(\ppp, \qqq)$.
Let $\hhh$ be any  $\theta$ and $\sigma$-stable Cartan subalgebra of $\gggg$ contained in $\ppp\cap \qqq$. 
 For $\alpha\in \Delta(\gggg,\hhh)$, we denote by $\gggg_\alpha$ (resp.\ $s_\alpha$) the root space (resp.\ the reflection) corresponding to $\alpha$.
Since $\hhh$ is $\theta$-stable, $\theta$  and $\sigma$ induce actions on $\Delta(\gggg,\hhh)$.
We easily see $\theta\alpha=-\sigma\alpha$ for any $\alpha\in\Delta(\gggg,\hhh)$.

For a subspace $U$ in $\gggg$,
put $\Delta(U)=\{\alpha\in\Delta(\gggg,\hhh)\mid \gggg_\alpha\subseteq
U\}$ and $\rho(U)=\frac{1}{2}\sum_{\alpha\in\Delta(U)}\in \hhh^\ast$.

We put
\begin{gather*}
 \mmm=\hhh+\sum_{\alpha\in \Delta(\ppp)\cap(-\Delta(\ppp))}\gggg_\alpha, \,\,\,\,
\nnn=\sum_{\alpha\in \Delta(\ppp)-\Delta(\mmm)}\gggg_\alpha,  \,\,\,\,
 \bnn=\sum_{\alpha\in \Delta(\nnn)}\gggg_{-\alpha}, \\
\llll=\hhh+\sum_{\alpha\in \Delta(\qqq)\cap(-\Delta(\qqq))}\gggg_\alpha, \,\,\,\,
 \uuu=\sum_{\alpha\in \Delta(\qqq)-\Delta(\llll)}\gggg_\alpha, \,\,\,\,
 \buu=\sum_{\alpha\in \Delta(\uuu)}\gggg_{-\alpha}.
\end{gather*}
We immediately see
$\qqq=\llll+\uuu$ (resp.\ $\ppp=\mmm+\nnn$) is an orderly Levi decomposition of $\qqq$ (resp.\ $\ppp$) and the nilradical satisfies $\sigma(\uuu)=\buu$ (resp.\ $\theta(\nnn)=\bnn$).
Moreover, $\buu$ (resp.\ $\bnn$) is the opposite nilradical to $\uuu$ (resp.\ $\nnn$).

We denote by $L_\cpx$, $P_\cpx$, and $M_\cpx$ the analytic subgroups of $G_\cpx$ with respect to $\llll$, $\ppp$, and $\mmm$, respectively.
We put $L=L_\cpx\cap G$, $P=P_\cpx\cap G$, $M=M_\cpx\cap G$.


In the setting above, we easily have:
\begin{prop}\mbox{} 

(S3) \,\,\, $\llll\cap\ppp$ is a parabolic subalgebra of $\llll$ and $L\cap P$ is a parabolic subgroup of $L$.

(S4) \,\,\, $\mmm\cap\qqq$ is a parabolic subalgebra of $\mmm$.

(S5) \,\,\, $\llll\cap\mmm$ is a  $\theta$ and $\sigma$-stable Levi subalgebra of the both $\llll\cap\ppp$ and $\mmm\cap\qqq$.

\end{prop}



\medskip


For a Borel subalgebra $\bbb$, we also write $\Delta^+_\bbb$ for $\Delta(\bbb)$. 
$\Delta^+_\bbb$ is a positive system of $\Delta(\gggg,\hhh)$

Put $\tnn=(\uuu\cap\mmm)+\nnn$, $\tuu=(\nnn\cap\llll)+\uuu$, $\tpp=(\llll\cap\mmm)+\tnn$, and $\tqq=(\llll\cap\mmm)+\tuu$.
Then $\tpp$ (resp.\ $\tqq$) is a parabolic subalgebra of $\gggg$ with a Levi part $\llll\cap\mmm$ and the nilradical $\tnn$ (resp.\ $\tuu$).

We fix a Borel subalgebra $\bbb^0$ of $\llll\cap\mmm$ containing $\hhh$.
We put $\bbb_1=\bbb^0+\tnn$ and $\bbb_2=\bbb^0+\tuu$.
Obviously, $\bbb_1$ and $\bbb_2$ are Borel subalgebras of $\gggg$.
Let $\vvv$, $\vvv_1$, and $\vvv_2$ be the nilradical of $\bbb^0$, $\bbb_1$, and $\bbb_2$, respectively.
Put $\ddd=\vvv+\nnn\cap\llll+\uuu\cap\mmm+\uuu\cap\nnn$.
Then, we easily see $\vvv_1=\ddd\oplus(\nnn\cap\buu)$ and $\vvv_2=\ddd\oplus(\bnn\cap\uuu)$.

\begin{lem}
We have 
\[\dim \uuu\cap\kkk -\dim \uuu\cap\mmm\cap\kkk=\dim\uuu\cap\bnn.\]
\end{lem}

\pf
Since $\gggg=\mmm\oplus\bnn\oplus\nnn$ and $\uuu$ is $\theta$-stable, we have $\dim \uuu\cap\kkk -\dim \uuu\cap\mmm\cap\kkk=\dim ((\uuu\cap\bnn)\oplus(\uuu\cap\nnn))\cap\kkk$.
Let $p: (\uuu\cap\bnn)\oplus(\uuu\cap\nnn)\rightarrow \uuu\cap\bnn$ be the projection to the first factor.
Since $\nnn\cap\kkk=0$, the restriction of $p$ to $((\uuu\cap\bnn)\oplus(\uuu\cap\nnn))\cap\kkk$ is an injection.
On the other hand, for any $X\in \uuu\cap\bnn$, we have $X\oplus \theta X \in ((\uuu\cap\bnn)\oplus(\uuu\cap\nnn))\cap\kkk$.
So, the restriction of $p$ to $((\uuu\cap\bnn)\oplus(\uuu\cap\nnn))\cap\kkk$ is onto.
\,\,\,\, $\blacksquare$


\begin{lem}

Put $d=\dim\uuu\cap\bnn$.
There exits a sequence of complex roots $\alpha_1,....\alpha_{d} \in \Delta(\gggg,\hhh)$ satisfying the following conditions (1)-(4).
For $1\leqslant k \leqslant d$, we put $\Delta^+_k=s_{\alpha_k}\cdots s_{\alpha_1}\Delta_{\bbb_1}^+$.
We also put $\Delta^+_0=\Delta_{\bbb_1}^+$.

(1) \,\,\, For $1\leqslant k \leqslant d$, $\alpha_k\in\Delta(\nnn\cap\buu)$.

(2) \,\,\, For $1\leqslant k \leqslant d$, $\Delta(\ddd)\subseteq \Delta_{k}^+$.

(3) \,\,\, For $1\leqslant k \leqslant d$, $\alpha_k$ is simple with respect to $\Delta_{k-1}^+$.

(4) \,\,\, For $1\leqslant k \leqslant d$, $\theta\alpha_k\notin \Delta_{k-1}^+$.

(5) \,\,\, For $1\leqslant k \leqslant d$, $\alpha_k \in \Delta_{k-1}^+$ and $-\theta\alpha_k\in\Delta_{k-1}^+$.

(6) \,\,\, $\Delta_{d}^+=\Delta_{\bbb_2}^+$.

\end{lem}

\pf \mbox{(cf.\ [14], Lemma 11.128)}

 For a positive system $\Delta^+$ of $\Delta(\gggg,\hhh)$, we define $\mbox{ht}(\Delta^+)=\card(\Delta^+\cap\Delta(\bnn\cap\uuu))$.
We immediately see $\mbox{ht}(\Delta_{\bbb_1}^+)=0$ and $\mbox{ht}(\Delta_{\bbb_2}^+)=d$.

We construct the sequence $\alpha_1,...,\alpha_d$ inductively as follows.
Let $1\leqslant k\leqslant d$ and assume that $\alpha_1,...,\alpha_{k-1}$ are already defined so that the conditions in (1)-(5) above hold.
First, (1) and (3) imply $\mbox{ht}(\Delta_{k-1})=k-1$.

We have  a disjoint union $\Delta(\gggg,\hhh)=\Delta(\nnn\cap\buu)\sqcup\Delta(\vvv_2)\sqcup -\Delta(\ddd)$.
So, (2) implies $\Delta^+_{k-1}\subseteq \Delta(\nnn\cap\buu)\sqcup\Delta(\vvv_2)$.
If there is no simple root for $\Delta^+_{k-1}$ contained in $\nnn\cap\buu$, we have any simple root for $\Delta^+_{k-1}$ is contained in $\Delta(\vvv_2)=\Delta_{\bbb_2}^+$.
Hence we have $\Delta_{k-1}^+=\Delta_{\bbb_2}^+$.
However, it contradicts $\mbox{ht}(\Delta_{k-1}^+)=k-1<d=\Delta_{\bbb_2}^+$.
So, there exists some simple root $\alpha_k$ for $\Delta_{k-1}^+$ such that $\alpha_k\in\Delta(\nnn\cap\buu)$.
Since $\theta(\nnn)=\bnn$ and $\theta(\buu)=\buu$, we see $\alpha_k$ is complex and $\theta\alpha_k\in \Delta(\bnn\cap\buu)\subseteq -\Delta(\ddd)\subseteq -\Delta_{k-1}^+$.
Hence, we see $\alpha_k$ satisfies the conditions in the above (1)-(5).
If $\Delta(\ddd)\subseteq \Delta^+$ and $\mbox{ht}(\Delta^+)=d$, then clearly $\Delta^+=\Delta_{\bbb_2}$.
So, we have $\Delta_d^+=\Delta_{\bbb_2}$, since $\mbox{ht}(\Delta_{d}^+)=d$.
Thus, we have (6). \,\, $\blacksquare$

We immediately see:

\begin{cor}

 The complex roots $\alpha_1,....\alpha_{d}$ in the above Lemma 1.2.4
 are distinct from each other and we have $\Delta(\nnn\cap\buu)=\{\alpha_1,....\alpha_{d}\}$.

\end{cor}

{\bf  CAUTION} \,\,\, The above numeration $\{\alpha_1,...,\alpha_d\}$
of $\Delta(\nnn\cap\buu)$ may depend on the choice of $\bbb^0$.


\subsection{Change of polarization}

\mbox{}
In this section, we fix a $\sigma\theta$ pair $(\ppp, \qqq)$.
Let $\mmm,\llll,....$ be as in 2.1.

Let $L^\sim$ (resp.\ $(L\cap M)^\sim$) be the metaplectic double covering of $L$ (resp.\ $L\cap M$) with respect to  $\delta(\uuu)$ (resp.\ $\delta(\uuu\cap\mmm)$).

\begin{lem} On $\llll\cap\mmm$, we have
\[ \delta(\uuu)-\delta(\uuu\cap\mmm)=\delta(\bnn\cap\llll)+\delta(\nnn)+2\delta(\uuu\cap\bnn).\]
\end{lem}

\pf
Remark that $\delta(\bnn\cap\uuu)=-\delta(\nnn\cap\buu)$, $\delta(\bnn\cap\llll)=-\delta(\nnn\cap\llll)$, etc.
So, we have the lemma from the computation below.
\begin{align*}
\delta(\uuu)+\delta(\nnn\cap\llll)&=\delta(\uuu\cap\mmm)+\delta(\uuu\cap\bnn)+\delta(\uuu\cap\nnn)+\delta(\nnn\cap\llll)\\
&=\delta(\uuu\cap\mmm)+2\delta(\uuu\cap\bnn)+\delta(\buu\cap\nnn)+\delta(\nnn\cap\llll)+\delta(\uuu\cap\nnn)\\
&=\delta(\uuu\cap\mmm)+2\delta(\uuu\cap\bnn)+\delta(\nnn) \,\,\,\,\,\,\, \blacksquare
\end{align*}

We define a one dimensional representation $\xi_{\ppp,\qqq}$  of  $L\cap M$ on a space $\cpx_{\ppp,\qqq}$ by
\[ \xi_{\ppp,\qqq}(\ell)=\xi_{\delta(\bnn\cap\llll)}(\ell)\xi_{\delta(\nnn)}(\ell)
\xi_{2\delta(\bnn\cap\uuu)^\prime}(\ell) \,\,\,\,\,\, (\ell\in L\cap M).\]

From Lemma 2.2.1, we easily see:

\begin{lem}
Assigning $(\ell,z) \in(L\cap M)^\sim$ to $(\ell, z\xi_{\ppp.\qqq}(\ell))$, we have an embedding of the group $(L\cap M)^\sim \hookrightarrow L^\sim$.
\end{lem}

Let $(P\cap L)^\sim$ be the parabolic subgroup of $L^\sim$ which is the pull-back of $P\cap L$ to $L^\sim$.
Under the identification by the embedding in  Lemma 2.2.2, we can regard $(L\cap M)^\sim$ as a Levi subgroup of $(P\cap L)^\sim$. 

\medskip

Following is the main result of the section.

\begin{thm}  \mbox{}

(1) \,\,\,\, Let $Z$ be a Harish-Chandra $(\llll\cap \mmm, L\cap M\cap K)$-module with an infinitesimal character $\lambda\in\hhh^\ast$.
 Assume
 $\langle\lambda-\delta(\uuu\cap\mmm)-\delta(\nnn),\alpha\rangle\geqslant0$ for all $\alpha\in \Delta(\uuu)$ such that $2\frac{\langle\lambda-\delta(\uuu\cap\mmm)-\delta(\nnn),\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\itg$.

Then, we have
\eqb [{}^u\Ind_P^G(({}^u\rca^{\mmm, M\cap K}_{\qqq\cap\mmm, L\cap M\cap K})^{\dim \uuu\cap\mmm\cap\kkk}(Z))]=[({}^u\rca^{\gggg, K}_{\qqq, L\cap K})^{\dim\uuu\cap\kkk}({}^u\Ind_{P\cap L}^L(Z\otimes\cpx_{2\delta(\uuu\cap\bnn)^\prime}))] \tag{$*$} 
\end{equation}

(2) \,\,\,\, Let $Z$ be a Harish-Chandra $(\llll\cap \mmm, L\cap M\cap K)$-module with an infinitesimal character $\lambda\in\hhh^\ast$.
We assume $\langle\lambda-\delta(\uuu\cap\mmm),\alpha\rangle\geqslant 0$ for all
$\alpha\in \Delta(\uuu)$ such that $2\frac{\langle\lambda-\delta(\uuu\cap\mmm),\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\itg$.
Then, we have
\eqb [\Ind_P^G(({}^u\rca^{\mmm, M\cap K}_{\qqq\cap\mmm, L\cap M\cap K})^{\dim \uuu\cap\mmm\cap\kkk}(Z))]=[({}^u\rca^{\gggg, K}_{\qqq, L\cap K})^{\dim\uuu\cap\kkk}(\Ind_{P\cap L}^L(Z\otimes\cpx_{\ppp,\qqq}))] \tag{$**$} 
\end{equation}

(3) \,\,\,\, Let $\tilde{Z}$ be a Harish-Chandra $(\llll\cap \mmm, (L\cap M\cap K)^\sim)$-module with an infinitesimal character $\lambda\in\hhh^\ast$ such that $Z=\tilde{Z}\otimes\cpx_{\delta(\uuu\cap\mmm)^\prime}$ is reduced to 
a Harish-Chandra $(\llll\cap \mmm, L\cap M\cap K)$-module.
We assume $\langle\lambda,\alpha\rangle\geqslant 0$ for all $\alpha\in
\Delta(\uuu)$ such that $2\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in\itg$.
Then, 
\eqb [\Ind_P^G(({}^n\rca^{\mmm, M\cap K}_{\qqq\cap\mmm, L\cap M\cap K})^{\dim \uuu\cap\mmm\cap\kkk}(\tilde{Z}))]=[({}^n\rca^{\gggg, K}_{\qqq, L\cap K})^{\dim\uuu\cap\kkk}(\Ind_{(P\cap L)^\sim}^{L^\sim}(\tilde{Z}))] \tag{$***$} 
\end{equation}

\end{thm}


\pf

(2),(3) are rephrasings of (1).
We remark that characters of standard modules form a basis of the Grothendieck group of the category of Harish-Chandra modules.
Taking account of additivity of cohomological inductions, it suffices to show ($*$) in case of $Z$ is a standard module.

As in 2.1, we fix a $\theta$ and $\sigma$ stable Cartan subalgebra $\hhh$ of $\llll\cap \mmm$ and a Borel subalgebra $\bbb^0$ of $\llll\cap\mmm$ containing $\hhh$.
We denote by $\vvv$ the nilradical of $\bbb^0$.
Let $H_\cpx$ be the analytic subgroup of $G_\cpx$ and put $H=H_\cpx\cap G$.
Let $Y$ be a one-dimensional $H$-representation whose differential is just $\lambda$.
We consider the case of $Z=(\rca^{\llll\cap\mmm, L\cap M\cap K}_{\bbb, T})^{\dim\vvv\cap\kkk}(Y)$.
Put $\bbb_1=\bbb+\uuu\cap\mmm+\nnn$ and $\bbb_2=\bbb+\nnn\cap\llll+u$.
Then, $\bbb_1$ and $\bbb_2$ are Borel subalgebras of $\gggg$.
From [14] Corollary 11.86 (Induction-by-stage formula), we have
\[  {}^u\Ind_P^G(({}^u\rca^{\mmm, M\cap K}_{\qqq\cap\mmm, L\cap M\cap K})^{\dim \uuu\cap\mmm\cap\kkk}((\rca^{\llll\cap\mmm, L\cap M\cap K}_{\bbb, T})^{\dim\vvv\cap\kkk}(Y)))\cong (({}^u\rca^{\gggg, K}_{\bbb_1, T})^{\dim \uuu\cap\mmm\cap\kkk+\dim\vvv\cap\kkk}(Y), \]
\[({}^u\rca^{\gggg, K}_{\qqq, L\cap K})^{\dim\uuu\cap\kkk}({}^u\Ind_{P\cap L}^L((\rca^{\llll\cap\mmm, L\cap M\cap K}_{\bbb, T})^{\dim\vvv\cap\kkk}(Y) \otimes\cpx_{2\delta(\uuu\cap\bnn)^\prime}))\cong
({}^u\rca^{\gggg, K}_{\bbb_2, T})^{\dim \uuu\cap\kkk+\dim\vvv\cap\kkk}(Y\otimes\cpx_{2\delta(\uuu\cap\bnn)^\prime}). \]
So, we have only to show
\eqb
(({}^u\rca^{\gggg, K}_{\bbb_1, T})^{\dim \uuu\cap\mmm\cap\kkk+\dim\vvv\cap\kkk}(Y)\cong
({}^u\rca^{\gggg, K}_{\bbb_2, T})^{\dim \uuu\cap\kkk+\dim\vvv\cap\kkk}(Y\otimes\cpx_{2\delta(\uuu\cap\bnn)^\prime}).\tag{$\circ$} \end{equation}
However, we have Lemma 2.1.3 and 2.1.4.
So, ($\circ$) can be obtained by the successive application of the transfer theorem ([14] Theorem 11.87).
\,\, $\blacksquare$



\subsection{{ Derived functor modules; complex case}}

For complex connected reductive groups, irreducible unitary representation with regular integral infinitesimal character is a parabolic induction from a one-dimensional unitary representation ([4]).
Moreover, Enright proved they have non-trivial $(\gggg,K)$-cohomologies.
On the other hands, for general reductive Lie groups, Vogan-Zuckerman
proved that any irreducible unitary representation with regular integral
infinitesimal character and with  non-trivial $(\gggg,K)$-cohomology is
nothing but a derived functor module. ([37])
Here we give an explanation of this phenomenon in viewpoint of the change of polarization.

Let $G$ be a complex connected reductive Lie group and we fix a Cartan involution $\theta$.
Here, we denote by $\gggg_0$ the real Lie algebra of $G$.
Then the complexification of $\gggg_0$ can be identified with $\gggg_0\times\gggg_0$.
Let $\ppp_0$ be any parabolic subalgebra of $\gggg_0$ with a Levi decomposition $\ppp_0=\mmm_0+\nnn_0$ such that $m_0$ is $\theta$ stable.
If we choose the identification appropriately, then the complexification $\ppp$ of $\ppp_0$ can be identified with $\ppp_0\times\ppp_0\subseteq \gggg_0\times\gggg_0$.
On the other hand, if we put $\qqq=\ppp_0\times\bpp_0$, $\qqq$ is a $\theta$-stable parabolic algebra.
Here, $\bpp_0$ means the opposite parabolic subalgebra to $\ppp_0$.
We immediately see $(\ppp,\qqq)$ is a $\sigma\theta$-pair and $\ppp$ and $\qqq$ have a common Levi part $\mmm_0\times\mmm_0$.
Applying the Theorem 2.2.3, we see that, for complex connected reductive groups, derived functor modules are actually certain irreducible degenerate principal series representations.

\bigskip

\subsection{{Derived functor modules;  general case}}

For $G=\GL(n,\rel)$, derived functor modules are parabolic induction from the external tensor product of some copies of distinguished derived functor modules so-called Speh representations and possibly a one-dimensional representation. ([25])

We examine this phenomenon in viewpoint of the change of polarization.
Here, we use notations as in 1.2, such as $G$, $G_\cpx$, $K$, $K_\cpx$, $\gggg$, $\gggg_0$, $\theta$, $\sigma$, etc.
Let $\qqq$ be a $\theta$-stable  parabolic subalgebra with an orderly  Levi decomposition $\qqq=\llll+\uuu$. 
Let $L$ be the Levi subgroup corresponding to $\llll$ defined as in 1.1.

Let $\aaa$ be the $-1$-eigenspace with respect to $\theta$ in the center of $\llll$.
We call $\qqq$  pure imaginary if $\aaa$ is contained the center of $\gggg$.

Let $\mmm$ be the centralizer of $\aaa$ in $\gggg$.
Then $\mmm$ is a Levi subalgebra of a $\sigma$-stable parabolic subgroup $\ppp$.
Obviously $(\ppp, \qqq)$ is a $\sigma\theta$-pair and  $\llll\subseteq \mmm$.
$\qqq$ is pure imaginary if and only if $\mmm=\gggg$ holds.

Conversely, we assume that there is a $\sigma$-stable parabolic subalgebra $\ppp$ of $\gggg$
such that $(\ppp, \qqq)$ is a $\sigma\theta$-pair and there is an orderly Levi decomposition $\ppp=\mmm+\nnn$ such that $\llll\subseteq \mmm\neq\gggg$.
Then, we have $\qqq$ is not pure imaginary since the $-1$-eigenspace with respect to $\theta$ in the center of $\mmm$ also centralize $\llll$.

From Theorem 2.2.3, we have:
\begin{prop}
Let $\qqq$ be a $\theta$-stable  parabolic subalgebra with an orderly  Levi decomposition $\qqq=\llll+\uuu$.
Assume that $\qqq$ is not pure imaginary.
Then,  there is a $\sigma$-stable  parabolic subalgebra $\ppp$ of $\gggg$ with an orderly Levi decomposition $\ppp=\mmm+\nnn$ such that the derived functor modules of $\gggg$ with respect to $\qqq$ is isomorphic to the parabolic induction from a derived functor module of $\mmm$.
\end{prop}

Obviously, if $G$ has a compact Cartan subgroup, any  $\theta$-stable parabolic subalgbra is pure imaginary.

We interpret Speh's result as follows.
So, for a while, we put $G=\GL(n, \rel)$
We fix a Cartan involution $\theta(g)={}^tg^{-1}$ of $G$. So, we put $K=\Oo(n)$ here.
For a positive integer $k$, we put $J_k=\begin{pmatrix} 0 & -I_k \\ I_k & 0 \end{pmatrix}$.
First, we assume $n$ is even and write $n=2k$.
Put 
\begin{gather*}
\llll(k)=\left\{\left. \begin{pmatrix} A & -B \\ B & A \end{pmatrix}\in
 \gl(2k,\cpx)\right| A,B\in M_k(\cpx)\right\}, \\
\uuu(k)=\left\{ \left. \begin{pmatrix} \sqrt{-1}S & S \\ S & -\sqrt{-1}S \end{pmatrix}\in \gl(2k,\cpx) \right| S\in M_k(\cpx)\right\}, \\
\qqq(k)=\llll(k)+\uuu(k).
\end{gather*}
Then, $\qqq(k)$ is a $\theta$-stable parabolic subalgebra of $\gl(2k,\cpx)$ and $\qqq(k)=\llll(k)+\uuu(k)$ is a Levi decomposition such that $\llll(k)$ is a $\theta$ and $\sigma$-stable Levi part.
The derived functor module with respect to $\qqq(k)$ is a Speh representation of $\GL(2k,\cpx)$.
Actually, we have:
\begin{prop}
If $n$ is odd, there is no proper pure imaginary $\theta$-stable parabolic subalgebra.

If $n$ is even, any proper pure imaginary $\theta$-stable parabolic subalgebra is $\SO(n)$-conjugate to $\qqq(\frac{n}{2})$.
\end{prop}

Next, we consider general $\theta$-stable parabolic subalgebras.
For a sequence of positive integers $\arn=(n_1,...,n_\ell)$ such that $0\leqs n-2n_1+\cdots+2n_\ell$,
we put $q=n-2n_1+\cdots+2n_\ell$
\[\ttt(\arn)=\{\diag(t_1J_{n_1},...,t_{\ell}J_{n_\ell},0_{q})\in \gl(n, \cpx)| t_1,...,t_\ell \in \cpx\} \].
We denote by $\llll(\arn)$ the centralizer of $\ttt(\arn)$ in $\gl(n,\cpx)$.
Then we have
\[ \llll(\arn)=\{\diag(A_1,..., A_\ell,D)\in\gl(n,\cpx)|A_i,\in\llll(n_i) \,\,\, (1\leqs i\leqs \ell), D\in \gl(q,\cpx)\}.\]
and
$ \llll_0(\arn)=\llll(\arn)\cap \gl(n,\rel)$ is a real form of $\llll(\arn)$ and 
\[\llll_0(\arn)\cong\gl(n_1,\cpx)\times\cdots\times\gl(n_\ell,\cpx)\times\gl(q,\rel).\]
Put
\[\mmm(\arn)=\{\diag(A_1,....,A_\ell,D)\in\gl(n,\cpx)|A_i\in GL(2n_i,\cpx) \,\,\, (1\leqs i\leqs \ell), D\in \gl(q,\cpx)\} . \]
There is a $\theta$-stable parabolic subalgebra $\qqq(\arn)$ such that 
\[\mmm(\arn)\cap\qqq(\arn)=\{\diag(A_1,..., A_\ell,D)\in\gl(n,\cpx)|A_i,\in\qqq(n_i) \,\,\, (1\leqs i\leqs \ell), D\in \gl(q,\cpx)\}. \]
Any $\theta$-stable parabolic subalgebra in $\gl(n,\cpx)$ is $\Oo(n,\cpx)$-conjugate to some $\qqq(\arn)$.
Let $\nnn$ be the Lie algebra of the upper triangular matrices in $\gl(n,\cpx)$ and put $\ppp(\arn)=\mmm(\arn)+\nnn$.
We denote by $\nnn(\arn)$ the nilradical of $\ppp(\arn)$.
Then, $(\ppp(\arn), \qqq(\arn))$ is a $\sigma\theta$-pair.
Applying Theorem 2.2.3 to the $\sigma\theta$-pair, we get Speh's
result([25] Theorem 4.2.2.).

Next, we consider the case of $G=\GL(k,\qrt)$. Write $\qrt=\cpx+j\cpx$.
This case we put 
$K=\Spp(n)=\{g\in \GL(k,\qrt)| {}^t\bar{g}g=I_k \}$.
Then we regard $\gl(k,\cpx)$ as a real Lie subalgebra of $\gl(k,\qrt)$.
For $\ell\in\itg$ and $t\in\sqrt{-1}\rel$, we define a one-dimensional unitary representation
$\xi_{\ell,t}$ of $\GL(k,\cpx)$ as follows.
\[
 \xi_{\ell, t}(g)=\left(\frac{\det(g)}{|\det(g)|}\right)^\ell|\det(g)|^{t}.
\]
Let $\qqq(k)$ be a $\theta$-stable  parabolic subalgebra with an orderly Levi decomposition $\qqq(k)=\llll(k)+\uuu(k)$.
We choose the nilradical $\nnn(k)$ so that $\xi_{\ell,t}$ is good with respect
to $\qqq(k)$ for sufficient large $\ell$.
Derived functor modules with respect to $\qqq(k)$ are called quaternionic Speh representations.

For $t\in \sqrt{-1}\rel$, there is a one-dimensional unitary representation
$\tilde{\xi}_t$ of $\GL(k,\qrt)$ whose restriction to $\GL(k,\cpx)$ is
$\xi(0,t)$.

We put  

\begin{dfn}

\begin{equation}
 A_k(\ell,t)=({}^u\rca_{\qqq(k),\Oo(k)}^{\gl(k,\qrt)\otimes_\rel\cpx,
\Spp(k)})^{k(k+1)}(\xi_{\ell+2k,t})   \,\,\,\,\,  \,\, (\ell \in\itg).\tag{$\ast$}
\end{equation}

We also put
\[
 A_k(-\infty,t)=\tilde{\xi}_t
\]

\end{dfn}

For $\ell\in\itg$, $A_k(\ell,t)$ is derived functor module in the good (resp.\ weakly fair) range
in the sense of [35] if and only if $\ell\geqslant 0$ (resp.\
$\ell\geqslant -k$).

We immediately see:
\[
 A_k(\ell,t)\cong A_k(\ell,0)\otimes\tilde{\xi}_t.
\]
We easily have:

\begin{prop}
Any proper pure imaginary $\theta$-stable parabolic subalgebra is $\Spp(k)$-conjugate to $\qqq(k)$.
\end{prop}

As in the case of $\GL(k, \rel)$, any derived functor module of
$\GL(k,\qrt)$ is a parabolic induction from the external tensor product
of some copies of quaternionic Speh representations and possibly a
one-dimensional representation.
(cf.\ [34])

Next, we consider the case of $G=\SO_0(2p+1,2q+1)$.
In this case,  a Levi part of a non-pure imaginary $\theta$-stable
parabolic  subalgebra $\qqq$ is isomorphic to $\so(1,1)\oplus \uuu(p_1,q_1)\oplus
\cdots \uuu(p_k,q_k)$.
Here, $p_1+\cdots+p_k=p$ and  $q_1+\cdots+q_k=q$.

Let $\ppp$ be a maximal cuspidal parabolic subalgebra whose Levi part is
isomorphic to $\so(2p,2q)\oplus \so(1,1)$.
The derived functor module with respect to the above $\qqq$ is a
parabolic induction with respect to $\ppp$ from a derived functor
module of $\so(2p,2q)$ with respect to a $\theta$-stable parabolic
subalgebra whose Levi part is isomorphic to $\uuu(p_1,q_1)\oplus
\cdots \uuu(p_k,q_k)$.

Among the exceptional real simple Lie algebras, only E I and E IV have
non pure imaginary $\theta$-stable parabolic subalgebras.



\setcounter{section}{3}
\section*{\S\,\,3.\,\,\,\, Application of the change-of-polarization to
$\SO^\ast(2n)$ and $\Spp(p,q)$}
\setcounter{subsection}{0}

Throughout this section, we assume $G$ is either $\SO^\ast(2n)$  or
$\Spp(n-q,q)$ with $2q\leqslant n$. For $G=\Spp(n-q,q)$, put $p=n-q$.
For $G=\SO^\ast(2n)$, we put $q=\left[\frac{n}{2}\right]$.
In both cases $G=\Spp(p,q)$ and $G=\SO^\ast(2n)$, $q$
is the real rank of $G$.

\subsection{{Root systems}}

We fix a maximal compact subgroup $K$ of $SO^\ast(2n)$ (resp.\
 $\Spp(p,q)$), which is isomorphic to $U(n)$ (resp.\ $\Spp(p)\times\Spp(q)$).
We denote by $G_\cpx$ the complexification of $G$ as in 1.2.  So,
$G_\cpx$ is isomorphic to $\SO(2n,\cpx)$ or $\Spp(n,\cpx)$.
We denote by $\theta$ the Cartan involution corresponding to $K$ as in 1.2.
We fix a $\theta$-stable maximally split Cartan subgroup $\sH$ of $G$.
We remark that all the Cartan subgroups of $G$ are connected.
We stress that we use notations introduced in \S \,1.

First, we consider the root system $\Delta(\gggg,\shh)$ for $G=\SO^\ast(2n)$.
Then we can choose an orthonormal basis $e_1,...,e_n$ of $\shh^\ast$ such that 
\[\Delta(\gggg,\shh) = \{ \pm e_i\pm e_j\mid 1\leqslant i< j\leqslant n\}.\]
If $n$ is even, we write $n=2q$.  In this case, we choose the above $e_1,...,e_n$ so that
$\theta(e_{2i-1})=-e_{2i}$ and $\theta(e_{2i})=-e_{2i-1}$ for all $1\leqslant i\leqslant q$.
If $n$ is odd, we write $n=2q+1$.  In this case, we choose the above $e_1,...,e_n$ so that
$\theta(e_{2i-1})=-e_{2i}$ and $\theta(e_{2i})=-e_{2i-1}$ for all $1\leqslant i\leqslant q$ and $\theta(e_{2q+1})=e_{2q+1}$.
We immediately see that $\pm(e_{2i-1}-e_{2i})$  (resp.\ $\pm (e_{2i-1}+e_{2i})$) \,\, $(1\leqslant i\leqslant q)$ are compact imaginary (resp.\ real) and the other roots are complex.

If $G=\Spp(n-q,q)$, put $p=n-q$ and  choose $e_1,...e_n$ such that 
\[ \Delta(\gggg,\shh) = \{ \pm e_i\pm e_j\mid 1\leqslant i< j\leqslant n\}\cup\{\pm 2e_i\mid 1\leqslant i\leqslant n\},\]
$\theta(e_{2i-1})=-e_{2i}, \theta(e_{2i})=-e_{2i-1}$ for $1\leqslant
i\leqslant q$,
and
$\theta(e_i)=e_i$ for $2q< i\leqslant n$.

We fix a simple system for $\Delta(\gggg,\shh)$ as follows.
If $G=\SO^\ast(2n)$, then put $\Pi=\{e_1-e_2,...,e_{n-1}-e_n, e_{n-1}+e_n\}$.
If $G=\Spp(p,q)$, then put $\Pi=\{e_1-e_2,...,e_{n-1}-e_n, 2e_n\}$.

We denote by $\Delta^+$ the corresponding positive system of $\Delta(\gggg,\shh)$.
Let $E_1,....,E_n$ be the dual basis of $\shh$ to $e_1,....,e_n$.



\subsection{{Square Quadruplets}}

One famous realization of $\Spp(p,q)$ is as the automorphism group of an
 indefinite Hermitian form on a $\qrt$-vector space.
Namely,
\begin{equation*}
 \Spp(p,q)=\left\{g\in\GL(p+q,\qrt)|
	    {}^t\bar{g}I_{p,q}g=I_{p,q}\right\}.
\tag{$\bigstar$}
\end{equation*}
Here, $I_{p,q}=\begin{pmatrix} I_p & 0 \\ 0 & -I_q\end{pmatrix}$.
Similarly, we consider complex indefinite unitary group.
\[
 \U(p,q)=\left\{g\in\GL(p+q,\cpx)| {}^t\bar{g}I_{p,q}g=I_{p,q}\right\}.
\]
$\U(p,q)$   is regarded as a subgroup of $\Spp(p,q)$  in the obvious
way.
We fix a maximal compact subgroup of $\Spp(p,q) $ as follows.
\[
 K= \Spp(p)\times\Spp(q)= \left\{\left.\begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix}\right|  A\in \Spp(p), B\in\Spp(q)\right\}.
\]
We denote by $\theta$ the corresponding Cartan involution.

For the case of $p=q$, we also consider another realization:
\[
 \Spp(q,q)=\left\{g\in\GL(2q,\qrt)| {}^t\bar{g}J_qg=J_q\right\}.
\]
We put $n=2q$.
Here, $J_{q}=\begin{pmatrix} 0 & I_q \\ I_q & 0 \end{pmatrix}$.
Then,  identifying $\GL(q,\qrt)$ with the following group, we regard
$\GL(q,\qrt)$  as a subgroup of $\Spp(q,q)$:
\[
 \left\{\left. \begin{pmatrix}A & 0 \\ 0 & {}^t\bar{A}\end{pmatrix}\right|  A\in
 \GL(q,\qrt)\right\}.
\]
We  consider $\U(q,q)\cap \GL(q,\qrt)$ as a subgroup of
$\Spp(q,q)$.
This group is 
\[
 \left\{\left.\begin{pmatrix}A & 0 \\ 0 & {}^t\bar{A}\end{pmatrix}\right|  A\in \GL(q,\cpx)\right\}.
\]
We identify it with $\GL(q,\cpx)$ and obtain the following ``square quadruplet.''

\begin{equation}
\begin{array}{ccc}
\GL(q,\qrt) &   \subseteq  & \Spp(q,q) \\
\cup\mid  &     & \cup \mid \\
\GL(q,\cpx) &  \subseteq & \U(q,q)   
\end{array}\tag{A}
\end{equation} 
In (A),  each  inclusion gives a symmetric pair.
We easily see $\U(q,q)$, $\GL(q,\qrt)$, and $\GL(q,\cpx)$ are the
centralizers in $\Spp(q,q)$ of their centers, respectively.
Since $\GL(q,\cpx)$ has the same rank and the same real rank as
$\Spp(q,q)$,
we can choose $\theta$-stable maximally split Cartan subgroup $\sH$ of
$\Spp(q,q)$ which is contained in $\GL(q,\cpx)$. 
We denote by $\shh$ the complexified Lie algebra of $\sH$.
We may apply the notations on the root system for $\Delta(\gggg,\shh)$

We choose the standard Borel subalgebra $\bbb_1(q)$ of
${\gggg}={\mathfrak s}{\mathfrak p}(n,\cpx)$ corresponding to
$\Delta^+$ in 3.1.
We define a subset $S(q)=\{e_i-e_{i+1}\mid 1\leqslant i \leqslant n \}$
of $\Pi$.
We denote by $\tpp(q)$ the standard parabolic subalgebra corresponding
to $S(q)$, namely $\bbb_1(q)\subseteq \tpp(q)$ and
$\Delta(\tpp(q),\shh)=\Delta^+\cup(\itg S(q)\cap\Delta(\gggg,\shh))$
Then, we easily see $\GL(q,\qrt)$ is the $\theta$-stable Levi subgroup
for $\tpp(q)$.

Next, we consider another simple system $\Pi_u$ of $\Delta(\gggg,\shh)$
as follows.
\[
 \Pi_u=\{e_i-e_{i+2}\mid 1\leqslant i\leqslant n-2\}\cup\{e_{n-1}+e_n\}\cup\{-2e_2\}.
\]
We also put $S_u(q)=\Pi_u-\{-2e_2\}$.
We we choose the standard Borel subalgebra $\bbb_2(q)$ of
$\gggg=\fsp(n,\cpx)$ corresponding to
$\Pi_u$ and denote by $\tqq(q)$ the parabolic subalgbra of $\gggg$
containing $\bbb_2(q)$ and corresponding to $S_u(q)$.
Since, $\theta(S_u(q))=-S_u(q)$ and $\theta(-2e_2)\equiv -2e_2 \,\,\,
(\mod \itg S_u(q))$, $\tqq(q)$ is $\theta$-stable.
We easily see $\U(q,q)$ is a Levi subgroup for $\tqq(q)$.
$\U(q,q)$, $\GL(q,\qrt)$, and $\GL(q,\cpx)$ are the
centralizers of their centers in $\Spp(q,q)$.
In fact, the Lie algebra of the center of
$\U(q,q)$ (resp.\ $\GL(q,\qrt)$) is spanned by $\sum_{i=1}^q
(E_{2i-1}-E_{2i})$ (resp.\ $E_1+\cdots+E_n$).
Here, $n=2q$ and we follows the notations in 3.1.
The center of $\U(q,q)$ (resp.\ $\GL(q,\qrt)$) is compact (resp.\
real split) and $\theta$-stable, and $\U(q,q)$ (resp.\ $\GL(q,\qrt)$) is
a Levi subgroup for a maximal
$\theta$-stable (resp.\ $\sigma$-stable) parabolic subalgebra (say
$\tqq(q)$ (resp.\ $\tpp(q)$)) of
$\fsp(2q,\cpx)=\fsp(q,q)\otimes_\rel\cpx$.

Since $\shh\subseteq \tpp(q)\cap\tqq(q)$, 
$(\tpp(q),\tqq(q))$ forms a $\sigma\theta$-pair.
Put $\ppp(q)=\tpp(q)\cap (\uuu(q,q)\otimes_\rel\cpx)$ and
$\qqq(q)=\tqq(q)\cap (\gl(q,\qrt)\otimes_\rel\cpx)$.

Similarily, $\GL(q,\cpx)$ is the centralizer of the split (resp.\
compact) part of its center in $\U(q,q)$ (resp.\ $\GL(q,\qrt)$).
$\GL(q,\cpx)$ is a Levi subgroup for a maximal
$\sigma$-stable (resp.\ $\theta$-stable) parabolic subalgebra 
$\ppp(q)$ (resp.\ $\qqq(q)$) of
$\gl(2q,\cpx)=\uuu(q,q)\otimes_\rel\cpx$ (resp.\
$\gl(2q,\cpx)=\gl(q,\qrt)\otimes_\rel\cpx$).

$\ppp(q)$ is usually called a Siegel parabolic subalgebra and $\qqq(q)$
is the one defined in 2.4, the unique  (up to $\Spp(q)$-conjugacy) pure
imaginary $\theta$-stable parabolic subalgebra.
We denote by $P(q)$ the Siegel parabolic subgroup of $G$  corresponding
to $\ppp(q)$.
For $\ell\in\itg$ and $t\in\sqrt{-1}\rel$, we define a one-dimensional unitary representation
$\xi_{\ell,t}$ of $\GL(n,\cpx)$ as follows.
\[
 \xi_{\ell, t}(g)=\left(\frac{\det(g)}{|\det(g)|}\right)^\ell |\det(g)|^{t}.
\]
We define the degenerate unitary principal series with respect to $P(q)$ as
follows:
\begin{equation}
 I_q(\ell,t)=\Ind_{P(q)}^{\U(q,q)}(\xi_{\ell,t}) \,\,\,\,\, (\ell\in \itg, t\in\sqrt{-1}\rel)
\tag{$\dagger$}
\end{equation}

We introduce similar structure for $\SO^\ast(4q)$ as follows.
\begin{equation}
\begin{array}{ccc}
\GL(q,\qrt) &   \subseteq  & \SO^\ast(4q) \\
\cup\mid  &     & \cup \mid \\
\GL(k,\cpx) &  \subseteq & \U(q,q)   
\end{array}\tag{B}
\end{equation}

In fact, as in the case of $\Spp(q,q)$, $\U(q,q)$ (resp.\ $\GL(q,\qrt)$)
above is the centerilzer of $\sum_{i=1}^q
(E_{2i-1}-E_{2i})$ (resp.\ $E_1+\cdots+E_n$)  in $\SO^\ast(4q)$. 
(Here, $n=2q$.) $GL(q,\cpx)$ is the intersection of $\U(q,q)$ and
$\GL(q,\qrt)$.
For $q\geqslant 2$, we define
\begin{align*}
 \Pi_u & =\{e_i-e_{i+2}\mid 1\leqslant i\leqslant n-2\}\cup\{e_{n-1}+e_n\}\cup\{-e_2-e_4\},
\\
S_u(q)& =\Pi_u-\{-e_2-e_4\}.
\end{align*}
If $q=1$, put $\Pi_u=\{e_1+e_2, e_1-e_2\}$ and $S_u(1)=\{e_1+e_2\}$.
We define $\tpp(q)$ and $\tqq(q)$ in a similar manner to the case of $G=\Spp(q,q)$.
In this case, situation is quite similar to the case of $\Spp(q,q)$.



\subsection{{Maximal parabolic subgroups}}

Let $k$ be a positive integer  such that $k  \leq  q$.
If $G=\Spp(p,q)$, put  $p^\prime=p-k$ and
$q^\prime=q-k$.
If $G=\SO^\ast(2n)$, put $r=n-2k$.
We put
 $A=\sum_{j=1}^{k}E_{j}$.
Then we have $\theta(A)=-A$ .
We denote by $\aaa_{(k)}$  the one-dimensional Lie subalgebra of
$\shh$ spanned by $A$.

We define a subset $S(k)$ of $\Pi$ as follows.
If $G=\Spp(p,q)$, we define
\[ S(k)=\left\{\begin{array}{ll} 
\Pi-\{e_{2k}-e_{2k+1}\} & \mbox{if $p^\prime>0$},\\
\Pi-\{2e_{n}\} & \mbox{if $p^\prime=0$}
\end{array}\right. .\]
If $G=\SO^\ast(2n)$, we define
\[ S(k)=\left\{\begin{array}{ll} 
\Pi-\{e_{2k}-e_{2k+1}\}& \mbox{if $r>0$},\\
\Pi-\{e_{n-1}+e_{n}\} & \mbox{if $r=0$}
\end{array}\right. .\]
We denote by $M_{(k)}$ (resp.\ $\mmm_{(k)}$) the standard maximal Levi
subgroup (resp.\ subalgebra) of $G$ (resp.\ $\gggg$) corresponding to
$S(k)$.
Namely $M_{(k)}$ is the centralizer of $\aaa_{(k)}$ in $G$.
Let $P_{(k)}$ be a parabolic subgroup of $G$ whose
$\theta$-invariant Levi part is $M_{(k)}$.
We denote by $N_{(k)}$ the nilradical of $P_{(k)}$.
We denote by $\ppp_{(k)}$, $\mmm_{(k)}$, and $\nnn_{(k)}$ the complexified Lie
algebra of $P_{(k)}$, $M_{(k)}$, and $N_{(k)}$,
respectively.
We choose $P_{(k)}$ so that $\{\alpha\in\Delta\mid\gggg_\alpha\subseteq
\nnn_{(k)}\}\subseteq\Delta^+$.  

Formally, we denote by $\Spp(0,0)$ and $\SO^\ast(0)$ the trivial group $\{1\}$.
Then, we have
\[ M_{(k)}\cong 
\left\{\begin{array}{ll} 
\GL(k,\qrt)\times \Spp(p^\prime,q^\prime) & \,\,\,\, \mbox{if $G=\Spp(p,q)$}\\
\GL(k,\qrt)\times \SO^\ast(2r) & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}
\end{array}\right. .\]
Often, we identify $\GL(k,\qrt)$, $\Spp(p^\prime,q^\prime)$, $\SO^\ast(2r)$ with subgroups of $M_{(k)}$ in obvious ways.
We call such identifications the standard identifications.
The Cartan involution $\theta$ induces  Cartan involutions on $M_{(k)}$, $\GL(k,\qtr)$, $\Spp(p^\prime, q^\prime)$, and $SO^\ast(2r)$ and we denote them by the same letter $\theta$.
We put $M^\circ_{(k)}=\Spp(p^\prime,q^\prime)$ (resp.\ $\SO^\ast(2r)$), if
$G=\Spp(p,q)$ (resp.\ $\SO^\ast(2n)$).

We denote by $\mmm^\circ_k$ the complexified Lie
algebra of $M^\circ_k$.

Later, we treat various $\Spp(p,q)$'s and $\SO^\ast(2n)$'s at the same
time.
So, sometimes we write $P_{(k)}(p,q)$ (resp.\ $P_{(k)}^\ast(2n)$) for
$P_{(k)}$ if  $G=\Spp(p,q)$ (resp.\ $G=\SO^\ast(2n)$).

We define a basis $\Pi_u^{(k)}$ of $\Delta(\gggg,\shh)$  as follows.
If $2k=n$, then we put $\Pi_u^{(k)}=\Pi_u$, where $\Pi_u$ is defined in
3.2.
If $2k<n$, then we put
\[
 \Pi_u^{(k)}=\{e_i-e_{i+2}\mid 1\leqslant i\leqslant
 2k-2\}\cup\{e_{2k-1}+e_{2k}, -e_2-e_{2k+1}\}\cup\{\gamma\in\Pi\mid
 \gamma(E_i)=0 (1\leqslant i\leqslant 2k)\}.
\]
Here, $\Pi$ is the basis of $\Delta(\gggg,\shh)$ defined in 3.1.
We denote by $\bbb_{(k)}$ the standard Borel subalgebra of $\gggg$
Put $S_u^{(k)}=\Pi_u^{(k)}-\{-e_2-e_{2k+1}\}$.
Let $\qqq_{(k)}$ be the parabolic subalgebra of $\gggg$ containing
$\bbb_{(k)}$ corresponding to $S_u^{(k)}$.
Since, $\theta(\itg S_u^{(k)})=\itg S_u^{(k)}$ and
$\theta(-e_2-e_{2k+1})\equiv -e_2-e_{2k+1} \,\,\,
(\mod \itg S_u^{(k)})$, $\qqq_{(k)}$ is $\theta$-stable.
We easily see that $U(k,k)\times  M^\circ_{(k)}$ is  a Levi subgroup
(say $L_{(k)}$)for
$\qqq_{(k)}$.
We denote by $\llll_{(k)}$ the complexifed Lie algebra of $L_{(k)}$.

Since $\shh\subseteq \ppp_{(k)}\cap\qqq_{(k)}$, $(\ppp_{(k)},
\qqq_{(k)})$ is a $\sigma\theta$-pair.

We denote by $G_{(k)}$ the centralizer of $\{E_i\mid 2k<i\leqslant n\}$
in $G$.
(If $2k=n$, we put $G_{(k)}=G$.)
If $G=\Spp(p,q)$ (resp.\ $G=\SO^\ast(2n)$), then $G_{(k)}$ is isomorphic
to $\Spp(k,k)$ (resp.\ $\SO^\ast(4k)$). 
We have the following diagram:
\begin{equation}
\begin{array}{ccc}
M_{(k)} &   \subseteq  & G_{(k)}M^\circ_{(k)}\\
\cup\mid  &     & \cup \mid \\
M_{(k)}\cap L_{(k)} &  \subseteq & L_{(k)}   
\end{array}\tag{C}
\end{equation} 
Taking intersection of $G_{(k)}$ and each term of (C),
we have a square quadruplet in the sense of 3.2:
\begin{equation}
\begin{array}{ccc}
\GL(k,\qrt) &   \subseteq  & G_{(k)} \\
\cup\mid  &     & \cup \mid \\
\GL(k,\cpx) &  \subseteq & \U(k,k)   
\end{array}\tag{D}
\end{equation} 
Put $a_G=1$ (resp.\ $a_G=-1$), if $G=\Spp(p,q)$ (resp.\ $G=\SO^\ast(2n)$).
We have the following result. The proof is straightfoward.
\begin{lem}
Define $\xi_{\ppp_{(k)},\qqq_{(k)}}$ as in  2.2.
For $1\leqslant i\leqslant n$, we have:
\[
\xi_{\ppp_{(k)},\qqq_{(k)}}(E_i)=
\left\{\begin{array}{ll} (-1)^{i+1}\frac{2n-3k+a_G}{2}& \,\,\,\, \mbox{if $1\leqslant
 i\leqslant 2k$}  \\
0 & \,\,\,\, \mbox{otherwise}
\end{array}\right. .
\]
\end{lem}

We denote by $\shh_{(k)}$ (resp.\ $\shh^{(k)}$) the $\cpx$-linear span
of $\{E_i\mid 2k<i\leqslant n\}$ (resp.\ $\{E_i\mid 1\leqslant
i\leqslant 2k\}$ ).
Using the direct sum decomposition $\shh=\shh^{(k)}\oplus\shh_{(k)}$,
we have $\shh^\ast=(\shh^{(k)})^\ast\oplus\shh_{(k)}^\ast$.

Let $\pi$ be an irreducible unitary representation of $M^\circ_{(k)}$.
Since $\shh_{(k)}=\shh\cap\mmm^\circ_{(k)}$,  $\shh_{(k)}$ is a Cartan
subalgebra of $\mmm^\circ_{(k)}$.
Let $\lambda_\pi\in\shh_{(k)}^\ast$ be the infinitesimal character of
$\pi$.
($\lambda_\pi$ is determined up to the Weyl group action.)

For $\ell\in\itg$ and $t\in\sqrt{-1}\rel$, we consider the one-dimensional
representation $\xi_{\ell,t}$ of $\GL(k,\cpx)$ defined in 3.2.
We consider the representation $\xi_{\ell,t}\boxtimes\pi$ of
$\GL(k,\cpx)\times M^\circ_{(k)}$.
Let $\lambda_{\ell,t,\pi}$ be the infinitesimal character of
$\xi_{\ell,t}\boxtimes\pi$.
Then we have:
\begin{align*}
 \lambda_{\ell,t,\pi}(E_{2i-1})& = \frac{k-1+\ell+t}{2}-i+1  \,\,\,\,
 (1\leqslant i\leqslant k),
\\
\lambda_{\ell,t,\pi}(E_{2i})& = \frac{k-1-\ell+t}{2}-i+1  \,\,\,\,
 (1\leqslant i\leqslant k),
\\
\lambda_{\ell,t,\pi}|_{\shh_{(k)}}& =\lambda_\pi
\end{align*}

We define:
\[
 c_{\ell,t,\pi}=\max\left(\{0\}\cup\left\{|\lambda_\pi(E_i)|\left|
 n-2k<i\leqslant n,
 \left\{\pm\lambda(E_i)-\frac{\ell+t-1}{2}\right\}\cap\itg\right. \neq\emptyset\right\}\right).
\]
Applying Theorem 2.2.3 (2) to the $\sigma\theta$-pair
$(\ppp_{(k)},\qqq_{(k)})$, we have:

\begin{prop}
Let $\pi$ be an irreducible unitary representation of $M^\circ_{(k)}$.
Let $\ell\in\nat$ and $t\in\sqrt{-1}\rel$.
We assume $\ell \geqslant 2c_{\ell,t,\pi}-1$.
Put $S=k(n-2k+1)$ (resp.\ $S=k(2n-3k)$), if $G=\Spp(p,q)$  (resp.\ if $ G=\SO^\ast(2n)$).

Then,
\[
 \Ind_{P_{(k)}}^G(A_k(\ell,t)\boxtimes\pi)\cong (\rca_{\qqq_{(k)},K\cap L_{(k)}}^{\gggg, K})^S(I_k(\ell+2n-k+a_G,t)\boxtimes\pi).
\]
Here, $A_k(\ell,t)$ (resp.\ $I_k(\ell,t)$) is a quaternionic Speh
 representation (resp.\ a degenerate principal series representatiion)
 defined in 2.4 ($\ast$) (resp.\ 3.2 ($\dagger$)).
\end{prop}

\subsection{{$\theta$-stable parabolic subalgebras}}

We retain the notations in 3.1 and 3.3.
The classifications of $K$-conjugate class of $\theta$-stable
parabolic subalgebras with respect to real classical groups are more or less well-known.
Here, we review the classification for $G=\U(p,q), \Spp(p,q), \SO^\ast(2n)$.
First, we discuss $\theta$-stable parabolic subalgebras with respect to $\U(p,q)$
(cf.\ [36] Example 4.5).

Let $\ell$ be a positive integer. Put
\[\dbP_\ell(p,q)=\left\{((p_1,...,p_\ell),(q_1,...,q_\ell))\in \nat^{\ell}\times\nat^{\ell}\mid \sum_{i=1}^\ell p_i=p, \sum_{i=1}^\ell q_i=q, p_j+q_j>0 \;\; \mbox{for all $1\leq j\leq \ell$}\right\},\] 
We also put
$\dbP(p,q)=\bigcup_{\ell> 0}\dbP_\ell(p,q)$ and 
$\dbP(0,0)=\dbP_0(0,0)=\{((\emptyset),(\emptyset))\}$.
If $(\bbpp,\bpq)\in\dbP(p,q)$ satisfies $(\bbpp,\bpq)\in\dbP_\ell(p,q)$,
we call $\ell$ the length of $(\bbpp,\bpq)$. 
For $(\bbpp,\bpq)\in\dbP(p,q)$, we define
$ I_{(\bbpp,\bpq)}=\diag(I_{p_1,q_1},...,I_{p_\ell,q_\ell})$.
Then we have
\[
 \U(p,q)=\left\{g\in\GL(p+q,\cpx)| {}^t\bar{g}I_{(\bbpp,\bpq)}g=I_{(\bbpp,\bpq)}\right\}.
\]
Let $\theta$ be the Cartan involution given by the conjugation by $I_{(\bbpp,\bpq)}$.
In this realization, we denote by $\qqq(\bbpp,\bpq)$ the
block-upper-triangular parabolic subalgebra of
$\gl(p+q,\cpx)=\uuu(p,q)\otimes_\rel\cpx$ with blocks of sizes
$p_1+q_1,....,p_\ell+q_\ell$ along the diagonal.
Then, $\qqq(\bbpp,\bpq)$ is a $\theta$-stable parabolic subalgebra.
The corresponding Levi subgroup $\U(\bbpp,\bpq)$ consists of diagonal
blocks.
\[
\U(\bbpp,\bpq)\cong \U(p_1,q_1)\times\cdots\times\U(p_\ell,q_\ell). 
\]
We denote by $\uuu(\bbpp,\bpq)$ the Lie algebra of $\U(\bbpp,\bpq)$.
Via the above construction of $\qqq(\bbpp,\bpq)$, $K$-conjugate class of $\theta$-stable parabolic subalgebras with
respect to $\U(p,q)$  is classified by $\dbP(p,q)$.

For $G=\Spp(p,q),\SO^\ast(2n)$, we put
\[
 \dbP_G=
\left\{\begin{array}{ll} 
\bigcup_{\substack{p^\prime\leqslant p \\ q^\prime\leqslant q}}\dbP(p^\prime,q^\prime) & \,\,\,\,  \mbox{if $G=\Spp(p,q)$,} \\
\bigcup_{p^\prime+q^\prime\leqslant n}\dbP(p^\prime,q^\prime) & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}
\end{array}\right. ,
\]
$K$-conjugate class of $\theta$-stable parabolic subalgebras with
respect to $G$  is classified by $\dbP_G$.
We give a construction of $\theta$-stable parabolic subalgebra
$\tqq(\bbpp,\bpq)$ for $(\bbpp,\bpq)\in\dbP_G$.

First, we assume $G=\Spp(p,q)$ , $(\bbpp,\bpq)\in
\dbP_\ell(p^\prime,q^\prime)$, $0\leqslant p^\prime\leqslant p$, and
$0\leqslant q^\prime\leqslant q$.
Put $p_0=p-p^\prime$ and $q_0=q-q^\prime$.
Then we have a symmetric pair $(\Spp(p,q), \Spp(p^\prime,q^\prime)\times\Spp(p_0,q_0))
$.
Taking account of the realization of $\Spp(p^\prime,q^\prime)$ as the  automorphism group of an
 indefinite Hermitian form on a $\qrt$-vector space (3.2 ($\bigstar$)),
 we see that $\U(p^\prime,q^\prime)\subseteq \Spp(p^\prime,q^\prime)$.
Hence we have $\U(p^\prime,q^\prime)\times\Spp(p_0,q_0)\subseteq
\Spp(p,q)$.
Put $L_{(p^\prime,q^\prime)}(p,q)=\U(p^\prime,q^\prime)\times\Spp(p_0,q_0)$
Since the centralizer in $\Spp(p,q)$ of the center of
$\U(p^\prime,q^\prime)$ is $L_{(p^\prime,q^\prime)}(p,q)$,
$L_{(p^\prime,q^\prime)}(p,q)$ is a Levi subgroup of a
$\theta$-stable maximal parabolic subalgebra $\tqq_{(p^\prime,q^\prime)}(p,q)$
of $\fsp(p+q,\cpx)$.
We denote by $\tuu_{(p^\prime,q^\prime)}(p,q)$ the nilradical of
$\tqq_{(p^\prime,q^\prime)}(p,q)$.
In fact there are two possibilities of the choice of $\tuu_{(p^\prime,q^\prime)}(p,q)$.
Our choice should be compatible with the construction in 3.4.
Namely, we should choose $\tuu_{(p^\prime,q^\prime)}(p,q)$ so that
$\tqq_{(p^\prime,q^\prime)}(p,q)=\tqq_{(k)}$ , if $p^\prime=q^\prime=k$.
Such a choice is determined as follows.
For $\ell\in\itg$, we define a character $\eta_\ell$ of
$\U(p^\prime,q^\prime)$ by
\[
 \eta_\ell(g)=\det(g)^\ell \,\,\,\, (g\in\U(p^\prime,q^\prime)).
\]
Let $\pi$ be any irreducible unitary representation of  $\Spp(p_0,q_0)$.
Then, we choose $\tuu_{(p^\prime,q^\prime)}(p,q)$ so that
$\eta_\ell\boxtimes\pi$
is good with respect to $\tqq_{(p^\prime,q^\prime)}(p,q)$ for a sufficiently
large $\ell$.

We denote by $\llll_{(p^\prime,q^\prime)}(p,q)$ the complexfied Lie algebra of
$L_{(p^\prime,q^\prime)}(p,q)$.
Let $\qqq{(\bbpp,\bpq)}(p,q)$ be the $\theta$-stable parabolic subgroup of
$\uuu(p^\prime,q^\prime)\otimes_\rel\cpx$ defined as above.
Since $L_{(p^\prime,q^\prime)}(p,q)=\U(p^\prime,q^\prime)\times\Spp(p_0,q_0)$,
$\qqq(\bbpp,\bpq)\oplus\fsp(p_0+q_0,\cpx)$ is a $\theta$-stable
parabolic subalgebra of $\llll_{(p^\prime,q^\prime)}$.
Define 
\[
 \tqq_{(\bbpp,\bpq)}(p,q)=(\qqq(\bbpp,\bpq)\oplus\fsp(p_0+q_0,\cpx))+
\tuu_{(p^\prime,q^\prime)}(p,q).
\]
Then $\tpp_{(\bbpp,\bpq)}(p,q)$ is a $\theta$-stable parabolic subalgebra of
$\fsp(p+q,\cpx)$.
The corresponding Levi subgroup is
$L_{(\bbpp,\bpq)}(p,q)=\U(\bbpp,\bpq)\times\Spp(p_0,q_0)$.

Next, we consider the case $G=\SO^\ast(2n)$.
Assume $(\bbpp,\bpq)\in
\dbP_\ell(p^\prime,q^\prime)$, $p^\prime+q^\prime\leqslant n$.
Put $n^\prime=p^\prime+q^\prime$ and $n_0=n-n^\prime$.
Then we have a symmetric pair $(\SO^\ast(2n), \SO^\ast(2n^\prime)\times\SO^\ast(2n_0))
$.
There is a symmetric pair $(\U(p^\prime,q^\prime),
\SO^\ast(2n^\prime))$.
Put $L_{(p^\prime,q^\prime)}^\ast(2n)=\U(p^\prime,q^\prime)\times\SO^\ast(2n_0)$.
Since the centralizer in $\SO^\ast(2n)$ of the center of
$\U(p^\prime,q^\prime)$ is $L_{(p^\prime,q^\prime)}^\ast(2n)$,
$L_{(p^\prime,q^\prime)}^\ast(2n)$ is a Levi subgroup of a
$\theta$-stable maximal parabolic subalgebra $\tqq_{(p^\prime,q^\prime)}^\ast(2n)$
of $\so(2n,\cpx)$.
Now that we can construct a $\theta$-stable parabolic subalgebra
$\tpp_{(\bbpp,\bpq)}^\ast(2n)$ of $\so(2n,\cpx)$ in the same way as the case of
$G=\Spp(p,q)$.
In this case the Levi subgroup $L_{(\bbpp,\bpq)}^\ast(2n)$ is isomorphic to
$\U(\bbpp,\bpq)\times\SO^\ast(2n_0)$.

\subsection{{A rearrangement  formula}}

First, we consider the case of  $G=\Spp(p,q)$.
Let $p^\prime$ and $q^\prime$ be non-negative integers such that $p^\prime+q^\prime>0$.
Moreover, we assume that $p^\prime\leqslant p$ and
$q^\prime\leqslant q$.
Put $p_0=p-p^\prime$ and $q_0=q-q^\prime$.
We consider $\theta$-stable maximal parabolic subalgebra
$\tqq_{(p^\prime,q^\prime)}(p,q)$ defined in 3.4.


Let $\hhh(p_0,q_0)$ (resp.\ $\hhh_{(p^\prime,q^\prime)}$)  be a $\theta$
and $\sigma$-stable compact Cartan subalgebra for
$\Spp(p_0,q_0)$  (resp.\ $\U(p^\prime,q^\prime)$).

Taking account of
$L_{(p^\prime,q^\prime)}(p,q)=\U(p^\prime,q^\prime)\times\Spp(p_0,q_0)$,
we put 
 \[
 \hhh(p,q)=\hhh_{(p^\prime,q^\prime)}\oplus\hhh(p_0,q_0)\subseteq \llll
 _{(p^\prime,q^\prime)}(p,q)\subseteq \fsp(p+q,\cpx).
\]
Then, $\hhh(p,q)$ is a $\theta$
and $\sigma$-stable compact Cartan subalgebra for $\Spp(p,q)$.
Using the above direct sum decomposition, we regard
$\hhh_{(p^\prime,q^\prime)}^\ast$
and $\hhh(p_0,q_0)^\ast$ as a subspace of $\hhh(p,q)^\ast$.
We introduce an orthonormal basis $\{f_1,...,f_{p^\prime+q^\prime}\}$
(resp.\ $\{f_{p^\prime+q^\prime+1},..., f_{p+q}\}$) of  $\hhh_{(p^\prime,q^\prime)}^\ast$
(resp.\ $\hhh(p_0,q_0)^\ast$ ) such that

\begin{align*}
\Delta(\fsp(p+q,\cpx), \hhh(p,q)) &=\{ \pm f_i\pm f_j\mid 1\leqslant  i< j\leqslant
 p+q\}\cup\{\pm 2f_i\mid 1\leqslant i\leqslant  p+q\}, 
\\
\Delta(\uuu(p^\prime, q^\prime)\otimes_\rel\cpx,
 \hhh_{(p^\prime,q^\prime)}) &=\{ f_i - f_j\mid 1\leqslant  i,j\leqslant
 p^\prime+q^\prime,  i\neq j\},
\\ 
\Delta(\fsp(p_0+q_0,\cpx), \hhh(p_0,q_0)) &=\{ \pm f_i\pm f_j\mid
 p^\prime+q^\prime<  i< j\leqslant
 p+q\}\cup\{\pm 2f_i\mid  p^\prime+q^\prime<  i\leqslant p+q\}, 
\\
\Delta(\tuu_{(p^\prime,q^\prime)}(p,q), \hhh(p,q)) &=\{ f_i\pm
 f_j\mid 1\leqslant  i\leqslant p^\prime+q^\prime< j\leqslant
 p+q\}\cup\{ f_i+f_j\mid 1\leqslant  i\leqslant j\leqslant
 p^\prime+q^\prime \}, 
\end{align*}

We denote by $F_1,...,F_{p+q}$ the basis of $\hhh(p,q)$ dual to
$f_1,...,f_{p+q}$.
We have
\[
\delta(\tuu_{(p^\prime,q^\prime)}(p,q))(F_i)=
\left\{\begin{array}{ll} \frac{2p+2q-p^\prime-q^\prime+1}{2} & \,\,\,\,  \mbox{if $1\leqslant
 i\leqslant p^\prime+q^\prime$,} \\
0 & \,\,\,\, \mbox{otherwise}
\end{array}\right. 
\]

For $\ell\in\itg$, we consider the one-dimensional  unitary
representation $\eta_\ell$ of $\U(p^\prime,q^\prime)$
defined  in 3.5. 

Let $Z$ be any Harish-Chandra module for $\Spp(p_0,q_0)$ with an infinitesimal
character $\lambda\in\hhh(p_0,q_0)^\ast\subseteq\hhh(p,q)^\ast$.
$\lambda$ is unique up to the Weyl group action.
Put $||\lambda||=\max(\{0\}\cup\{|\lambda(F_i)|\mid p^\prime+q^\prime<i
\leqslant p+q,
\lambda(F_i)\in\itg)\})$.
$||\lambda||$ is invariant under the Weyl group action on $\lambda$, so
we write $||Z||=||\lambda||$.

$\eta_\ell\boxtimes
Z$ has an infinitesimal character $[\ell,\lambda]\in\hhh(p,q)^\ast$ such
that
\[
[\ell,\lambda](F_i)=
\left\{\begin{array}{ll} \ell+\frac{p^\prime+q^\prime+1}{2}-i & \,\,\,\,  \mbox{if $1\leqslant
 i\leqslant p^\prime+q^\prime$,} \\
\lambda(F_i)& \,\,\,\, \mbox{if $p^\prime+q^\prime<i \leqslant p+q$}
\end{array}\right.  
\]
We denote by $\hca(\Spp(p,q))_\mu$ the category of Harish-Chandra
modules for $\Spp(p,q)$ with an infinitesimal character $\mu$.

\begin{dfn}
For $\ell\in\itg$ and $Z\in \hca(\Spp(p_0,q_0))_\lambda$, put
\[
 \rca_{p^\prime, q^\prime}^{p,q}(\ell)(Z)=\left(\rca^{\fsp(p+q,\cpx), K}_{\tqq_{(p^\prime,q^\prime)}(p,q) , L_{(p^\prime,q^\prime)}(p,q)\cap K}\right)^{S}((\eta_\ell\boxtimes Z)\otimes\cpx_{2\delta(\tuu_{(p^\prime,q^\prime)}(p,q))}),
\]
where $S=\frac{p^\prime(4p-3p^\prime+1)+q^\prime(4q-3q^\prime+1)}{2}$.
If $\ell\geqslant ||\lambda||-(p_0+q_0)$, then the above cohomological
 induction is in good range and we have an exact functor
\[
 \rca_{p^\prime, q^\prime}^{p,q}(\ell ): \hca(\Spp(p_0,q_0))_\lambda  \rightarrow \hca(\Spp(p,q))_{[\ell,\lambda]+\delta(\tuu_{(p^\prime,q^\prime)}(p,q))}
\]
\end{dfn}


Next, we consider the following setting. 
Let $k$ be a positive integer such that $k\leqslant p$ and $k\leqslant q$.
Let $p^\prime$ and $q^\prime$ be non-negative integers such that $p^\prime+q^\prime>0$.
Moreover, we assume that $p^\prime+k\leqslant p$ and
$q^\prime+k\leqslant q$.
We consider $\theta$-stable parabolic subalgebra
$\tqq_{(p^\prime,q^\prime)}(p-k,q-k)$ of
$\mmm^\circ_{(k)}=\Spp(p-k,q-k)$ defined in 3.5.

So, the Levi subgroup $L_{(p^\prime,q^\prime)}(p-k,q-k)$ of $\tqq_{(p^\prime,q^\prime)}(p-k,q-k)$ is written as
$U(p^\prime, q^\prime)\times \Spp(p-p^\prime-k, q-q^\prime-k)$.
Put $\sH^{(k)}=\exp(\shh^{(k)})\cap G$.
Then $\sH^{(k)}$ is a maximally split Cartan subgroup of $\GL(k,\qrt)$.
(Here, we consider the decomposition $M_{(k)}=\GL(k,\qrt)\times
M_{(k)}^\circ$.)
We fix a compact Cartan subgroup $\uH_{(k,p^\prime,q^\prime)}$ of
$L_{(p^\prime,q^\prime)}(p-k,q-k)$  and put $H(k,p^\prime,q^\prime)=\sH^{(k)}\times\uH_{(k,p^\prime,q^\prime)}$.
We denote by $\hhh(k,p^\prime,q^\prime)$ the complexified Cartan
subalgebra of $H(k,p^\prime,q^\prime)$.
We denote by $L^\prime_{(k,p^\prime,q^\prime)}$ the centralizer of the center
of $\U(p^\prime,q^\prime)$ in $G$.
Then, we have 
$L^\prime(k,p^\prime,q^\prime)\cong \U(p^\prime,q^\prime)\times
\Spp(p-p^\prime,q-q^\prime)$.
Let $\tqq^\prime(k, p^\prime,q^\prime)$ be a $\theta$-stable parabolic
subalgebra of $\Spp(p,q)$ with the Levi subgroup $L^\prime_{(k,p^\prime,q^\prime)}$.
Let $\phi$ be any irreducible unitary representation of
$\Spp(p-p^\prime,q-q^\prime)$.
We choose $\tqq^\prime(k, p^\prime,q^\prime)$ so that
$\eta_\ell\boxtimes\phi$ is good with respect to $\tqq^\prime(k,
p^\prime,q^\prime)$ for sufficiently large $\ell$.
$\tqq^\prime(k, p^\prime,q^\prime)$ is $K$-conjugate to
$\tqq_{(p^\prime,q^\prime)}(p,q)$
defined in 3.4.

Since $\hhh(k,p^\prime,q^\prime)\subseteq\tqq^\prime(k,
p^\prime,q^\prime)\cap\ppp_{(k)}(p,q)$, $(\ppp_{(k)}(p,q),\tqq^\prime(k,
p^\prime,q^\prime))$ is a $\sigma\theta$-pair.

Since $\shh^{(k)}\subseteq\shh$ has a basis $E_1,...,E_{2k}$, for any
$\lambda\in(\shh^{(k)})^\ast$, we define 
\[
 ||\lambda||=\max(\{0\}\cup \{|\lambda(E_i)|\mid 1\leqslant i\leqslant
 2k, |\lambda(E_i)|\in\itg\}).
\]
For any Harish-Chandra module $V$ with an infinitesimal character
$\lambda\in(\shh^{(k)})^\ast$,
we put $||V||=||\lambda||$.
This is well-defined, since $||\lambda||$ is invariant under the Weyl
group action.
For example, we easily have:

\begin{lem}\mbox{}

(1) \,\,\,\, If $\chi$ is a one-dimensional unitary representation of
 $\GL(k,\qrt)$, then $||\chi||=0$.

(2) \,\,\,\, For $\ell\in\itg$ and $t\in\sqrt{-1}\rel$, we have
\[
 ||A_k(\ell,t)||=
\left\{\begin{array}{ll} 
\frac{2k+\ell-1}{2} & \,\,\,\,  \mbox{if $\ell$ is odd and $t=0$,} \\
0 & \,\,\,\, \mbox{otherwise}
\end{array}\right. 
\]
\end{lem}

Applying Theorem 2.2.3 to the $\sigma\theta$-pair $(\ppp_{(k)}(p,q),\tqq^\prime(k,
p^\prime,q^\prime))$, we have:
\begin{thm}
\mbox{(a rearrangement formula for $\Spp(p,q)$)}

Let $k$ be a positive integer such that $k\leqslant p$ and $k\leqslant q$.
Let $p^\prime$ and $q^\prime$ be non-negative integers such that $p^\prime+q^\prime>0$.
Moreover, we assume that $p^\prime+k\leqslant p$ and
$q^\prime+k\leqslant q$.
Let $V$ (resp.\ $Z$) be a Harish-Chandra module with an infinitesimal
 character for $\GL(k,\qrt)$ (resp.\ $\Spp(p-p^\prime-k, q-q^\prime-k)$).
Let $\ell$ be an integer such that $\ell\geqslant \max\{||V||,||Z||\}-(p-p^\prime-k)-(q-q^\prime-k)$.
Then we have
\[
\left[\Ind^{\Spp(p,q)}_{P_{(k)}(p,q)}\left(V\boxtimes \rca_{p^\prime,q^\prime}^{p-k,q-k}(\ell)(Z)\right)\right]=
\left[\rca_{p^\prime,q^\prime}^{p,q}(\ell-2k)\left(\Ind^{\Spp(p-p^\prime,q-q^\prime)}_{P_{(k)}(p-p^\prime,q-q^\prime)}(V\boxtimes Z)\right)\right].
\]
The above cohomological inductions are in the good region.
\end{thm}

Next, we consider the case of $\SO^\ast(2n)$.

Put $n_0=n-p^\prime-q^\prime$.
We consider $\theta$-stable maximal parabolic subalgebra
$\tqq_{(p^\prime,q^\prime)}^\ast(2n)$ defined in 3.5.


Let $\hhh(2n_0)$ (resp.\ $\hhh_{(p^\prime,q^\prime)}$)  be a $\theta$
and $\sigma$-stable compact Cartan subalgebra for
$\SO^\ast(2n_0)$  (resp.\ $\U(p^\prime,q^\prime)$).

Taking account of
$L_{(p^\prime,q^\prime)}^\ast(2n)=\U(p^\prime,q^\prime)\times\SO^\ast(2n_0)$,
we put 
 \[
 \hhh(2n)=\hhh_{(p^\prime,q^\prime)}\oplus\hhh(2n_0)\subseteq \llll
 _{(p^\prime,q^\prime)}^\ast(2n)\subseteq \so(2n,\cpx).
\]
Then, $\hhh(2n)$ is a $\theta$
and $\sigma$-stable compact Cartan subalgebra for $\SO^\ast(2n)$.
Using the above direct sum decomposition, we regard
$\hhh_{(p^\prime,q^\prime)}^\ast$
and $\hhh(2n_0)^\ast$ as a subspace of $\hhh(2n)^\ast$.
We introduce an orthonormal basis $\{f_1,...,f_{p^\prime+q^\prime}\}$
(resp.\ $\{f_{p^\prime+q^\prime+1},..., f_{2n}\}$) of  $\hhh_{(p^\prime,q^\prime)}^\ast$
(resp.\ $\hhh(2n_0)^\ast$ ) such that

\begin{align*}
\Delta(\so(2n,\cpx), \hhh(2n)) &=\{ \pm f_i\pm f_j\mid 1\leqslant  i< j\leqslant
 p+q\}, 
\\
\Delta(\uuu(p^\prime, q^\prime)\otimes_\rel\cpx,
 \hhh_{(p^\prime,q^\prime)}) &=\{ f_i\-f_j\mid 1\leqslant  i,j\leqslant
 p^\prime+q^\prime,  i\neq j\},
\\ 
\Delta(\so(2n_0,\cpx), \hhh(2n_0)) &=\{ \pm f_i\pm f_j\mid
 p^\prime+q^\prime<  i< j\leqslant
 p+q\}, 
\\
\Delta(\tuu_{(p^\prime,q^\prime)}^\ast(2n), \hhh(2n)) &=\{ f_i\pm
 f_j\mid 1\leqslant  i\leqslant p^\prime+q^\prime< j\leqslant
 p+q\}\cup\{ f_i+f_j\mid 1\leqslant  i<j\leqslant
 p^\prime+q^\prime \}, 
\end{align*}

We denote by $F_1,...,F_{p+q}$ the basis of $\hhh(2n)$ dual to
$f_1,...,f_{2n}$.

We have
\[
\delta(\tuu_{(p^\prime,q^\prime)}^\ast(2n))(F_i)=
\left\{\begin{array}{ll} \frac{2n-p^\prime-q^\prime-1}{2} & \,\,\,\,  \mbox{if $1\leqslant
 i\leqslant p^\prime+q^\prime$,} \\
0 & \,\,\,\, \mbox{otherwise}
\end{array}\right. 
\]

Let $Z$ be any Harish-Chandra module for $\SO^\ast(2n_0)$ with an infinitesimal
character $\lambda\in\hhh(2n_0)^\ast\subseteq\hhh(2n)^\ast$.
$\lambda$ is unique up to the Weyl group action.
Put $||\lambda||=\max(\{0\}\cup\{|\lambda(F_i)|\mid p^\prime+q^\prime<i
\leqslant 2n,
\lambda(F_i)\in\itg)\})$.
$||\lambda||$ is invariant under the Weyl group action on $\lambda$, so
we write $||Z||=||\lambda||$.

$\eta_\ell\boxtimes
Z$ has an infinitesimal character $[\ell,\lambda]\in\hhh(2n)^\ast$ such
that
\[
[\ell,\lambda](F_i)=
\left\{\begin{array}{ll} \ell+\frac{p^\prime+q^\prime+1}{2}-i & \,\,\,\,  \mbox{if $1\leqslant
 i\leqslant p^\prime+q^\prime$,} \\
\lambda(F_i)& \,\,\,\, \mbox{if $p^\prime+q^\prime<i \leqslant p+q$}
\end{array}\right.  
\]
We denote by $\hca(\SO^\ast(2n))_\mu$ the category of Harish-Chandra
modules for $\SO^\ast(2n)$ with an infinitesimal character $\mu$.

\begin{dfn}
For $\ell\in\itg$ and $Z\in \hca(\SO^\ast(2n_0))_\lambda$, put
\[
 \rca_{p^\prime, q^\prime}^{2n}(\ell)(Z)=\left(\rca^{\so(2n,\cpx), K}_{\tqq_{(p^\prime,q^\prime)}^\ast(2n) , L_{(p^\prime,q^\prime)}^\ast(2n)\cap K}\right)^{S}((\eta_\ell\boxtimes Z)\otimes\cpx_{2\delta(\tuu_{(p^\prime,q^\prime)}^\ast(2n))}),
\]
where $S=(p^\prime+q^\prime)(n-p^\prime-q^\prime)+p^\prime q^\prime$.
If $\ell\geqslant ||\lambda||-n_0+1$, then the above cohomological
 induction is in good range and we have an exact functor
\[
 \rca_{p^\prime, q^\prime}^{2n}(\ell ): \hca(\SO^\ast(2n_0))_\lambda  \rightarrow \hca(\SO^\ast(2n))_{[\ell,\lambda]+\delta(\tuu_{(p^\prime,q^\prime)}^\ast(2n))}
\]
\end{dfn}

In a similar way to the case of $\Spp(p,q)$, we have:

\begin{thm}
\mbox{(a rearrangement formula for $\SO^\ast(2n)$)}

Let $k$ be a positive integer such that $k\leqslant p$ and $k\leqslant q$.
Let $p^\prime$ and $q^\prime$ be non-negative integers such that $p^\prime+q^\prime>0$.
Moreover, we assume that $p^\prime+q^\prime+2k\leqslant n$.
Let $V$ (resp.\ $Z$) be a Harish-Chandra module with an infinitesimal
 character for $\GL(k,\qrt)$ (resp.\ $\SO^\ast(2(n-p^\prime-q^\prime-2k))$).
Let $\ell$ be an integer such that $\ell\geqslant \max\{||V||,||Z||\}-(n-p^\prime-q^\prime-2k)-1$.
Then we have
\[
\left[\Ind^{\SO^\ast(2n)}_{P_{(k)}^\ast(2n)}\left(V\boxtimes \rca_{p^\prime,q^\prime}^{2(n-2k)}(\ell)(Z)\right)\right]=
\left[\rca_{p^\prime,q^\prime}^{2n}(\ell-2k)\left(\Ind^{\SO^\ast(2(n-p^\prime-q^\prime))}_{P_{(k)}^\ast(2(n-p^\prime-q^\prime))}(V\boxtimes Z)\right)\right].
\]
The above cohomological inductions are in the good region.
\end{thm}

\subsection{{Decomposition formulas}}

We define:
\begin{dfn}
Let $k$ be a positive integer and $\ell$ be an integer such that
 $\ell+k\in2\itg$.
Let $i$ be an integer such that $0\leqslant i\leqslant k$.
We define the following derived functor module for $U(k,k)$.
\[
 B^{(i)}_k(\ell)=\left({}^u\rca_{\qqq((i,k-i),(k-i,i)),\U(i)\times\U(k-i)\times\U(k-i)\times U(i)}^{\uuu(k,k)\otimes_\rel\cpx, \U(k)\times\U(k)}\right)^{2i(k-i)}(\eta_{\frac{\ell+k}{2}}\boxtimes\eta_{\frac{\ell-k}{2}}).
\]
\end{dfn}

$B^{(i)}_k(\ell)$ is not in the good region. 
In fact, it is an irreducible unitary representation located at the end
of the weakly fair region in the sense of [35].

We quote the following reducibility result of the degenerate principal
series.

\begin{thm}
\mbox{\rm (Kashiwara-Vergne, Johnson,...)}

Let $\ell\in\itg$ and $t\in\sqrt{-1}\rel$.

(1)  \,\,\,\, If $\ell+k\in2\itg$, then
\[
 I_k(\ell,0)=\bigoplus_{i=0}^k B_k^{(i)}(\ell)
\]

(2)  \,\,\,\, If $t\neq 0$  or $\ell+k+1\in2\itg$, then $I_k(\ell,t)$
 is irreducible.
\end{thm}

Some remarks are in order.
The reducibility of $I_k(\ell,0)$ is established by [9].
The irreducibility result is due to [7].
Identifying irreducible components in (1) as derived functor modules is
an easy conclusion from [2] and it has been  more or less
known by experts.
For example, a proof is given in [21] 3.4.

Combining Theorem 3.6.2 and Proposition 3.3.2, we have:

\begin{prop}\mbox{}

(1)  \,\,\,\,  
Let $p,q$ be positive integers such that $q\leqslant p$.
Let $G=\Spp(p,q)$   and let $k$ be a positive integer such
 that $k\leq q$.
Let $V$ be an irreducible unitary representation of $\Spp(p-k,q-k)$.
Let $m$ be an integer such that $m\geqslant ||V||+k-1$.
Then we have
\[
 \Ind^{\Spp(p,q)}_{P_{(k)}(p,q)}(A_k(2m+1,0)\boxtimes V)\cong
\bigoplus_{i=0}^k \rca^{p,q}_{i,k-i}(m-n+k)\left(\rca^{p-i,q-k+i}_{k-i,i}(m-n+2k)(V)\right).
\]

(2)  \,\,\,\,  
Let $n$ be a positive integer.
Let $G=\SO^\ast(2n)$ and let $k$ be a positive integer such
 that $2k\leq n$.
Let $V$ be an irreducible unitary representation of $\SO^\ast(2(n-2k))$.
Let $m$ be an integer such that $m\geqslant ||V||+k-1$.
Then we have
\[
 \Ind^{\SO^\ast(2n)}_{P_{(k)}^\ast(2n)}(A_k(2m+1,0)\boxtimes V)\cong
\bigoplus_{i=0}^k \rca^{2n}_{i,k-i}(m-n+k+1)\left(\rca^{2(n-k)}_{k-i,i}(m-n+2k+1)(V)\right).
\]
\end{prop}

We introduce notations for derived functor modules.

First, we assume $G=\Spp(p,q)$ , $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime\leqslant p$, and
$0\leqslant q^\prime\leqslant q$.
Put $p_0=p-p^\prime$ and $q_0=q-q^\prime$.
We consider the derived functor modules with respect to
$\tpp_{(\bbpp,\bpq)}(p,q)$.
For $1\leqslant i\leqslant m$, we put $p_i^\ast=p_1+\cdots+p_i$ and $q_i^\ast=q_1+\cdots+q_i$.
Let $\ell,...,\ell_m$ be integers and put
\begin{multline*}
 A^{p,q}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m)=
\\
\rca^{p,q}_{p_1,q_1}(\ell_1)\left(\rca^{p-p_1,q-q_1}_{p_2,q_2}(\ell_2)\left(\cdots
\left(\rca^{p-p^\ast_{i-1},q-q^\ast_{i-1}}_{p_i,q_i}(\ell_i)\left(\cdots
\left(\rca^{p_0+p_m, q_0+q_m}_{p_m,q_m}(\ell_m)(1_{\Spp(p_0,q_0)})\right)\cdots\right)\right)\cdots\right)\right).
\end{multline*}
Here, $1_{\Spp(p_0,q_0)}$ is the trivial representation of $\Spp(p_0,q_0)$.
In this setting, we define as follows.
\begin{align*}
\delta_i &=p+q-p_i^\ast-q_i^\ast-\frac{p_i+q_i-1}{2} \,\,\,\,
 \,\,\,\,\, (1\leqslant i  \leqslant m),
\\
\tilde{\ell}_i&=\ell_i+\delta_i   \,\,\,\, \,\,\,\,\,  (1\leqslant i  \leqslant m).
\end{align*}

Then, $A^{p,q}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m)$ is in good (resp.\
weakly fair) region if and only if
$\ell_1\geqslant\ell_2\geqslant\cdots\geqslant\ell_m\geqslant 0$ (resp.\
$\tilde{\ell}_1\geqslant\tilde{\ell}_2\geqslant\cdots\geqslant\tilde{\ell}_m\geqslant
0$).


Next, we assume $G=\SO^\ast(2n)$ , $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime+q^\prime\leqslant n$.
Put $n_0=n-p^\prime-q^\prime$.
We consider the derived functor modules with respect to
$\tpp_{(\bbpp,\bpq)}^\ast(2n)$.
For $1\leqslant i\leqslant m$, we put $p_i^\ast=p_1+\cdots+p_i$ and $q_i^\ast=q_1+\cdots+q_i$.
Let $\ell,...,\ell_m$ be integers and put
\begin{multline*}
 A^{2n}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m)=
\\
\rca^{2n}_{p_1,q_1}(\ell_1)\left(\rca^{2n-p_1-q_1}_{p_2,q_2}(\ell_2)\left(\cdots
\left(\rca^{2n-p^\ast_{i-1}-q^\ast_{i-1}}_{p_i,q_i}(\ell_i)\left(\cdots
\left(\rca^{n_0+p_m+q_m}_{p_m,q_m}(\ell_m)(1_{\SO^\ast(2n_0)})\right)\cdots\right)\right)\cdots\right)\right).
\end{multline*}
Here, $1_{\SO^\ast(2n_0)}$ is the trivial representation of $\SO^\ast(2n_0)$.
In this setting, we define as follows.
\begin{align*}
\delta_i &=p+q-p_i^\ast-q_i^\ast-\frac{p_i+q_i-1}{2}-1 \,\,\,\,
 \,\,\,\,\, (1\leqslant i  \leqslant m),
\\
\tilde{\ell}_i&=\ell_i+\delta_i   \,\,\,\, \,\,\,\,\,  (1\leqslant i  \leqslant m).
\end{align*}

Then, $A^{2n}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m)$ is in good (resp.\
weakly fair) region if and only if
$\ell_1\geqslant\ell_2\geqslant\cdots\geqslant\ell_m\geqslant 0$ (resp.\
$\tilde{\ell}_1\geqslant\tilde{\ell}_2\geqslant\cdots\geqslant\tilde{\ell}_m\geqslant
0$).

Combining Theorem 3.5.3, Theorem 3.5.5, and Proposition 3.6.3 ,  we have:
\begin{thm}\mbox{}

(1) \,\,\,\,\,
Let $p,q$ be positive integers such that $q\leqslant p$.
We consider the setting of  $G=\Spp(p,q)$.
We assume $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime\leqslant p$, and
$0\leqslant q^\prime\leqslant q$.
Let $k$ be a positive integer.
Put $n=p+q$ and put $n^\prime_j=(p_j+q_j)+\cdots+(p_m+q_m)+2k$
for $1\leqslant i\leqslant m$.
Let $s$ be a non-negative integer.

Let $\ell_1,...,\ell_m$ be integers such that
 $\ell_1\geqslant\ell_2\geqslant\cdots\geqslant\ell_m\geqslant 0$.
Moreover, we assume there is some $1\leqslant j\leqslant m$ such that 
$\ell_{j-1}\geqslant s-n^\prime_j+3k$ and $s-n^\prime_j+2k\geqslant \ell_j$.
(Here, we  put, formally, $\ell_0=+\infty$.)
Put $\bbpp_i^\prime=(p_1,...,p_{j-1},i,k-i,p_j...,p_m)$ and
 $\bpq_i^\prime=(q_1,...,q_{j-1},k-i,i,q_j,...,q_m)$ for $1\leqslant i\leqslant k$.
Then we have
\begin{multline*}
 \Ind_{P_{(k)}(p+k,q+k)}^{\Spp(p+k,q+k)}(A_k(2s+1)\boxtimes A^{p,q}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m))
\\
\cong
\bigoplus_{i=0}^kA^{p+k,q+k}_{(\bbpp_i^\prime,\bpq_i^\prime)}
(\ell_1-2k,...,\ell_{j-1}-2k,
 s-n^\prime_j+k,s-n^\prime_j+2k,\ell_j,...,\ell_1).\tag{$A$}
\end{multline*}

(2) \,\,\,\,\,
Let $n$ be positive integer and 
we consider the setting of  $G=\SO^\ast(2n)$.
We assume $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime+q^\prime\leqslant n$.
Let $k$ be a positive integer.
Put $n^\prime_j=(p_j+q_j)+\cdots+(p_m+q_m)+2k$
for $1\leqslant i\leqslant m$.
Let $s$ be a non-negative integer.

Let $\ell_1,...,\ell_m$ be integers such that
 $\ell_1\geqslant\ell_2\geqslant\cdots\geqslant\ell_m\geqslant 0$.
Moreover, we assume there is some $1\leqslant j\leqslant m$ such that 
$\ell_{j-1}\geqslant s-n^\prime_j+3k+1$ and $s-n^\prime_j+2k+1\geqslant \ell_j$.
(Here, we  put, formally, $\ell_0=+\infty$.)
Put $\bbpp_i^\prime=(p_1,...,p_{j-1},i,k-i,p_j...,p_m)$ and
 $\bpq_i^\prime=(q_1,...,q_{j-1},k-i,i,q_j,...,q_m)$ for $1\leqslant i\leqslant k$.
Then we have
\begin{multline*}
 \Ind_{P_{(k)}^\ast(2(n+2k))}^{\SO^\ast(2(n+2k))}(A_k(2s+1)\boxtimes A^{2n}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m))
\\
\cong
\bigoplus_{i=0}^kA^{2(n+2k)}_{(\bbpp_i^\prime,\bpq_i^\prime)}
(\ell_1-2k,...,\ell_{j-1}-2k,
 s-n^\prime_j+k+1,s-n^\prime_j+2k+1,\ell_j,...,\ell_1).\tag{$A^\prime$}
\end{multline*}

(3) \,\,\,\,\,
The derived functor modules in the right hand side of (A) and
 ($A^\prime$) are all non-zero
 and irreducible. (Actually, they are good-range cohomological induction
 form non-zero irreducible modules.)
\end{thm}

Here, we apply the translation principle in weakly fair range in [35] to
the above result and obtain:
\begin{thm}\mbox{}

(1) \,\,\,\,\,
Let $p,q$ be positive integers such that $q\leqslant p$.
We consider the setting of  $G=\Spp(p,q)$.
We assume $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime\leqslant p$, and
$0\leqslant q^\prime\leqslant q$.
Let $k$ be a positive integer.
Put $n=p+q$ and put $n^\prime_j=(p_j+q_j)+\cdots+(p_m+q_m)+2k$
for $1\leqslant i\leqslant m$.
Let $s$ be an integer such that $2s+1\geqslant -k$.
Let $\ell_1,...,\ell_m$ be integers such that
 $\tilde{\ell}_1\geqslant\tilde{\ell}_2\geqslant\cdots\geqslant\tilde{\ell}_m\geqslant 0$.
We choose any $1\leqslant j\leqslant m$ such that 
$\tilde{\ell}_{j-1}\geqslant s+\frac{k+1}{2}\geqslant \tilde{\ell}_j$.
(Here, we  put, formally, $\ell_0=+\infty$.)
Put $\bbpp_i^\prime=(p_1,...,p_{j-1},i,k-i,p_j...,p_m)$ and
 $\bpq_i^\prime=(q_1,...,q_{j-1},k-i,i,q_j,...,q_m)$ for $1\leqslant i\leqslant k$.
Then we have
\begin{multline*}
 \Ind_{P_{(k)}(2(p+k,q+k)}^{\Spp(p+k,q+k)}(A_k(2s+1)\boxtimes A^{p,q}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m))
\\
\cong
\bigoplus_{i=0}^kA^{p+k,q+k}_{(\bbpp_i^\prime,\bpq_i^\prime)}
(\ell_1-2k,...,\ell_{j-1}-2k,
 s-n^\prime_j+k,s-n^\prime_j+2k,\ell_j,...,\ell_1). \tag{$B$}
\end{multline*}

(2) \,\,\,\,\,
Let $n$ be positive integer and 
we consider the setting of  $G=\SO^\ast(2n)$.
We assume $(\bbpp,\bpq)\in
\dbP_m(p^\prime,q^\prime)$, $0\leqslant p^\prime+q^\prime\leqslant n$.
Let $k$ be a positive integer.
Put $n^\prime_j=(p_j+q_j)+\cdots+(p_m+q_m)+2k$
for $1\leqslant i\leqslant m$.

Let $s$ be an integer such that $2s+1\geqslant -k$.
Let $\ell_1,...,\ell_m$ be integers such that
 $\tilde{\ell}_1\geqslant\tilde{\ell}_2\geqslant\cdots\geqslant\tilde{\ell}_m\geqslant 0$.
We choose any $1\leqslant j\leqslant m$ such that 
$\tilde{\ell}_{j-1}\geqslant s+\frac{k+1}{2}\geqslant \tilde{\ell}_j$.
(Here, we  put, formally, $\ell_0=+\infty$.)
Put $\bbpp_i^\prime=(p_1,...,p_{j-1},i,k-i,p_j...,p_m)$ and
 $\bpq_i^\prime=(q_1,...,q_{j-1},k-i,i,q_j,...,q_m)$ for $1\leqslant i\leqslant k$.
Then we have
\begin{multline*}
 \Ind_{P_{(k)}^\ast(2(n+2k))}^{\SO^\ast(2(n+2k))}(A_k(2s+1)\boxtimes A^{2n}_{(\bbpp,\bpq)}(\ell_1,....,\ell_m))
\\
\cong
\bigoplus_{i=0}^kA^{2(n+2k)}_{(\bbpp_i^\prime,\bpq_i^\prime)}
(\ell_1-2k,...,\ell_{j-1}-2k,
 s-n^\prime_j+k+1,s-n^\prime_j+2k+1,\ell_j,...,\ell_1). \tag{$B^\prime$}
\end{multline*}



\end{thm}

\pf
The proof is similar to the arguments in [21] 3.3.
We consider the case of $G=\Spp(p,q)$.
(The case of $G=\SO^\ast(2n)$ is similar.)
For an integer $a$, we denote by $\eta_a$  one dimensional
representation of $\GL(h,\cpx)$ defined by 
 $\eta_{a}(g)=\det(g)^a$.
Let $a_1,...,a_m$ and $b$ be non-negative integers and consider a one
dimensional representation
$\eta=\eta_{a_1}\boxtimes\cdots\boxtimes\eta_{a_{j-1}}\boxtimes\eta_b\boxtimes\eta_b\boxtimes\eta_{a_j}\cdots\boxtimes\eta_{a_m}\boxtimes
1_{\Spp(p_0+q_0,\cpx)}$
of  $\GL(p_1+q_1,\cpx)\times\cdots\times\GL(p_{j-1}+q_{j-1},\cpx)\times\GL(k,\cpx)\times\GL(k,\cpx)\times\GL(p_j+q_j,\cpx)\times\cdots\times\GL(p_m+q_m,\cpx)\times\Spp(p_0+q_0,\cpx)$.
If $\eta_1\geqslant\cdots\geqslant a_{j-1}\geqslant b\geqslant
a_j\geqslant\cdots\geqslant a_m$, then there is
an irreducible finite dimensional representation $V$ of $G_\cpx$ which
contains $\eta$ as the  highest weight space .
If we choose $a_1 \geqslant a_2\geqslant\cdots\geqslant a_{j-1}\gg b \gg
 a_j\geqslant\cdots\geqslant a_m$ suitably, we have
$s^\prime=s+b$, $\ell^\prime_r=\ell_r+a_r$ \,\,\,\,\, $(1\leqslant
r\leqslant m)$ satisfy the regularity assumption in Theorem 3.6.4.
So, we have:

\begin{multline*}
 \Ind_{P_{(k)}(p+k,q+k)}^{\Spp(p+k,q+k)}(A_k(2s^\prime+1)\boxtimes A^{p,q}_{(\bbpp,\bpq)}(\ell^\prime_1,....,\ell^\prime_m))
\\
\cong
\bigoplus_{i=0}^kA^{p+k,q+k}_{(\bbpp_i^\prime,\bpq_i^\prime)}
(\ell^\prime_1-2k,...,\ell^\prime_{j-1}-2k,
 s^\prime-n^\prime_j+k,s^\prime-n^\prime_j+2k,\ell^\prime_j,...,\ell^\prime_1).\tag{$C$}
\end{multline*}

Let $T$ the translation functor from the infinitesimal character of the
modules in (C) to that of (B).
If we apply $T$ to the both sides of $C$, we obtain (B) above.
The argument is the same as [21] Lemma 3.3.3.
The main ingredient is [35]  Proposition 4.7 .
(We may apply similar argument to non-elliptic cohomological induction
by [29] Lemma 7.2.9 (b).)

\,\,\,\, Q.E.D.

{\it Remark} \,\,\,
In Theorem 3.6.5, a choice of $j$ need not be unique. 
So, depending on the choices of $j$, we have apparently different
formulas.
Their compatibility is assured by [21] Theorem 3.3.4, which
is an easy conclusion of [2] Theorem 4.2.
The derived functor modules in the right hand side of (B) and ($B^\prime$)are all in the
weakly fair region. 


\bigskip

\setcounter{section}{4}
\section*{\S\,\,4.\,\,\,\, Reduction of irreducibilities}
\setcounter{subsection}{0}

\subsection{{Comparison of Hecke algebra module structures}}

Let $G$ be a connected real reductive linear Lie group as in 1.2.
Moreover, we assume that all the Cartan subgroups of $G$ are connected.
This assumption is satisfied for the groups $\Spp(p,q)$, $\SO^\ast(2n)$,
and their Levi subgroups.
It will allow us to simplify the description of coherent families, which we now recall. 

Under this assumption, we may write the regular character
$(H,\Gamma,\lambda)$ as $(H,\lambda)$, since  $\Gamma$ is uniquely determined by $\lambda$.
We fix a regular weight $\spl\lambda\in\shh^\ast$.  
Put $\Lambda=\spl\lambda+\pca_G$.

We denote by $W_{\spl\lambda}$ (resp.\ $\Delta_{\spl\lambda}$) the integral Weyl group (resp.\ the integral root system) for $\lambda$.
Namely, we put
$W_{\spl\lambda}=\{ w\in W\mid w\spl\lambda-\spl\lambda\in \qca \}$,
$\Delta_{\spl\lambda}=\left\{ \alpha\in \Delta\left| \frac{\langle\alpha,\spl\lambda\rangle}{\langle\alpha,\alpha\rangle}\in\itg \right\}\right.$.
We put
$\Delta_{\spl\lambda}^+=\{ \alpha\in \Delta_{\spl\lambda}\mid \langle\alpha,\spl\lambda\rangle>0\}$.
Then, $\Delta^+_{\spl\lambda}$ is a positive system for $\Delta_{\spl\lambda}$.
We denote by $\Pi_{\spl\lambda}$ the set of simple roots in $\Delta^+_{\spl\lambda}$.

A map $\Theta$ of $\Lambda$ to the space of invariant eigendistributions
on $G$ is called a coherent family on $\Lambda$ if it satisfies the
following conditions. 
(Our assumption that all the Cartan subgroups are connected makes the
definition of a coherent family much simpler. For the formulation in the
general setting, see
[29].)

(C1) \,\, For all $\eta\in \Lambda$, $\Theta(\eta)$ is a complex linear combination of  the distribution characters of Harish-Chandra modules with infinitesimal character $\eta$.

(C2) \,\, For any finite dimensional representation $E$, we have
\[ [E]\Theta(\eta)=\sum_{\mu\in\pca_G}[\mu:E]\Theta(\eta+\mu)   \,\,\,\, \,\,\,\, (\eta\in \Lambda).\]
Here, $[\mu:E]$ means the multiplicity of the weight $\mu$ in $E$.

We denote by $\cca(\Lambda)$ the set of coherent families on $\Lambda$.
For $w\in W_{\spl\lambda}$ and $\Theta\in\cca(\Lambda)$, we define $w\cdot\Theta$ by $(w\cdot\Theta)(\eta)=\Theta(w^{-1}\eta)$.
We see $\cca(\Lambda)$ is a $W_{\spl\lambda}$-representation.
This representation is called the coherent continuation representation for $\Lambda$.

For any Harish-Chandra $(\gggg, K)$-module $V$ with an infinitesimal character $\spl\lambda$, there is a unique coherent family $\Theta_V$ such that $\Theta_V(\spl\lambda)=[V]$.
For a regular character $\gamma=(H,\lambda)$ such that $\chi_{\lambda}=\chi_{\spl\lambda}$, we put $\Theta^G_\gamma=\Theta_{\pi_G(\gamma)}$
and $\bar{\Theta}^G_\gamma=\Theta_{\bar{\pi}_G(\gamma)}$.
If $\eta\in \Lambda$ is regular and dominant (with respect to $\Delta^+_{\spl\lambda}$), then $(H,\bi_{\spl\lambda,\lambda}(\eta))$ is a regular character and we have $\Theta^G_{\gamma}(\eta)=[\pi_G(H,\bi_{\spl\lambda,\lambda}(\eta))]$
and $\bar{\Theta}^G_{\gamma}(\eta)=[\bar{\pi}_G(H,\bi_{\spl\lambda,\lambda}(\eta))]$.
Put ${\bf St}_G(\spl\lambda)=\{\Theta^G_{\gamma}\mid \gamma\in R_G(\spl\lambda)\}$ and ${\bf Irr}_G(\spl\lambda)=\{\bar{\Theta}^G_{\gamma}\mid \gamma\in R_G(\spl\lambda)\}$.
We define a bijection $\Theta\leadsto \bar{\Theta}$ of ${\bf St}_G(\spl\lambda)$ onto ${\bf Irr}_G(\spl\lambda)$ by $\bar{\Theta^G_\gamma}=\bar{\Theta}^G_\gamma$ for $\gamma\in R_G(\spl\lambda)$.
${\bf St}_G(\spl\lambda)$ forms a basis of $\cca(\Lambda)$ and so does
${\bf Irr}_G(\spl\lambda)$. 

We write $\bar{\Theta}^G_\gamma=\sum_{{\Theta}\in {\bf St}_G(\spl\lambda)}M_G(\gamma,{\Theta}){\Theta}$ and $M_G(\gamma,\delta)=M_G(\gamma,{\Theta}_\delta)\in \cpx$.
For $\gamma=(H,\lambda)\in R_G(\spl\lambda)$ and $w\in W_{\spl\lambda}$, the cross product is defined as follows.
\[ w\times\gamma=(H,\bi_{\spl\lambda,\lambda}(w)^{-1}\lambda).\]
Then, we have $w\times\gamma\in R_G(\spl\lambda)$.
Moreover, for any $\gamma,\gamma^\prime\in R_G(\spl\lambda)$ such that $\Theta^G_\gamma=\Theta^G_{\gamma^\prime}$, we have $\Theta^G_{w\times\gamma}=\Theta^G_{w\times\gamma^\prime}$ for all $w\in W_{\spl\lambda}$.
So, we put $w\times\Theta^G_\gamma=\Theta^G_{w\times\gamma}$.


Let $H$ be a $\theta$-stable Cartan subgroup of $G$ and let $\hhh$ be its complexified Lie algebra.
For a non-compact imaginary root $\beta\in\Delta(\gggg,\hhh)$,
we denote by $c_\beta\in Ad(G_\cpx)$ the Cayley transform associated with
$\beta$ (see [13] p419).
For a real  root $\alpha\in\Delta(\gggg,\hhh)$, we denote by $c^\alpha$
the (inverse)Cayley transform associated with
$\alpha$ ( In [13] p420, Knapp denote $c^\alpha$ by $d_\alpha$.)

We recall the Cayley transforms of regular characters (cf.\ [32]).
Fix $\gamma=(H,\lambda)\in R_G(H,\spl\lambda)$, and choose $\alpha\in\Delta_{\spl\lambda}$ such that $\alpha$ is noncompact imaginary with respect to $\gamma$.
we put $c_\alpha(\gamma)=(Ad(c_{\bi_{\spl\lambda,\lambda}(\alpha)})(H), \lambda\cdot\Ad(c_{\bi_{\spl\lambda,\lambda}(\alpha)})^{-1})$.
Then, we have $c_\alpha(\gamma)\in R_G(\spl\lambda)$ and $\alpha$ is real with respect to $c_\alpha(\gamma)$. 
It is easy to see $c_\alpha(\Theta^G_\gamma)=\Theta^G_{c_\alpha(\gamma)}$ is well-defined.

Conversely, consider $\gamma\in R_G(\spl\lambda)$ and $\alpha\in\Delta_{\spl\lambda}$ which is real with respect to $\gamma$.
We call $\alpha$ satisfies the parity condition with respect to $\gamma$, if there is some $\gamma^\prime\in R_G(\spl\lambda)$ such that $\alpha$ is noncompact imaginary with respect to $\gamma^\prime$ and $\gamma=c_\alpha(\gamma^\prime)$.
If $\alpha$ satisfies the parity condition with respect to $\gamma$, there are just two regular characters in $R_G(\Ad(c^{\bi_{\spl\lambda,\lambda}(\alpha)})(H),\spl\lambda)$, say $c_+^\alpha(\gamma)$ and $c_-^\alpha(\gamma)$, in the preimage of $\gamma$ with respect to $c_\alpha$.
Since we assume that all the Cartan subgroups of $G$ are connected, $c_\pm^\alpha(\gamma)$ are not $K$-conjugate to each other.
It is easy to see $c_\pm^\alpha(\Theta^G_\gamma)=\Theta^G_{c_\pm^\alpha(\gamma)}$ is well-defined.

We denote by $H(W_{\spl\lambda})$ the Iwahori-Hecke algebra for
$W_{\spl\lambda}$.
We denote by $q$ the indeterminant appearing in the definition of  $H(W_{\spl\lambda})$.
 
Put $\cca(\Lambda)_q=\cca(\Lambda)\otimes_{\cpx}\cpx[q]$.
We introduce $H(W_{\spl\lambda})$-module structure on $\cca(\Lambda)_q$
as in [32] p239.
The important thing is that the Hecke algebra module structure is completely determined by the action of cross product and Cayley transforms on the $K$-conjugacy classes of regular characters in $R_G(\spl\lambda)$.

If we consider the specialization at $q=1$ of this Hecke algebra module $\cca(\Lambda)_q$, then we have a $W_{\spl\lambda}$-representation on $\cca(\Lambda)$.
The relation to the coherent continuation representation is given as follows.

\begin{thm}
([29], [30])

We have an isomorphism
\[ \mbox{(Specialization of $\cca(\Lambda)_q$ at $q=1$)} \cong \mbox{(Coherent continuation representation)}\otimes \mbox{sgn},\]
where $\mbox{sgn}$ means the signature representation of $W_{\spl\lambda}$.
This isomorphism preserves the basis ${\bf St}_G(\spl\lambda)$.

\end{thm}


The following result is crucial in our proof.

\begin{thm}(see [32],  [1] Chapter 16)

For $\gamma,\delta\in R_G(\spl\lambda)$, the complex number
 $M(\gamma,\delta)$ is computed from an algorithm
 (the Kazhdan-Lusztig type algorithm) which depends only on the Hecke
 algebra structure on $\cca(\Lambda)_q$.

\end{thm}

Let $P$ be a parabolic subgroup of $G$ with $\theta$-stable Levi part $L$ such that $\sH\subseteq L$.
(We remark that all the Cartan subgroups of $L$ are connected.)
We fix a regular character $\spl\lambda\in\spl\hhh^\ast$ as above.
Put $\Lambda_L=\spl\lambda+\pca_L$ and $\Lambda_G=\spl\lambda+\pca_G$.
Then, we easily see $\Lambda_G\subseteq\Lambda_L$.
Let $\Theta$ be a coherent family on $\Lambda_L$.
For fixed $\nu\in\Lambda_G$, we write $\Theta(\nu)=\sum_{i=1}^na_i[V_i]$, where $V_i$ are certain Harish-Chandra $(\llll, K\cap L)$-modules with infinitesimal character $\nu$ and $a_i$ are complex numbers.
We write $\Ind_L^G(\Theta)(\nu)=\sum_{i=1}^na_i[\Ind^G_P(V_i)]$.
The above definition is independent of the choice of the linear combination, since the parabolic induction is exact.
From a property of induction, the above definition depends only on $L$ and does not depend on $P$.
Moreover, $\nu\leadsto\Ind_L^G(\Theta)(\nu)$ forms a coherent family on $\Lambda_G$, thanks to a version of Mackey's tensor product theorem ([26] Lemma 5.8) for induction and the exactness of the induction.

Let $H$ be a $\theta$-stable Cartan subgroup of $L$.  Hence $H$ is also a Cartan subgroup of $G$.
Let $\gamma=(H,\lambda)$ be a regular character for $L$ with an infinitesimal character $\spl\lambda$.
Then $\gamma$ is also a regular character for $G$.
We easily see $\Ind_L^G(\Theta_\gamma^L)=\Theta_\gamma^G$.

Next we describe a result on the comparison of Hecke module
structures.
Besides $G$ we also consider another real reductive linear Lie group $G^\prime$ whose Cartan subgroups are all connected.
We denote the objects with respect to $G^\prime$ by attaching the ``prime'' to the notations for the corresponding objects for $G$.
For example, we fix a Cartan involution $\theta^\prime$ for $G^\prime$ and fix a $\theta^\prime$-invariant maximally split Cartan subgroup $\sH^\prime$, etc.
We fix a regular weight $\spl\lambda\in\shh^\ast$ and put $\Lambda=\spl\lambda+\pca_G$.
Moreover, we assume the following conditions on $G$ and $G^\prime$.

(C1) \,\,\,\, There is a linear isomorphism $\psi : \shh^\ast \rightarrow(\shh^\prime)^\ast$ such that  $\psi(\Delta_{\spl\lambda})=\Delta^\prime$.
Here, $\Delta^\prime$ means the root system with respect to $(\gggg^\prime, \shh^\prime)$.
Moreover, $\psi(\spl\lambda)$ is regular integral with respect to $\Delta^\prime$ and $\psi(\pca_G)\subseteq\pca_{G^\prime}$.
$\psi$ induces an isomorphism $\psi_\sharp : W_{\spl\lambda}\rightarrow W^\prime$.
Here, $W^\prime$ is the Weyl group for $\Delta^\prime$.

(C2) \,\,\,\, There is a bijection $\Psi$ of the $K$-conjugacy classes of $\spl\lambda$-integral $\theta$-invariant Cartan subgroups of $G$ to the $K^\prime$-conjugacy classes of $\psi(\spl\lambda)$-integral $\theta^\prime$-invariant Cartan subgroups of $G^\prime$.

(C3) \,\,\,\, There is a bijection $\tilde{\Psi} : {\bf St}_G(\spl\lambda) \rightarrow {\bf St}_{G^\prime}(\psi(\spl\lambda))$ which is compatible with $\Psi$ in (C2).

(C4) \,\,\,\, For $\Theta\in{\bf St}_G(\spl\lambda)$, we have $\psi\circ\theta_\Theta=\theta_{\tilde{\Theta}}\circ\psi$.   
Hence, for $\alpha\in \Delta_{\spl\lambda}$, we have $\alpha$ is imaginary, complex, real with respect to $\Theta$ if and only if $\psi(\alpha)$ is imaginary, complex, real, respectively, with respect to $\tilde{\Psi}(\Theta)$.

(C5) \,\,\,\, Let $\alpha\in \Delta_{\spl\lambda}$ and $\Theta\in{\bf St}_G(\spl\lambda)$. 
If $\alpha$ is imaginary, we have $\alpha$ is compact  with respect to $\Theta$ if and only if $\psi(\alpha)$ is compact with respect to $\tilde{\Psi}(\Theta)$.
If $\alpha$ is real, we have $\alpha$ satisfies the parity condition  with respect to $\Theta$ if and only if $\psi(\alpha)$ satisfies the parity condition with respect to $\tilde{\Psi}(\Theta)$.

(C6) \,\,\,\, $\tilde{\Psi}$ is compatible with the cross actions.
Namely, for $w\in W_{\spl\lambda}$ and $\Theta\in {\bf St}_G(\spl\lambda)$ we have $\psi_\sharp(w)\times\tilde{\Psi}(\Theta)=\tilde{\Psi}(w\times \Theta)$.

(C7) \,\,\,\, $\tilde{\Psi}$ is compatible with the Cayley transform.
Namely, if $\Theta\in{\bf St}_G(\spl\lambda)$ and if $\alpha\in\Delta_{\spl\lambda}$ is noncompact imaginary with respect to $\Theta$,
then we have $\tilde{\Psi}(c_\alpha(\Theta))=c_{\psi(\alpha)}(\tilde{\Psi}(\Theta))$.
Moreover, if $\Theta\in{\bf St}_G(\spl\lambda)$ and if $\alpha\in\Delta_{\spl\lambda}$ is real and satisfies the parity condition with respect to $\Theta$, we have $\tilde{\Psi}(c^\alpha_\pm(\Theta))=c^{\psi(\alpha)}_\pm(\tilde{\Psi}(\Theta))$.

Put $\Lambda^\prime=\psi(\spl\lambda)+\pca_{G^\prime}$.
Since ${\bf St}_G(\spl\lambda)$ (resp.\ ${\bf St}_{G^\prime}(\psi(\spl\lambda))$) forms a basis of $\cca(\Lambda)$ (resp.\ $\cca(\Lambda^\prime)$), $\tilde{\Psi}$ in (C3) extends to a linear (resp.\ $\cpx[q]$-module) isomorphism of $\cca(\Lambda)$ (resp.\ $\cca(\Lambda)_q$) onto $\cca(\Lambda^\prime)$ (resp.\ $\cca(\Lambda^\prime)$).
We denote these isomorphisms of complex vector spaces and $\cpx[q]$-modules by the same letter $\tilde{\Psi}$.
If we identify $W_{\spl\lambda}$ and $W^\prime$ via the isomorphism $\psi_\sharp$ in(C1) above, we can regard $\cca(\Lambda^\prime)$ (resp.\ $\cca(\Lambda^\prime)_q$) as a $W_{\spl\lambda}$-representation (resp.\ a $H(W_{\spl\lambda})$-module).

Examining the definition of the Hecke algebra module structures ([32] p239), we easily see the conditions (C4)-(C7) imply $\tilde{\Psi}$ is $H(W_{\spl\lambda})$-module isomorphism of $\cca(\Lambda)_q$ onto $\cca(\Lambda^\prime)_q$.
From Theorem 4.1.1, we also see $\tilde{\Psi} : \cca(\Lambda)\rightarrow\cca(\Lambda^\prime)$ is an isomorphism between coherent continuation representations.

From Theorem 4.1.2 (the Kazhdan-Lusztig type algorithm for Harish-Chandra modules), we see
$\overline{\tilde{\Psi}(\Theta)}=\tilde{\Psi}(\bar{\Theta})$ for all $\Theta\in {\bf St}_G(\spl\lambda)$.
Here, $\Theta\leadsto\bar{\Theta}$ is a bijection of ${\bf St}_G(\spl\lambda)$ (resp.\ ${\bf St}_{G^\prime}(\psi(\spl\lambda))$) onto ${\bf Irr}_G(\spl\lambda)$ (resp.\ ${\bf Irr}_{G^\prime}(\psi(\spl\lambda))$).

We have:
\begin{lem} 
In the setting above, let $\eta\in\Lambda$ and let $\Xi\in\cca(\Lambda)$.
Assume that there exists an irreducible Harish-Chandra $(\gggg^\prime, K^\prime)$-module $V^\prime$ such that $\tilde{\Psi}(\Xi)(\psi(\eta))=[V^\prime]$. 
Then, there is some irreducible Harish-Chandra $(\gggg,K)$-module $V$ with the infinitesimal character $\eta$ such that $\Xi(\eta)=[V]$.
\end{lem}
{\it Proof} \,\,\,\, 
There is some $w\in W_{\spl\lambda}$ such that $\langle \alpha, w\eta\rangle \geq 0$ for all $\alpha\in\Delta^+_{\spl\lambda}$.
We write $w\Xi=\sum_{\bar{\Theta}\in{\bf Irr}_G(\spl\lambda)}c_{\bar{\Theta}}\bar{\Theta}.$
Since $\tilde{\Psi}(\Xi)(\psi(\eta))=\tilde{\Psi}(w\Xi)(\psi(w\eta))$, we have
$[V^\prime]=\sum_{\bar{\Theta}\in{\bf Irr}_G(\spl\lambda)}c_{\bar{\Theta}}\tilde{\Psi}(\bar{\Theta})(\psi(w\eta))$.
It is known that there is a unique $\bar{\Upsilon}_0\in{\bf Irr}_{G^\prime}(\spl\lambda)$ such that $\bar{\Upsilon}_0(\psi(w\eta))=[V^\prime]$ (cf.\ [29] Theorem 7.2.7).
Put $\bar{\Theta}_0=\tilde{\Psi}^{-1}(\bar{\Upsilon})$.
For any $\bar{\Theta}\in{\bf Irr}_{G^\prime}(\spl\lambda)$ either $\bar{\Theta}(\psi(w\eta))=0$ or $\bar{\Theta}(\psi(w\eta))=[X]$ for some irreducible Harish-Chandra module $X$ (cf.\ [32] Theorem 7.6).
Hence, we have $c_{\bar{\Theta}_0}=1$ and if $c_{\bar{\Theta}}\neq 0$ and $\bar{\Theta}\neq\bar{\Theta}_0$ then $\tilde{\Psi}(\bar{\Theta})(\psi(w\eta))=0$.
From [32] Theorem 7.6 (also see [32] Definition 5.3), the above (C1)-(C7) imply that $\tilde{\Psi}(\bar{\Theta})(\psi(w\eta))=0$ if and only if $\bar{\Theta}(w\eta)=0$ for all $\bar{\Theta}\in{\bf Irr}_G(\spl\lambda)$.
Hence, we have $\bar{\Theta}(w\eta)=0$ if $c_{\bar{\Theta}}\neq 0$ and 
 $\bar{\Theta}\neq\bar{\Theta}_0$.
Moreover, there is an irreducible Harish-Chandra $(\gggg, K)$-module $V$ such that $\bar{\Theta}_0(w\lambda)=[V]$.
Therefore $\Xi(\eta)=(w\Xi)(w\eta)=\sum_{\bar{\Theta}\in{\bf Irr}_G(\spl\lambda)}c_{\bar{\Theta}}\bar{\Theta}(\psi(w\eta))=\bar{\Theta}_0(w\eta)=[V]$.
\,\,\,\,\, Q.E.D.


\subsection{Standard parabolic subgroups}

In this section, let $G$ be either $\Spp(n-q,q)$ with $2q\leqslant n$ or $\SO^\ast(2n)$.
Fix  $\theta$, $\sH$, etc. as in 3.1.

We also fix some particular orthonormal basis $e_1,...,e_n$ of
$\shh^\ast$, as in 3.1.
We fix a simple system $\Pi$ of $\Delta(\gggg,\shh)$ as in 3.1.

Let $\kappa=(k_1,...,k_s)$ be a finite sequence of positive integers
such that 
\[
 k_1+\cdots+k_s\leqslant \left\{ \begin{array}{ll} q & \,\,\,\, \mbox{if $G=\Spp(p,q)$},
\\
\frac{n}{2} & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}.
\end{array}
\right. .
\]
We put $k^\ast_i= k_1+\cdots k_i$ for $1\leqslant i\leqslant s$ and $k^\ast_0=0$.
If $G=\Spp(p,q)$, put  $p^\prime=p-k^\ast_s$ and
$q^\prime=q-k^\ast_s$.
If $G=\SO^\ast(2n)$, put $r=n-2k^\ast_s$.
We put
$A_i=\sum_{j=1}^{2k_i}E_{k^\ast_{i-1}+j} \,\,\,\,\, (1\leqslant
 i\leqslant s)$,
Then we have $\theta(A_i)=-A_i$ for $1\leqslant
 i\leqslant s$.
We denote by $\aaa_\kappa$  the Lie subalgebra of
$\shh$ spanned by $\{A_i\mid 1\leqslant
 i\leqslant s\}$.
We define a subset $S(\kappa)$ of $\Pi$ as follows.
If $G=\Spp(p,q)$, we define
\[ S(\kappa)=\left\{\begin{array}{ll} 
\Pi-\{e_{2k^\ast_i}-e_{2k^\ast_i+1}| 1\leqslant i\leqslant s\} & \mbox{if $p^\prime>0$},\\
\Pi-(\{e_{2k^\ast_i}-e_{2k^\ast_i+1}| 1\leqslant i\leqslant s-1\}\cup \{2e_{n}\}) & \mbox{if $p^\prime=0$}
\end{array}\right. .\]
If $G=\SO^\ast(2n)$, we define
\[ S(\kappa)=\left\{\begin{array}{ll} 
\Pi-\{e_{2k^\ast_i}-e_{2k^\ast_i+1}| 1\leqslant i\leqslant s\} & \mbox{if $r>0$},\\
\Pi-(\{e_{2k^\ast_i}-e_{2k^\ast_i+1}| 1\leqslant i\leqslant s-1\}\cup \{e_{n-1}+e_{n}\}) & \mbox{if $r=0$}
\end{array}\right. .\]
We denote by $M_\kappa$ (resp.\ $\mmm_\kappa$)  the centralizer of
$\aaa_\kappa$ in $G$ (resp.\ $\gggg$).
$M_\kappa$ is a Levi subgroup of $G$.
Let $P_\kappa$ be a parabolic subgroup of $G$ whose
$\theta$-invariant Levi part is $M_\kappa$.
We denote by $N_\kappa$ the nilradical of $P_\kappa$.
We denote by $\ppp_\kappa$, $\mmm_\kappa$, and $\nnn_\kappa$ the complexified Lie
algebra of $P_\kappa$, $M_\kappa$, and $N_\kappa$, respectively.
We choose $P_\kappa$ so that $\{\alpha\in\Delta\mid
\gggg_\alpha\subseteq\nnn_\kappa\}\subseteq\Delta^+$. 
Formally, we denote by $\Spp(0,0)$ and $\SO^\ast(0)$ the trivial group $\{1\}$ and we denote by $\GL(\kappa,\qtr)$ a product group $\GL(k_1,\qtr)\times\cdots\times \GL(k_s,\qtr)$.
Then, we have
\[ M_\kappa\cong 
\left\{\begin{array}{ll} 
\GL(\kappa,\qrt)\times \Spp(p^\prime,q^\prime) & \,\,\,\, \mbox{if $G=\Spp(p,q)$}\\
\GL(\kappa,\qrt)\times \SO^\ast(2r) & \,\,\,\, \mbox{if $G=\SO^\ast(2n)$}
\end{array}\right. .\]
Often, we identify $\GL(\kappa,\qrt)$, $\Spp(p^\prime,q^\prime)$, $\SO^\ast(2r)$ with subgroups of $M_\kappa$ in obvious ways.
We call such identifications the standard identifications.
The Cartan involution $\theta$ induces  Cartan involutions on $M_\kappa$, $\GL(\kappa,\qtr)$, $\Spp(p^\prime, q^\prime)$, and $SO^\ast(2r)$ and we denote them by the same letter $\theta$.
We put $M^\circ_\kappa=\Spp(p^\prime,q^\prime)$ if $G=\Spp(p,q)$ and put
$M^\circ_\kappa=\SO^\ast(2r)$ if $G=\SO^\ast(2n)$.
We denote by $\mmm^\circ_\kappa$ the complexified Lie
algebra of $M^\circ_\kappa$.

For $\tau\in{\mathfrak S}_s$ and $\kappa=(k_1,...,k_s)$, we define
$\kappa^\tau=(k_{\tau(1)},..., k_{\tau(s)})$.
Let $\xi$ be an irreducible unitary representation of $M_\kappa$.
$\xi$ can be written as $\xi=\xi_1\boxtimes
\cdots\boxtimes\xi_s\boxtimes\xi_0$, where for $1\leqslant i\leqslant s
$ (resp.\ for $i=0$) $\xi_i$ is an irreducible unitary representation of
$\GL(k_i,\qrt)$ (resp.\ $M^\circ_\kappa $).
For $\tau\in {\mathfrak S}_s$,  we denote by $\xi^\tau$ an irreducible
unitary representation of $L_{\kappa^\tau}$,
$\xi_{\tau(1)}\boxtimes\cdots\boxtimes\xi_{\tau(s)}\boxtimes\xi_0$.
The following is a special case of a well-known result of Harish-Chandra.
\begin{lem}
\mbox{\rm (Harish-Chandra)}

Let $\kappa=(k_1,...,k_s)$ and $\tau\in{\mathfrak S}_s$ be as above.
Let $\xi$ be an irreducible unitary representation of $M_\kappa$.
Then we have
$\Ind_{P_\kappa}^G(\xi)\cong\Ind_{P_{\kappa^\tau}}^G(\xi^\tau)$.
\end{lem}

Let $A_k(\ell,t)$ \,\,\,\,$(\ell\in\{\ell^\prime\in\itg\mid \ell^\prime\geqslant
-k\}\cup\{-\infty\})$  be the representation of $\GL(n, \qrt)$ defined in Definition
2.4.3.
If $\ell\geqslant -k$, $A_k(\ell,t)$ is a quaternionic Speh
representation in the weakly fair range.
$A_k(-\infty,t)$ is a unitary one-dimensional representation.

Any derived functor module  is a parabolic induction from an external
tensor product of some $A_k(\ell,t)$'s.
So, the unitarily induced module from a derived functor module (in
weakly fair range) can be
written as:
\begin{equation*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z). \tag{$\circledast$}
\end{equation*}

Here, $Z$ is a derived functor module of $M_\kappa^\circ$ in the weakly
fair range.
Moreover,  $\ell_i\in \{\ell\in\itg\mid \ell\geqslant
-k_i\}\cup\{-\infty\}$, and $t_i\in\sqrt{-1}\rel$ for $1\leqslant i\leqslant s$.
Using well-known Harish-Chandra's result,we may assume
$\sqrt{-1}t_i\geqslant 0$ for all $1\leqslant i\leqslant s$.

We assume that $\ell_i+1\in 2\itg$ and $t_i=0$ for some $1\leqslant
i\leqslant s$.
Then, using Lemma 4.2.1, we may assume $i=s$.
Let $\kappa^\prime=(k_1,...,k_{s-1})$.
Then from the induction-by-stage, we have
\begin{multline*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z)
\\
\cong
\Ind_{P_\kappa^\prime}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes A_{k_{s-1}}(\ell_{s-1},t_{s-1})
 \boxtimes\Ind_{P_{(k_s)}}^{M_{\kappa^\prime}^\circ}(A_{k_s}(\ell_s,0)\boxtimes Z)).
\end{multline*}

Applying the decomposition formula Theorem 3.7.5, we see that the above
induced module is a direct sum of the induced modules of the form like
\[
 \Ind_{P_\kappa^\prime}^G(A_{k_1}(\ell_1,t_1)\boxtimes\cdots\boxtimes A_{k_{s-1}}(\ell_{s-1},t_{s-1})
 \boxtimes Z^\prime)).
\]
Here, $Z^\prime$ is a derived functor module of $M_{\kappa^\prime}^\circ$ in the weakly
fair range.
Assume that we understand the reducibility of $Z^\prime$'s.
Then, applying the above argument, we can reduce the irreducible
decomposition of the above $\circledast$ to the following.
\begin{equation*}
 \Ind_{P_\kappa}^G(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)\boxtimes
A_{k_{h+1}}(\ell_{h+1},t_{h+1})\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z). \tag{$\lozenge$}
\end{equation*}
Here, $\ell_i$ is not odd integer if $1\leqslant i\leqslant h$,
$\sqrt{-1}t_i>0$ if $h<i\leqslant s$, and $Z$ is an irreducible
representation of  $M_\kappa^\circ$ whose infinitesimal character is in
$\pca_{M^\circ_\kappa}$.
Put $\tau=(k_1,...,k_h)$ and $\tau^\prime=(k_{h+1},...,k_s)$.
Also put $a=k_1+\cdots+k_h$ and  $b=k_{h+1}+\cdots+k_s$.

We state the main result of \S 4.
\begin{thm}
The following is equivalent.

(1) \,\,\,\,
The above $\lozenge$ is irreducible.

(2) \,\,\,\, The following induced module $\divideontimes$ is irreducible.
\begin{equation*} 
\Ind_{P_{\tau}}^{\SO^\ast(4a)}(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)). \tag{$\divideontimes$}
\end{equation*}
\end{thm}

{\it Remark}  \,\,\,\,\,
Under an appropriate regularity condition on $\ell_1,...,\ell_h$, we may
apply Proposition 3.3.2 to $\divideontimes$ successively, and we obtain
that $\divideontimes$ is a good-range elliptic cohomological induction
from an irreducible module like $I_{k_1}(\ell^\prime_1,0)\boxtimes\cdots\boxtimes
I_{k_h}(\ell^\prime_h,0)$.
Hence $\divideontimes$ is irreducible for such parameters.

In \S 5, we show $\divideontimes$ is irreducible if $\ell_1,...,\ell_h$
are all $-\infty$.

\subsection{Proof of Theorem 4.2.2}

We denote by $\shh_\kappa$ (resp.\ $\shh^\kappa$) the $\cpx$-linear span of $E_1,..., E_{2k^\ast_s}$ (resp.\ $E_{2k^\ast_s+1},..., E_n$).
Then, we can regard $\shh_\kappa$ (resp.\ $\shh^\kappa$) as the complexified Lie algebra of a $\theta$-invariant maximally split Cartan subgroup of $\GL(\kappa,\qtr)$ (resp.\ $\Spp(p^\prime,q^\prime)$ or $\SO^\ast(2r)$) via the standard identification.
We have a direct sum decomposition $\shh=\shh_\kappa\oplus\shh^\kappa$ and it induces $\shh^\ast=\shh^\ast_\kappa\oplus(\shh^\kappa)^\ast$.
Namely, we identify $\shh_\kappa^\ast$ (resp.\ $\shh_r^\ast$ ) the $\cpx$-linear span of $e_1,...,e_{2k^\ast_s}$ (resp.\ $e_{2k^\ast_s+1},...,e_n$).

We denote by $\rho$ the half sum of the roots in $\Delta^+$.
Let $\eta\in\shh^\ast$ be the infinitesimal character of   $A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)\boxtimes
A_{k_{h+1}}(\ell_{h+1},t_{h+1})\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z$.
We may (and do) assume $\Real(\eta)$ is in the closed Weyl chamber with
respect to $\Delta^+\cap\Delta(\mmm_\kappa,\shh)$.

We fix a sufficiently large integer $N$ and we put $\spl\lambda=2N\rho+\eta$ and $\Lambda=\spl\lambda+\pca_G$.
Then, we have $\eta\in\Lambda$.
Hence, $\Delta_\eta=\Delta_{\spl\lambda}$.
Moreover, we have $\spl\lambda$ is regular and $\Delta_{\spl\lambda}^+=\Delta^+\cap\Delta_{\spl\lambda}$.

We construct a surgroup $G^\prime$ of $M_\kappa$ as follows.
As a Lie group, $G^\prime$ is a product group $\SO^\ast(4a)\times\GL(b,\qtr)\times M^\circ_\kappa$.
The embedding of $M_\kappa=\GL(\tau,\qrt)\times\GL(\tau^\prime,\qrt)\times
M^\circ_\kappa$ into $G^\prime$ is induced from the inclusions
$\GL(\tau,\qrt)\subseteq\SO^\ast(4a)$ and $\GL(\tau^\prime,\qrt)\subseteq\GL(b,\qtr)$.
Easily see that we may fix a Cartan involution whose restriction to $M_\kappa$ is $\theta$.
We denote such a Cartan involution on $G^\prime$ by the same letter $\theta$ and denote by $K^\prime$ the corresponding maximal compact subgroup.
Since $M_\kappa$ is a Levi subgroup of both $G$ and $G^\prime$, $\sH$ is
a $\theta$-stable maximally split Cartan subgroup of $G^\prime$ as well
as $G$.

We denote by $\gggg^\prime$ the complexified Lie algebra of $G^\prime$ and denote by $\Delta^\prime$ the root system for $(\gggg^\prime, \shh)$.
From the construction of $G^\prime$, we have the integral root system $\Delta_{\spl\lambda}$ coincides with $\Delta^\prime_{\spl\lambda}$.

We want to apply Lemma 4.1.3 to $G$, $G^\prime$, and $\spl\lambda$ above.
In our setting, we put $\sH^\prime=\sH$ and $\spl\hhh^\prime=\spl\hhh$ and
put $\psi$ in (C1) to be the identity map. 
Hereafter, we denote by $G^\sharp$ any of $G$ and $G^\prime$. 
Similarly, we write $K^\sharp$, etc.

In order to define $\Psi$ and $\tilde{\Psi}$, we describe conjugacy classes of Cartan subgroups in $G$ and $G^\prime$.

First, we remark that there is one to one correspondence between $G^\sharp$-conjugacy classes of Cartan subgroups in $G^\sharp$ and $K^\sharp$-conjugacy classes of $\theta$-stable Cartan subgroups in $G^\sharp$ ([20]).
Second, a $G$-conjugacy class of Cartan subgroups of $G$ is determined by the dimension of the split part
and $GL(k,\qtr)$ has a unique $G$-conjugacy class of Cartan subgroups (cf.\ [27]).
Hence, we see a $K$-conjugacy class (resp.\ a $K^\prime$-conjugacy class) of $\theta$-stable (resp.\ $\theta^\prime$-stable) Cartan subgroups of $G$ (resp.\ $G^\prime$) is determined by the dimension of the split part.
We also see the same statement holds for $M_\kappa$.

Since there is obvious one to one correspondence between the conjugacy classes of Cartan subgroups and the conjugacy classes of the Cartan subgroups which is stable with respect to the complex conjugation, hereafter we consider Cartan subalgebras rather than Cartan subgroups.
In order to understand the Cayley transforms on Cartan subalgebras, we examine some particular Cartan subalgebras as follows.
Let $m$ be the greatest positive integer which is equal to or less than $\frac{h}{2}$.
For $1\leq i\leq m$, we put $\alpha_i=e_{2i-1}+e_{2i}$.
Then, $\{\alpha_1,....,\alpha_m\}$ is the entire collection of real roots in $\Delta^+$.
We define $c^{\alpha_i}\in G_\cpx$ as in 4.1.
Since $\alpha_1,...,\alpha_m$ are mutually orthogonal, we may regard $\alpha_i$ as a real root for $\Ad(c^{\alpha_j})(\spl\hhh)$.
So, we can regard $\Ad(c^{\alpha_i})(\Ad(c^{\alpha_j})(\spl\hhh)$ as a result of successive applications of Cayley transforms to $\spl\hhh$.
Because of the orthogonality of $\alpha_i$ and $\alpha_j$, we see $\Ad(c^{\alpha_i})(\Ad(c^{\alpha_j})(\spl\hhh))=\Ad(c^{\alpha_j})(\Ad(c^{\alpha^i})(\spl\hhh))$.

Let $J=\{\alpha_{r_1},....,\alpha_{r_k}\}
\subseteq \{\alpha_1,...,\alpha_m\}$.
Here, we assume $r_i\neq r_j$ for $i\neq j$.
Similarly as above. we can define successive applications of Cayley transforms as follows.
\[ \hhh_J=\Ad(c^{\alpha_{r_k}})(\Ad(c^{\alpha_{r_{k-1}}})(\cdots(\Ad(c^{\alpha_{r_1}})(\spl\hhh)\cdots)).\]
$\hhh_J$ only depends on $J$ and it is $\sigma$ and $\theta$-stable.
We denote by $H_J$ the corresponding Cartan subgroup of $G$ to $\hhh_J$.

Put $J_0= \{\alpha_{k^\ast_s+1},....,\alpha_m\}$.
If $J\subseteq J_0$, then $H_J\subseteq M_\kappa$ and $H_J$ is a $\theta$-stable Cartan subgroup of $M_\kappa$.

Since a $K^\sharp$-conjugacy (resp.\ $K\cap M_\kappa$-conjugacy) class
of $\theta$-stable Cartan subgroups of $G^\sharp$ (resp.\ $M_\kappa$) is
determined by the dimension of the split part, for $J_1,J_2\subseteq
J_0$, the following statements are equivalent.

(1) \,\, $H_{J_1}$ is $K$-conjugate to $H_{J_2}$.

(2) \,\, $H_{J_1}$ is $K^\prime$-conjugate to $H_{J_2}$.
 
(3) \,\, $H_{J_1}$ is $K\cap M_\kappa$-conjugate to $H_{J_2}$.

(4) \,\, $\card J_1=\card J_2$.

If $J\subseteq J_0$, $H_J$ is $\spl\lambda$-integral with respect to both $G$ and $G^\prime$.
Conversely, it is easy to check any $\spl\lambda$-integral $\theta$-stable Cartan subgroup of $G^\sharp$  is $K^\sharp$-conjugate to $H_J$ for some $J\subseteq J_0$. 
(For example, using a criterion for the parity condition [29], we may check  $\alpha_i$ satisfies the parity condition with respect to $\spl\lambda$ if and only if $\alpha_i\in J_0$.
The statement is deduced from this fact.)
We also remark that any $\theta$-stable Cartan subgroup of $M_\kappa$ is $K\cap M_\kappa$-conjugate to some $H_J$ with $J\subseteq J_0$.
Hence, there is a bijection $\Phi$ (resp.\ $\Phi^\prime$) of the set of the $K\cap M_\kappa$-conjugacy classes of $\theta$-stable Cartan subgroups of $M_\kappa$ to the set of $K$-conjugacy (resp.\ $K^\prime$-conjugacy) classes of $\spl\lambda$-integrable $\theta$-stable Cartan subgroups of $G$ (resp.\ $G^\prime$).
In fact $\Phi$ (resp.\ $\Phi^\prime$) is defined 
such that the image of the $K$-conjugacy class of $H_J$ under $\Psi$ is the $K^\prime$-conjugacy class of $H_J$ for any $J\subseteq J_0$.
We put $\Psi=\Phi^\prime\circ\Phi^{-1}$.
$\Psi$ is a bijection of the set of the $K$-stable conjugacy classes of $\spl\lambda$-integral $\theta$-stable Cartan subgroups of $G$ to the set of the $K^\prime$-conjugacy classes of $\spl\lambda$-integral $\theta^\prime$-stable Cartan subgroups of $G^\prime$.
$\Psi$ is compatible with Cayley transforms on (conjugacy classes of) Cartan subgroups, since $\Phi$ and $\Phi^\prime$ are.

Next, we consider the lift of $\Psi$ to the standard coherent families.

We put $J(i)=\{\alpha_{m},\alpha_{m-1},...,\alpha_{m-i+1}\}$ for $1\leq i\leq m-k^\ast_s$ and $J(0)=\emptyset$.
Put $H_i=H_{J(i)}$ for $0\leq i\leq m-k^\ast_s$.
Then, we easily see $H_1,....,H_{m-k^\ast_s}$ form a complete system of representatives  of the $K^\sharp$-conjugacy (resp.\ $K\cap M_\kappa$-conjugacy) classes of $\theta$-stable Cartan subgroups of $G^\sharp$ (resp.\ $M_\kappa$).
We denote by $\hhh_i$ the complexified Lie algebra of $H_i$ and by $W(\gggg^\sharp, \hhh_i)$ the Weyl group for $(\gggg^\sharp,\hhh_i)$.
We denote by $W(G^\sharp; H_i)$ the subgroup of $W(\gggg^\sharp,\hhh_i)$ consisting the elements of $W(\gggg^\sharp,\hhh_i)$ whose representatives can be chosen in $G^\sharp$.
We collect some of the useful facts:

\begin{lem} \,\,\,\,
For $1\leq i\leq m-k^\ast_s$, we have

(1) \,\,\, $W(\mmm_\kappa,\hhh_i)\subseteq W(\gggg^\prime,\hhh_i)\subseteq W(\gggg,\hhh_i)$,

(2) \,\,\, $W(G^\prime ; H_i)=W(\gggg^\prime,\hhh_i)\cap W(G; H_i)$,

(3) \,\,\, $R_{M_\kappa}(H_i,\spl\lambda)\subseteq R_{G^\prime}(H_i,\spl\lambda)\subseteq R_G(H_i,\spl\lambda)$.
\end{lem}


\medskip

(1) is easy to see from our construction of $G^\prime$.   (2) is easily checked using [30] Proposition 4.16.
(3) follows from (1).

\medskip

We define $\tilde{\Omega} : {\bf St}_{G^\prime}(\spl\lambda)\rightarrow {\bf St}_{G}(\spl\lambda)$ by $\tilde{\Omega}(\Theta^{G^\prime}_\gamma)=\Theta^{G}_\gamma$ for $\gamma\in R_{G^\prime}(H_i, \spl\lambda)$ for $1\leq i\leq m-k^\ast_s$.

We have remarked in section 1 that for $\gamma_1=(H_i,\lambda_1),
\gamma_2=(H_i,\lambda_2) \in R_{G^\sharp}(H_i,\spl\lambda)$, the
following statements are equivalent:

(a) \,\,\, $\gamma_1$ and $\gamma_2$ are $K^\sharp$-conjugate.

(b) \,\,\, There is some $w\in W(G^\sharp;H_i)$ such that $\lambda_1=w\lambda_2$.

(c) \,\,\, $\Theta^{G^\sharp}_{\gamma_1}=\Theta^{G^\sharp}_{\gamma_2}$.

Hence, from (2) and (3) of lemma 4.2.1, we see $\tilde{\Omega}$ is well-defined.

We have:
\begin{lem}\,\,\,\,
$\tilde{\Omega}$ is bijective.
\end{lem}
{\it Proof} \,\,\,\,
From lemma 4.3.1 (2) and the above remark, we see that the regularity of $\spl\lambda$ implies the injectivity of $\tilde{\Omega}$.
So, we show the surjectivity.

First, we fix some $1\leq i\leq m-k^\ast_s$.
Then $\Ad(c^{\alpha_{k^\ast_s+i}})\circ\Ad(c^{\alpha_{k^\ast_s+i-1}})\circ\cdots\circ\Ad(c^{\alpha_{k^\ast_s+1}})$ induces an linear isomorphism of $\hhh$ onto $\hhh_i$.
So, we also have an isomorphism $\hhh^\ast\cong\hhh_i^\ast$.
We denote by $\bar{e}_1,...,\bar{e}_n\in\hhh_i^\ast$ the image of $e_1,...,e_n\in\hhh^\ast$ under this isomorphism.
Then the Cartan involution acts on $\bar{e}_1,...,\bar{e}_n$ as follows.
\begin{gather*}
\theta(\bar{e}_{2i-1})=-\bar{e}_{2i}, \theta(\bar{e}_{2i})=-\bar{e}_{2i-1} \,\,\,\,\, (1\leq i\leq m-i),\\
\theta(\bar{e}_i)=\bar{e}_i \,\,\,\,\, (2(m-i)< i\leq n).
\end{gather*}
We also denote by $\lambda\in\hhh^\ast_i$ the image of $\spl\lambda$  under this isomorphism.
Write $\lambda=\sum_{j=1}^n\ell_j\bar{e}_j$.
Let $w\in W(\gggg,\hhh_i)$ and write 
$\lambda=\sum_{j=1}^n\bar{\ell}_j\bar{e}_j$.
Then $\bar{\ell}_1,...,\bar{\ell}_n$ is made from $\ell_1,...,\ell_n$ by a permutation of their indices and sign flips.
We assume that $\gamma_w=(H_i,w\lambda)\in R_G(\spl\lambda)$.
Then, it should satisfy the condition (R5) in 1.3.
So, we easily see:

(d1) \,\,\, $\bar{\ell}_j\in\itg \,\,\,\, (2(m-i)<j\leq n)$, 

(d2) \,\,\, $\bar{\ell}_{2j-1}-\bar{\ell}_{2j}\in\itg \,\,\,\, (1<j\leq m-i)$. 

We write $\sum_{k=1}^na_i\bar{e}_i\in\hhh_i^\ast$ by $(a_1,...,a_n)$.

From [30] Proposition 4.86, we easily see the following elements in $W(\gggg,\hhh_i)$ are contained in $W(G;H_i)$. 
\begin{gather*}
w_j(a_1,....,a_n)=(a_1,..,a_{2j-2},-a_{2j-1},-a_{2j},a_{2j+2},...,a_n)
 \,\,\,\, (1\leq j\leq m-i), \\
w_{b,c}(a_1,...,a_n)=(a_1,...,a_{2b-2},a_{2c-1},a_{2c},a_{2b+1},...,a_{2c-2},a_{2b-1},a_{2b},a_{2c+1},...,a_n)
 \,\,\, (1\leq b<c\leq m-i).
\end{gather*}
If we choose the product $w^\ast$ of suitable $w_j$'s and $w_{b,c}$'s above, we may have
$w^\ast\lambda=(d_1,...,d_n)$ satisfies:

(e1) \,\,\, $d_j\in\itg$ for all $k^\ast_s<j\leqslant  n$.

(e2) \,\,\, $d_j\not\in\rel$ for all $k^\ast_{h}<j\leqslant k^\ast_s$.

(e3) \,\,\, $d_j-\frac{1}{2}\in\itg$ for all $1\leqslant j\leqslant k^\ast_h$.

This means that $\gamma^\prime=(H_i,w^\ast w\lambda)\in R_{G^\prime}(\spl\lambda)$ and $\Theta^G_{\gamma_w}=\Theta^G_{\gamma^\prime}$.
Hence $\tilde{\Omega}$ is surjective.
\,\,\,\,\,\, $\Box$

We define $\tilde{\Psi} : {\bf St}_{G^\prime}(\spl\lambda)\rightarrow {\bf St}_G(\spl\lambda)$ by the inverse of $\tilde{\Omega}$.
From the above constructions, we easily see:
\begin{lem} \,\,\,\,
$\psi$, $\Psi$, and $\tilde{\Psi}$ defined above satisfy (C1)-(C7) in
 4.1.
\end{lem}
Now, we finish the proof of Theorem 4.2.2.
If $\gamma\in R_{M_\kappa}(\spl\lambda)$, then, taking account of $\gamma\in R_{G^\sharp}(\spl\lambda)$, we easily see the following:

(f1) \,\,\, $\Theta^{G^\prime}_\gamma=\Ind_{M_\kappa}^{G^\prime}(\Theta^{M_\kappa}_\gamma)$,

(f2) \,\,\, $\Theta^{G}_\gamma=\Ind_{M_\kappa}^G(\Theta^{M_\kappa}_\gamma)$,

(f3) \,\,\, $\tilde{\Psi}(\Theta^{G}_\gamma)=
\Theta^{G^\prime}_\gamma$.

Taking account of the additivity of induction, we see that for all $\bar{\Theta}\in {\bf Irr}_{M_\kappa}(\spl\lambda)$ we have $\tilde{\Psi}(\Ind_{M_\kappa}^G(\bar{\Theta}))=\Ind_{M_\kappa}^{G^\prime}(\bar{\Theta})$.
It is easy to see that there is some $\bar{\Theta}\in{\bf Irr}_{M_\kappa}(\spl\lambda)$ such that $\bar{\Theta}(\eta)=[A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0)\boxtimes
A_{k_{h+1}}(\ell_{h+1},t_{h+1})\boxtimes\cdots\boxtimes
 A_{k_s}(\ell_s,t_s)\boxtimes Z]$.
Hence, lemma 4.1.3 implies that the irreducibility of
$\lozenge$ is reduced to the irreducibility of a Harish-Chandra module which is the external product of the following:

(g1) \,\,\, $Z$, which is an irreducible Harish-Chandra module for $M^\circ_\kappa$,

(g2) \,\,\, $\Ind_{P_\tau}^{SO^\ast(4a)}(A_{k_1}(\ell_1,0)\boxtimes\cdots\boxtimes
  A_{k_h}(\ell_h,0))$,

(g3) \,\,\, Harish-Chandra modules for $\GL(b,\qtr)$  induced from irreducible unitary representations of their parabolic subgroups.

The irreducibilities of (g3) are found in [34] p502.
\,\,\,\, Q.E.D.


\bigskip

\setcounter{section}{5}
\section*{\S\,\, 5.\,\,\,\, Irreducibility representations
$SO^\ast(2n)$ and $\Spp(p,q)$  parabolically induced from one-dimensional unitary representations}
\setcounter{subsection}{0}

\subsection{{Some induced representations of $SO^\ast(4m)$}}

In this section we retain the notations in 3.1. and 4.2, and consider
the case of $G=\SO^\ast(2n)$.
 Moreover, we assume $n$ is even.  So, we write $n=2m$.

Since the universal covering group of $G_\cpx$ is a double cover, $\pca_G$ (cf.\ 1.2) is a subgroup of $\pca$ of index two.
Put $\Lambda=\pca-\pca_G$ (set theoretical difference).
$\Lambda$ is the other $\pca_G$ coset in $\pca$ than $\pca_G$ itself.
We fix a regular weight $\spl\lambda\in\Lambda$ as follows
\[\spl\lambda=\sum_{i=1}^n \frac{2n-2i+1}{2}e_i.\]
Hereafter, we simply write $W=W(\gggg,\shh)$ and $\Delta=\Delta(\gggg,\shh)$.
We have $W=W_{\spl\lambda}$, $\Delta=\Delta_{\spl\lambda}$, and
\[ \Pi_{\spl\lambda} =\{e_1-e_2,e_2-e_3, ....,e_{n-1}-e_{n}, e_{n-1}+e_n\}.\]
Let $\bbb$ be the Borel subalgebra of $\gggg$ such that $\shh\subseteq\bbb$ and the nilradical of $\bbb$ is the sum of the root spaces corresponding to the roots in $\Delta^+_{\spl\lambda}$.
We denote by $\rho$ the half sum of the positive roots in $\Delta^+_{\spl\lambda}$.

We consider a partition $\pi=(p_1,...,p_k)$ of a positive integer $m$ $(p_1,...,p_k)$ such that $0<p_1\leqslant p_2\leqslant \cdots\leqslant p_k$ and $p_1+p_2+\cdots+p_k=m$.
Let $PT(m)$ be the set of partitions of $m$.
As in 4.2, we consider the standard parabolic subgroup $P_\pi$ and its
Levi subgroup $M_\pi$ of $G$ corresponding to $\pi$.

Let $(\sigma^\pi_\lambda, \cpx^\pi_\lambda)$ be a one dimensional unitary representation of $M_\pi$ (or $\mmm_\pi$) such that the restriction to $\shh$ of the differential of $\sigma^\pi_\lambda$ is $\lambda\in \shh^\ast$.

We denote by $\rho_\pi$ the half sum of all the positive roots whose root space is in $\mmm_\pi$.
We put $\rho^\pi=\rho-\rho_\pi$.
The infinitesimal character of $\Ind_{P_\pi}^G(\cpx^\pi_\lambda)$ is $\rho_\pi+\lambda$.

It is easy to construct a nondegenerate $\gggg$-invariant pairing between $\Ind_{P_\pi}^G(\cpx^\pi_\lambda)$ and a generalized Verma module $M_{\ppp_\pi}(\lambda)=U(\gggg)\otimes_{U(\ppp_\pi)}\cpx^\pi_{-\lambda-\rho^\pi}$.


We are going to show the following our main result.

\begin{lem}
\,\,\,\,\,\, { Let $\pi$ be any partition of $m$.   Then, $Ind_{P_\pi}^G(\cpx^\pi_{0})$ is irreducible.}
\end{lem}

We prove this lemma in 5.3.

Combining Lemma 5.1.1 and Theorem 4.2.2, we have:

\begin{cor}
Representations of $SO^\ast(2n)$ and $\Spp(p,q)$ induced from
 one-dimensional unitary representations of their parabolic subgroups
 are irreducible.
\end{cor}

\subsection{{Coherent continuation representation for $SO^\ast(4m)$ with respect to $\Lambda$}}

We retain the notations in 5.1.

For a partition $\pi=(p_1,...,p_k)\in PT(m)$ of $m$, put
$p^\ast_i=\sum_{j=1}^i p_j$ for $1\leq i\leq k$ and define a subset
$S_\pi$ of $\Pi=\Pi_{\spl\lambda}$ as follows. ($P_\pi$ is the standard
parabolic subgroup corresponding to $S_\pi$.)
\[ S_\pi=\Pi -(\{e_{2p^\ast_i}-e_{2p^\ast_i+1})| 1\leq i\leq k-1\}\cup \{e_{2m-1}+e_{2m}\}.\]
For $\pi\in PT(m)$, we denote by $\sigma_\pi$ the MacDonald representation (cf.\ [3] p368) of $W$ with respect to $S_\pi\subseteq\Pi$.
From [19], $\sigma_\pi$ is a special
representation ([17], [18] also see [3] p374), which corresponds to the Richardson orbit in $\gggg$ with respect to the parabolic subalgebra $\ppp_\pi$ via the Springer correspondence.

There is another description of $\sigma_\pi$.
Since $W$ is the Weyl group of type $D_{2m}$, it is embedded into  the Weyl group $W^\prime$ of type $B_{2m}$.
It is well known that the irreducible representations of $W^\prime$ is parameterized by the pairs of partitions $(\kappa,\omega)$ such that $\kappa\in PT(k)$ and $\omega\in PT(2m-k)$ for some $0\leq k \leq 2m$.
Here, we regard $PT(0)$ consists of the empty partition $\emptyset$.
If $\kappa\neq\omega$, then the restriction of the representation corresponding to $(\kappa,\omega)$ is irreducible.
However, the restriction of the irreducible $W^\prime$-representation corresponding to $(\pi,\pi)$ \,\, ($\pi\in PT(m)$) to $W$ is decomposed into two irreducible $W$-representation, which are equidimensional.
From [3] p423 line 11-33, $\sigma_\pi$ is one of the irreducible constituent.

For each partition $\kappa\in PT(k)$, we denote by $\dim(\kappa)$ the dimension of the irreducible representation of ${\mathfrak S}_k$ corresponding to $\kappa$.
It is well-known that the dimension of the irreducible $W^\prime$-representation corresponding to $(\kappa,\omega)$ ($\kappa\in PT(k)$ and $\omega\in PT(2m-k)$) is $\frac{(2m)!\dim(\kappa)\dim(\omega)}{k!(2m-k)!}$.
(For example, see [10].)
So, we have :

\begin{lem} \,\,\,\, { For $\pi\in PT(m)$,}
\[\dim(\sigma_\pi)=\frac{(2m)!\dim(\pi)^2}{2(m!)^2}.\]
\end{lem}

We shall show:

\begin{thm} 
{ As a $W$-module the coherent continuation representation $\cca(\Lambda)$ is decomposed as follows.}
\[\cca(\Lambda)\cong\bigoplus_{\pi\in PT(m)}\sigma_\pi\]
\end{thm}

\medskip

First, we prove:

\begin{lem} 
{For each $\pi\in PT(m)$, the multiplicity of $\sigma_\pi$ in $\cca(\Lambda)$ is at least one.}
\end{lem}

{\it Proof} \,\,\,\,
We have only to show that there is an irreducible Harish-Chandra $(\gggg, K)$-module $V$ such that the infinitesimal character of $V$ is in $\Lambda$ and the character polynomial of $V$ ([11]) generates a $W$-representation isomorphic to $\sigma_\pi$.
First, we remark that $\Ind_{P_\pi}^G(\cpx_{\spl\lambda-\rho_\pi}^\pi)$ has a nondegenerate pairing with an irreducible  generalized Verma module $M_{\ppp_\pi}(-\spl\lambda-\rho_\pi)$ with the infinitesimal character $-\spl\lambda$.
Easily see that there is at least one irreducible constituent $V$ of $\Ind^G_{P_\pi}(\cpx_{\spl\lambda-\rho_\pi}^\pi)$ whose annihilator $I$ in $U(\gggg)$ is the dual of the annihilator of the generalized Verma module.
So the associated variety of $I$ is the closure of the Richardson orbit (say $\hol_\pi$) corresponding to $\ppp_\pi$.
The character polynomial with respect to $V$ is proportional to the Goldie rank polynomial of $I$ ([11]) and the $W$-representation generated by the Goldie rank polynomial is $\hol_\pi$.
So, we the lemma. \,\,\,\,\, Q.E.D.


{\it Proof of Theorem 5.2.2} \,\,\,\,
From Lemma 5.2.3, it suffices to show that 
\[
 \dim\cca(\Lambda)=\sum_{\pi\in PT(m)}\dim(\sigma_\pi).
\]
From Lemma 5.2.1, the right hand side is 
$\sum_{\pi\in
PT(m)}\frac{(2m)!(\dim(\pi)^2}{2(m!)^2}=\frac{(2m)!}{2\cdot m!}$, since
we have
\[
 \sum_{\pi\in PT(m)}(\dim (\pi))^2=\card {\mathfrak S}_m=m!.
\]
So, we have to show $\dim\cca(\Lambda)=\frac{(2m)!}{2\cdot m!}$.

$\dim\cca(\Lambda)$ is clearly, the number of $K$-conjugacy classes in the regular characters in $R_G(\spl\lambda)$.
Since only maximally split Cartan subgroups are $\spl\lambda$-integral, each $K$-conjugacy class has a representative in $R_G(\sH, \spl\lambda)$.
We denote by $W(G;\sH)$ the subgroup of $W$ consisting the elements $w$ of $W$ such that some representative of $w$ in $G_\cpx$ is in $G$ (or equivalently in $K$) and normalizes $\sH$.
Examining elements in $K$ which preserves $\sH$, we easily see $\dim\cca(\Lambda)=\card (W/W(G;\sH))$.
From [12] (also see [30], Proposition 4.16), $W(G;\sH)$ is generated by the following elements in $W$:

(1) \,\,\,\, $s_{e_{2i-1}-e_{2i}}$ \,\,\, ($1\leq i \leq m$) : reflections with respect to compact imaginary roots in $\Delta=\Delta(\gggg,\shh)$.

(2) \,\,\,\, $s_{e_{2i-1}+e_{2i}}$ \,\,\, ($1\leq i\leq m$) : reflections with respect to real roots in $\Delta$.

(3) \,\,\,\, $s_{e_{2i-1}-e_{2j-1}}s_{e_{2i}-e_{2j}}$ \,\,\, ($1\leq i<j\leq m$) 

So, we can easily see $W(G;\sH)$ is isomorphic to ${\mathfrak S}_m \times ((\itg/2\itg)^m\times(\itg/2\itg)^m)$.

So, we have $\dim\cca(\Lambda)=\frac{\card W}{\card W(G;\sH)}=\frac{(2m)!\cdot 2^{2m-1}}{m!\cdot 2^m\cdot 2^m}=\frac{(2m)!}{2\cdot m!}$ as desired.
\,\,\,\,\,\,\,\, Q.E.D.

We can interpret in terms of  cell structure of the coherent
continuation representation  $\cca(\Lambda)$.([2],
[30], [22])

A $W_{\spl\lambda}$-subrepresentation of $\cca(\Lambda)$ is called basal, if it is generated by a subset of ${\bf Irr}_G(\spl\lambda)$ as a $\cpx$-vector space.
For $\gamma\in R_G(\spl\lambda)$, we denote by $\mbox{Cone}(\gamma)$ the smallest basal subrepresentation of $\cca(\Lambda)$ which contains $\bar{\Theta}^G_\gamma$.
For $\gamma, \eta \in R_G(\spl\lambda)$, we write $\gamma\sim\eta$ (resp.\ $\gamma\leq\eta$) if $\mbox{Cone}(\gamma)=\mbox{Cone}(\eta)$ (resp.\ $\mbox{Cone}(\gamma)\supseteq\mbox{Cone}(\eta)$).
Obviously $\sim$ is an equivalence relation on $R_G(\spl\lambda)$.
For $\gamma\in R_G(\spl\lambda)$  let $s(\gamma)$ be the set of regular characters $\eta\in R_G(\spl\lambda)$ such that $\lambda\leq\eta$ and $\lambda\not\sim\eta$.
We define $\mbox{Cell}(\gamma)=\mbox{Cone}(\gamma)/\sum_{\eta\in s(\gamma)}\mbox{Cone}(\eta)$.
A cell (resp.\ cone) for $\cca(\Lambda)$ is a subquotient (resp.\ subrepresentation) of $\cca(\Lambda)$ of the form $\mbox{Cell}(\gamma)$ (resp.\ $\mbox{Cone}(\gamma)$) for some $\gamma\in R_G(\spl\lambda)$.

For each cell, we can associate a nilpotent orbit in $\gggg$ as follows.
For $\mbox{Cell}(\gamma)$, we consider an irreducible Harish-Chandra $(\gggg, K)$-module $\bar{\pi}(\gamma)$.  
The annihilator (say $I$) of $\bar{\pi}(\gamma)$ in $U(\gggg)$ is a primitive ideal of $U(\gggg)$ and its associated variety is the closure of a single nilpotent orbit in $\gggg$.
The nilpotent orbit constructed above is independent of the choice of $\gamma$ and we say it the associated nilpotent orbit for the cell $\mbox{Cell}(\gamma)$.
For each cone $\mbox{Cone}(\gamma)$, there is a canonical (up to scalar factor) $W_{\spl\lambda}$-homomorphism (say $\phi_{\gamma}$) of $\mbox{Cone}(\gamma)$ to the realization as a Goldie rank polynomial representation of the special $W_{\spl\lambda}$-representation corresponding to the associated nilpotent orbit via the Springer correspondence.
In fact this $\phi_\gamma$ factors to the cell $\mbox{Cell}(\gamma)$.
An important fact is $\phi_{\gamma}(\bar{\Theta}^G_\eta)$ is nonzero and
proportional to the Goldie rank polynomial of the annihilator of
$\bar{\pi}(\eta)$ in $U(\gggg)$ for all $\eta\sim\gamma$ ([11], [8]).
Hence, the multiplicity in $\mbox{Cell}(\gamma)$ of the special
$W$-representation corresponding to the associated nilpotent orbit via
the Springer correspondence is at least one.
McGovern proved that if $G$ is a classical group then the
multiplicity of the special representation is exactly one (cf.\ [22] Theorem 1).



In our particular setting, the proof of Lemma 5.2.3 tells us for each $\pi\in PT(m)$ there is at least one cell whose associated nilpotent orbit is $\hol_\pi$.

From Theorem 5.2.2, we have:
\begin{cor} \mbox{} 


(1) \,\,\,\, There is a one to one correspondence between the set of cells for $\cca(\Lambda)$ and $PT(m)$ induced from the above association of nilpotent orbits to cells.

(2) \,\,\,\, Each cell for $\cca(\Lambda)$ is irreducible and isomorphic to the special representation corresponding to the associated nilpotent orbit via the Springer correspondence.
\end{cor}

Harish-Chandra cells for classical groups are precisely studied by
McGovern ([22]). Almost all cases are treated in his paper.
For type D groups there are some exceptions (cf.\ p224 [22]).
Our result can be regarded as a supplement of his result.


From Corollary 5.2.4, we have:

 \begin{cor} 
{ Let $\lambda\in \Lambda$ and let $V_i$ ($i=1,2$) be irreducible Harish-Chandra $(\gggg, K)$-modules with an infinitesimal character $\lambda$.
Assume that the annihilator of $V_1$ in $U(\gggg)$ coincides with that of $V_2$.
Then, $V_1$ is isomorphic to $V_2$.}
\end{cor}

{\it Proof} \,\,\,\,
We may assume that $\langle\lambda,\alpha\rangle\geq 0$ for all $\alpha\in\Delta_{\spl\lambda}^+$.
It is known that for  each $i=1,2$ there is unique coherent family $\bar{\Theta}^G_{\gamma_i}\in {\bf Irr}_G(\spl\lambda)$ such that $[V_i]=\bar{\Theta}^G_{\gamma_i}(\lambda)$.
We show $\bar{\Theta}^G_{\gamma_1}=\bar{\Theta}^G_{\gamma_2}$.

First, we remark that the Goldie rank polynomial and the associated variety of the annihilator of $V_i$ in $U(\gggg)$ coincide with those of $\bar{\pi}(\gamma_i)$ for each $i$.
Hence, we have $\gamma_1\sim\gamma_2$ since there is at most one cell whose associated nilpotent orbit is  the unique dense orbit in the associated variety of $V_i$.
We consider the homomorphism $\phi_{\gamma_1} (=\phi_{\gamma_2})$ mentioned above.
Since $\phi_{\gamma_1}(\bar{\Theta}^G_{\gamma_i})$ is nonzero and proportional to the Goldie rank polynomial of the annihilator of $V_i$ in $U(\gggg)$ for each $i=1,2$, $\bar{\Theta}^G_{\gamma_1}$ is proportional to $\bar{\Theta}^G_{\gamma_2}$ modulo the kernel of $\phi_{\gamma_1}$.
Since the cell $\mbox{Cell}(\gamma_1)=\mbox{Cell}(\gamma_2)$ is irreducible, $\phi_{\gamma_1}$ induces an isomorphism of the cell $\mbox{Cell}(\gamma_1)$ to the corresponding Goldie rank polynomial representation.
This means that $\bar{\Theta}^G_{\gamma_1}$ is proportional to $\bar{\Theta}^G_{\gamma_2}$  modulo the subspace of $\mbox{Cone}(\gamma_1)$ generated as a $\cpx$-vector space by $\bar{\Theta}^G_\eta$ such that $\eta\geq\gamma_1$ and $\eta\not\sim\gamma_1$.
Since ${\bf Irr}_G(\spl\lambda)$ is a basis of $\cca(\Lambda)$, we have $\bar{\Theta}^G_{\gamma_1}=\bar{\Theta}^G_{\gamma_2}$ as desired.  \,\,\,\, Q.E.D.

\subsection{{Proof of Lemma 5.1.1}}

We need:

\begin{lem}
{The annihilator of $\Ind_{P_\pi}^G(\cpx^\pi_0)$ in $U(\gggg)$ is a maximal ideal for all $\pi\in PT(m)$.}
\end{lem}

{\it Remark} \,\,\,\, In fact, a more general result holds.  
So, we consider more general setting temporally.
Let $G$ be any connected real semisimple Lie group and $P$ be any
 parabolic subgroup of $G$. We denote by $M$ a Levi subgroup of $P$.
We denote by $\gggg$, $\mmm$, and $\ppp$ the complexified Lie algebras
 of $G$, $M$, and $P$, respectively. 
We denote by $\nnn$ the nilradical of $\ppp$
We denote by $1_M$ the trivial representation of $M$.
\begin{lem}
The annihilator of $\Ind_{P}^G(1_M)$ in $U(\gggg)$ is a maximal
 ideal. 
\end{lem}

As far as I know, such a result has not been published but is known by
experts (at least including D.\ A.\ Vogan).  For the comvenience for the
readers, we give a proof here.

{\it Proof} \,\,\,\, 
We denote by $\cpx_{-\rho^P}$ a one-dimensional representation of $\ppp$
defined by $\ppp\ni X\rightsquigarrow -\frac{1}{2}\tr(\ad(X)|_\nnn)$.
From the existence of nondegenerate pairing, it suffices to show that the annihilator of a generalized Verma module $M_\ppp(0)=U(\gggg)\otimes_{U(\ppp)}\cpx_{-\rho^P}$ is maximal.
We denote by $I$ the annihilator of $M_\ppp(0)$ in $U(\gggg)$.
We define
\[
 L(M_\ppp(0),M_\ppp(0))=\{\phi\in\End_\cpx(M_\ppp(0))\mid \dim \ad(U(\gggg))\phi<\infty\}.
\]
$L(M_\ppp(0),M_\ppp(0))$ has a obvious $U(\gggg)$-bimodule structure.
Then, $L(M_\ppp(0),M_\ppp(0))$ is isomorphic to a Harish-Chandra module
of an induced representation of $G_\cpx$ from a unitary one-dimensional
representation of $P_\cpx$.
Hence, $L(M_\ppp(0),M_\ppp(0))$ is completely reducible as
a $U(\gggg)$-bimodule.
Considering the action of $U(\gggg)$ on $M_\ppp(0)$, we have an embedding
of a $U(\gggg)$-bimodule $\U(\gggg)/I\hookrightarrow
L(M_\ppp(0),M_\ppp(0)) $.
Hence, $\U(\gggg)/I$ is also completely reducible as
a $U(\gggg)$-bimodule.
We consider the unit element $1$ of $U(\gggg)/I$.
Then, $\cpx 1$ is the
unique trivial $\ad (U(\gggg))$-type in $\U(\gggg)/I$.
So, the unit $1$ must contained in some irreducible component of $\U(\gggg)/I$
Since  $\U(\gggg)/I$ is generated by $1$  as
a $U(\gggg)$-bimodule,
$\U(\gggg)/I$ is irreducible as a $U(\gggg)$-bimodule.
This means that $I$ is maximal.

\,\,\,\, Q.E.D.

{\it Proof of Lemma 5.1.1} \,\,\,\,
From Corollary 5.2.5 and Lemma 5.3.1, we see that all the irreducible constituent of $\Ind_{P_\pi}^G(\cpx_0^\pi)$ is isomorphic to each other.
However, the multiplicity of the trivial $K$-representation in $\Ind^G_{P_\pi}(\cpx_0^\pi)$  is just one.
Hence $\Ind^G_{P_\pi}(\cpx_0^\pi)$ is irreducible as we desired.
\,\,\,\, Q.E.D.



\bigskip



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\end{document}
