Range characterization of the cosine transform on higher Grassmannians.
S.Alesker
J. Bernstein
{ \normalsize 69978 Tel Aviv, Israel }
We characterize the range of the cosine transform on real Grassmannians in terms of the decomposition under the action of the special orthogonal group $SO(n)$. Also a geometric interpretation of this image is given. The non- Archimedean analogues are discussed.
math0111031.tex
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\title{Range characterization of the cosine transform on higher Grassmannians. }
\date{}
\author{ S.Alesker , J. Bernstein
\\  { \normalsize Department of Mathematics, Tel Aviv University, Ramat Aviv}
 \\  { \normalsize 69978 Tel Aviv,
Israel }}
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\begin{document}
\maketitle
%\setcounter{section}{-1}
%\section{Introduction.}
\begin{abstract}
We characterize the range of the cosine transform on real
Grassmannians in terms of the decomposition under the action of
the special orthogonal group $SO(n)$. Also a geometric
interpretation of this image is given. The non- Archimedean
analogues are discussed.
\end{abstract}
\setcounter{section}{-1}
\section{Introduction.}
The main results of this paper are Theorems 1.2, 1.3, and 2.1
below. In Section 1 we study the image of the cosine transform.
The investigations of this problem were started by G. Matheron in
1974 \cite{matheron1}. Further references and known results in
this direction are discussed in Section 1 of this paper. Theorem
1.1 describes the image of the cosine transform (defined in
Section 1) acting from the space of functions on the Grassmannian
of real $j$- dimensional subspaces in $\RR^n$ to the space of
functions on the Grassmannian of real $i$- dimensional subspaces.
The description is given in terms of $K$- types, namely in terms
of the decomposition into irreducible subspaces under the action
of the special orthogonal group $SO(n)$. In Theorem 1.3 we
consider separately the case $i=j$; this case is especially
important since the study of other cases is reduced to this one.
It turns out that the cosine transform in this case can be
interpreted as an intertwining operator of $GL(n,\RR)$- modules.
This crucial observation allows us to use the representation
theory of reductive groups.
 We prove that this intertwining operator has an irreducible image.
 This step uses connection of the cosine transform with the theory
 of valuations on convex sets (discussed in Section 1) and earlier
 results of one of the authors about representation theoretical properties of valuations \cite{alesker1},
 \cite{alesker}.
 In order to describe the decomposition of the image of the cosine
 transform explicitly in terms of $K$- types, we use in a very
 essential way the results due to Howe and Lee \cite{howe-lee}.
In Section 2 we discuss an analogue of the cosine transform over
non-Archimedean local fields.

The cosine transform
 appears naturally in convex geometry, e.g.  in the study
of volumes of projections of convex sets, and related question of
integral and stochastic geometry (see \cite{goodey-howard1},
\cite{goodey-howard2}, \cite{goodey-howard-reeder},
\cite{goodey-zhang}, \cite{groemer},\cite{matheron1},
\cite{matheron2}, \cite{schneider70}). For the basic notions of
the representation theory of real reductive groups we refer to
\cite{wallach}.

We believe that there should exist a geometric interpretation of our results in the
non- Archimedean case.

{\bf Acknowledgements.} The first named author is grateful to A. Braverman,
D. Kazhdan, and D. Soudry for very useful discussions.

\section{Range of the cosine transform.}
First let us recall some notation. We will denote by $\gr_{k,n}$
the Grassmannian of real $k$- dimensional subspaces in the real
$n$- dimensional space $\RR^n$. Let us fix a Euclidean structure
on $\RR^n$. Let $E\in \gr_{i,n}, \, F\in \gr_{j,n}$. Assume that
$i\leq j$. Let us call by {\itshape cosine of the angle} between
$E$ and $F$ the following number:
$$|cos(E,F)|:=\frac{vol_i(Pr_F(A))}{vol_i(A)},$$
where $A$ is any subset of $E$ of non-zero volume, $Pr_F$ denotes
the orthogonal projection onto $F$, and $vol_i$ is the $i$-
dimensional measure induced by the Euclidean structure. (Note that
this definition does not depend on the choice of a subset
$A\subset E$). In the case $i\geq j$ we define the cosine of the
angle between them as cosine of the angle between their orthogonal
complements:
$$|cos(E,F)| :=|cos(E^\perp ,F^\perp )|.$$
(It is easy to see that if $i=j$ both definitions are equivalent.)


Let us call by {\itshape  sine of the angle} between $E$ and $F$
the cosine between $E$ and the orthogonal complement of $F$:
$$|sin(E,F)|:= |cos(E,F^\perp )|.$$
 The following properties are well known (and rather trivial):
$$|cos(E,F)|=|cos(F,E)|=|cos(E^\perp ,F^\perp)|,$$
$$|sin(E,F)|=|sin(F,E)|=|sin(E^\perp ,F^\perp)|,$$
$$0\leq |cos(E,F)|, \, |sin(E,F)| \leq 1.$$

For any $1\leq i, \, j \leq n-1$ one defines the cosine transform
$$T_{j,i}:C(\gr_{i,n}) \str C(\gr_{j,n})$$ as follows:
$$(T_{j,i}f)(E):= \int_{\gr_{i,n}} |cos(E,F)| f(F) dF,$$
where the integration is with respect to the Haar measure on the
Grassmannian.

{\bf Remark.} One should notice that very often in the literature
the cosine transform $T_{j,i}$ (resp. the Radon transform
$R_{j,i}$) is denoted by $T_{i,j}$ (resp. $R_{i,j}$), i.e. with
permutation of indexes. We prefer our notation since it is more
convenient to write the composition formulas like
$R_{j,i}=R_{j,k}\circ R_{k,i}$.

 Clearly the cosine transform  commutes with the
action of the orthogonal group  $O(n)$,  and hence its image is
$O(n)$- invariant subspace of functions. Our first main result
(Theorem 1.2) is the description of the image of the cosine
transform in terms of the decomposition of it with respect to the
action of $SO(n)$. Since the representation of $SO(n)$ in
functions on the Grassmannian is multiplicity free, it is
sufficient to list those irreducible representations of $SO(n)$
entering into the image of the cosine transform. Moreover it is
shown (Theorem 1.3) that for $i=j$ this image coincides with the
image of even translation invariant $i$- homogeneous continuous
valuations on convex sets under certain natural map (actually this
fact is proved first and then used in the proof of the more
explicit result).

Now let us recall standard facts on the representations of the
special orthogonal group $SO(n)$ (see e.g. \cite{zhelobenko}).
\begin{lemma}
 The isomorphism classes of irreducible representations of
$SO(n)$,
$n>2$ are parameterized by their highest weights, namely
sequences of integers $(m_1, m_2, \dots ,m_{[n/2]})$ which
satisfy:

(i) if $n$ is odd then
$m_1 \geq m_2  \geq \dots \geq  m_{[n/2]} \geq 0$:

(ii) if $n>2$ is even then
$m_1 \geq m_2 \geq \dots \geq  m_{n/2 -1} \geq |m_{n/2}|$.
\end{lemma}
%\newline
Recall also that for $n=2$ the representations of $SO(2)$ are
parameterized by a single integer number $m_1$. We will use the
following notation. Let us denote by $\Lambda^+$ the set of all
highest weights of $SO(n)$, and by $\Lambda^+_k$ the set of all
highest weights $\lambda =(m_1, m_2, \dots , m_{[n/2]})$ with
$m_i=0$ for $i>k$ and all $m_i$ are {\itshape even}.

Let us remind the decomposition of the space of functions on the
Grassmannian $\gr_{k,n}$ under the action of $SO(n)$ referring for
the proofs to \cite{sugiura}, \cite{takeuchi}. Since $\gr_{k,n}$
is a symmetric space, each irreducible representation enters with
multiplicity at most one. The representations which do appear have
highest weights precisely from $\Lambda^+_k \cap \Lambda^+_{n-k}$.
\newline
Now let us state our main result.
\begin{theorem}
Let $1\leq i, j \leq n-1$. Then the image of the cosine transform
$T_{j,i} :C(\gr_{i,n} )\str C(\gr_{j,n})$ consists of irreducible
representations of $SO(n)$ with highest weights $\lambda =(m_1,
\dots , m_{[n/2]})$ precisely with the following additional
conditions:

(i)$\lambda \in \Lambda^+_{i} \cap \Lambda^+_{n-i} \cap
\Lambda^+_{j} \cap \Lambda^+_{n-j}$;

(ii) $|m_2| \leq 2$;

Moreover every $C^\infty$- function on $\gr_{j,n}$ which belongs
to the closure of the sum of $SO(n)$- irreducible subspaces
satisfying the above conditions (i)-(ii), is an image under
$T_{j,i}$ of some $C^\infty$- function on $\gr_{i,n} $.
\end{theorem}
Note that as a corollary of this theorem we immediately get
exactly the same characterization of the image of the sine
transform. Theorem 1.2 was known for a long time for $i=j= 1$ (or
equivalently for $i=j=n-1$), see \cite{schneider70}
 or  \cite{groemer}. The case $n=4, \, i=j=2$ was described completely in
\cite{goodey-howard-reeder}; this paper contains also a partial information
on the general case.

The main case in the proof of this theorem is $i=j$. The cosine
transform for $i \ne j$ can be written as a composition of the
Radon transform between different Grassmannians and the cosine
transform for $i=j$. Thus to deduce the general case we use the
characterization of the image of the Radon transform
\cite{gelfand-graev-rosu}, see also \cite{grinberg}.  In order to
treat the case $i=j$ we first interpret the cosine transform as an
intertwining operator between certain representations on the
(non-compact) group $GL(n,\RR)$ induced from maximal parabolic
subgroups. Next we prove that the image of this operator is an
irreducible $GL(n,\RR)$- module . In order to do it we first show
(see Theorem 1.3 ) that the image is contained in the subspace
corresponding to even translation invariant valuations (see the
definitions below). But the last space is an irreducible
$GL(n,\RR)$- module  by the main result in \cite{alesker}. Hence
it coincides with the image of the cosine transform. The
decomposition of the space of even valuations with respect to the
action of $SO(n)$ was described in \cite{alesker} as an easy
corollary of the irreducibility and the computations due to Howe
and Lee \cite{howe-lee}.

Now let us describe the construction of the intertwining operator.
Let us denote by $L$ the line bundle over the Grassmannian $\gr_{i,n}$
whose fiber over a subspace $E\in \gr_{i,n}$ is the space of Lebesgue
measures on $E$ (which is denoted by $|\wedge ^i E^*|$). Clearly $L$ is $GL(n,\RR)$- equivariant
line bundle over $\gr_{i,n}$. Moreover if we fix the Euclidean structure
on $\RR^n$ we can identify $L$ with the trivial bundle in a way compatible with
the action of $SO(n)$. Let $M$ denote the line bundle
over the Grassmannian $\gr_{n-i,n}$ whose fiber over
$F\in  \gr_{n-i,n}$ is the space of Lebesgue measures on
the quotient space $\RR^n /F$ denoted by $|\wedge ^{i} (\RR^n /F)^*|$.
 Let $|\omega|$ denote the line bundle
of densities over $\gr_{n-i,n}$. Let $N:=L\otimes |\omega|$.
Define an intertwining operator $T$ from the space of continuous
sections of $N$ to the space of continuous sections of $L$
$$T:\Gamma (\gr_{n-i,n},N) \str \Gamma (\gr_{i,n}, L) $$
as follows. For $E\in \gr_{i,n}$ and $f\in \Gamma (\gr_{n-i,n},N)$ set
$$(Tf)(E)=\int _{F\in \gr_{n-i,n}} pr^*_{E,F} (f(F)) ,$$
where $pr_{E,F}$ denotes the natural map $E\str \RR^n /F$ and
$pr^*_{E,F}$ is the induced map $|\wedge ^{i} (\RR^n /F)^*| \str |\wedge ^i E^*|$.
Clearly $T$ is a non-trivial operator commuting with the action of $GL(n,\RR)$.
\begin{theorem}
The image of the operator $T$ is an irreducible $GL(n,\RR)$- module.
Moreover if we identify $L$ with the trivial bundle in an
$SO(n)$- equivariant way then the image of $T$ coincides
with the image of the cosine transform $T_{i,i}$.
\end{theorem}
Note that the second statement of the theorem easily follows
from the definitions. In order to prove the irreducibility of the image
we will need one more construction which we are going to
describe.

Let ${\cal K}^n$ denote the family of all convex compact subsets in $\RR^n$.
\begin{definition}
1) A function $\phi :{\cal K}^n \str \CC^n$
is called a valuation if for any $K_1, \, K_2 \in {\cal K}^n$
such that their union is also convex one has
$$\phi(K_1 \cup K_2)= \phi(K_1) +\phi(K_2) -\phi(K_1 \cap K_2).$$

2) A valuation $\phi$ is called continuous if it is continuous with
respect the Hausdorff metric on ${\cal K}^n$.

3) A valuation $\phi$ is called translation invariant if
 $\phi (K+x)=\phi(K)$ for every $x\in \RR^n$ and every $K$.

4) A valuation $\phi$ is called even if $\phi(-K)=\phi(K)$
for every $K\in {\cal K}^n$.

5) A valuation $\phi$ is called homogeneous of degree $k$
(or k-homogeneous) if for every $K\in {\cal K}^n$ and
every scalar $\lam \geq 0$
$\phi(\lam \cdot K)=\lam ^k \phi(K)$.
\end{definition}

We refer to further details on valuations to the surveys
\cite{mcmullen-schneider} and \cite{mcmullensurvey}.
We will need only few facts about them.
%**************************  Lemma 1.6  *******************************
\begin{lemma}\cite{hadwiger}
Every translation invariant $n$- homogeneous continuous
valuation on ${\cal K}^n$ is proportional to the Lebesgue measure on $\RR^n$.
\end{lemma}
Let us denote by $Val^{ev}_{n,i}$ the linear space of translation
invariant $i$- homogeneous even continuous valuations. It is a Frechet
space with respect to the topology of uniform
convergence on compact subsets of ${\cal K} ^n$.

There is a natural map
$$\gamma :\val \str \Gamma(\gr_{i,n}, L),$$
where $L$ is the line bundle defined  above. To define it, fix any
$\phi \in \val$. Take any $E\in \gr_{i,n}$. Consider the
restriction of $\phi$ to the class of convex compact subsets of
$E$. It is again translation invariant $i$- homogeneous continuous
valuation on $E$. Hence by Lemma 1.5 it is proportional to the
Lebesgue measure on $E$. Hence $\phi$ defines a continuous section
$\gamma (\phi)$ of $L$. This map was used in \cite{alesker1} and
\cite{alesker} in the proof of McMullen's conjecture. This map was
also independently considered by D. Klain \cite{klain2}. Note that
this map turns out to be injective by theorem of D. Klain
\cite{klain1}. The main fact on $\val$ we use is the following
result proved in \cite{alesker}.
%********************** Lemma 1.7 *******************************************
\begin{lemma}
For every integer $i$, $1\leq i \leq n-1$ the space $\val$ is an irreducible
$GL(n,\RR^n)$- module. Hence its image in $\Gamma (\gr_{i,n}, L)$
is an irreducible submodule.
\end{lemma}

{\bf Proof} of Theorem 1.3. By Lemma 1.6 it remains to show that
the image of $T$ is contained in $\gamma (\val)$. Fix any $f\in
\Gamma (\gr_{n-i,n}, N)$. Let us define a valuation $\phi$ as
follows. For every $K\in {\cal K}^n$ set
$$\phi (K):= \int _{F\in \gr_{n-i,n}} f(pr_{\RR^n /F} (K)),$$
where $pr_{\RR^n /F} :\RR^n \str \RR^n /F$ is the canonical map.
It is easy to see that $\phi \in \val$; and moreover $\gamma(\phi)
=T(f)$. Thus Theorem 1.3 is proved. \qed

{\bf Proof} of Theorem 1.2. First consider the case $i=j$. By
Theorem 1.3 and its proof the image of the cosine transform
coincides with the image under the map $\gamma$ of translation
invariant $i$- homogeneous continuous valuations. The explicit
decomposition of the last space under the action of $SO(n)$ was
given in \cite{alesker} (it was heavily based on \cite{howe-lee}).
Thus Theorem 1.2 is proved for $i=j$.
%****************************************************************************
%***************************** Corrections **********************************
%*****************************************************************************
Now consider the case $i \ne j$. Clearly we may assume that $j<i$.
One has the Radon transform
$$R_{j,i}: C(\gr_{i,n})\str C(\gr_{j,n})$$
defined by $(R_{j,i}f)(H)= \int_{F \supset H}  f(F) \cdot dF$. The
next lemma is well known, but we will present a proof for
convenience of the reader.
%******************************    Lemma 1.7 ***********************************************
\begin{lemma}
Let $1\leq j<i \leq n$. Then
$$T_{j,i}= c \cdot T_{j,j} R_{j,i},$$
here $c$ is a constant depending on $n,i,j$.
\end{lemma}
{\bf Proof.} Fix a subspace $E\in \gr_{j,n}$. Let $A$ be any
convex compact subset of $E$ of positive measure. Let $f$ be a
continuous function on $\gr_{i,n}$. Then by definition
\begin{equation}
 (T_{j,i}f)(E)=\int_{F\in \gr_{i,n}} \frac{vol_j (Pr_F(A))}{vol_j (A)} f(F) dF.
\end{equation}
Let us fix $F\in \gr_{i,n}$ for a moment. Let $B:= Pr_F(A)$.
 By the Cauchy- Kubota formula (see e.g. \cite{schneider-weil})
$$vol_j (B) =c \cdot \int _{H\in Gr_j(F)} vol_j (Pr_H (B) )dH,$$
where $ Gr_j(F)$ denotes the Grassmannian of $j$- dimensional
subspaces of $F$, and $c$ is a non-zero normalizing constant
depending on $i,j$, and $n$ only. Substituting this into (1) and
using the standard change of integration (normalizing all the Haar
measures to be probability measures) one gets:
$$  (T_{j,i}f)(E)=\frac{c}{vol_j(A)} \int_{F\in \gr_{i,n}} dF \cdot f(F) \left(
\int_{H\subset F} dH \cdot vol_j (Pr_F (A)) \right)=$$
$$ \frac{c}{vol_j(A)} \int_{H\in \gr_{j,n}} dH \cdot vol_j (Pr_H (A) )
\left( \int _{F\supset H} dF \cdot f(F) \right).$$ This identity
clearly proves the lemma. \qed

To finish the proof of Theorem 1.2 we need the following fact
proved in
  \cite{gelfand-graev-rosu} (see also \cite{grinberg}).

\begin{proposition}
For $j<i$ the Radon transform $R_{j,i}: C(\gr_{i,n})\str
C(\gr_{j,n})$ is injective iff $i+j \geq n$ and has a dense image
iff $i+j \leq n$.
\end{proposition}

This proposition, description of the decomposition of the space of
functions on the Grassmannians under the action of $SO(n)$, and
the characterization of the image of the cosine transform for
$i=j$ imply the first part of Theorem 1.2.

It remains to prove that if $f$ is a $C^\infty$- function on
$\gr_{i,n}$ belonging to the closure of the sum of $SO(n)$-
irreducible subspaces satisfying the conditions (i)-(ii) in the
statement of Theorem 1.2, then $f$
 is an image under $T_{j,i}$ of some $C^\infty$- function
on $\gr_{j,n} $. We may assume that $j<i$. By Lemma 1.7 $T_{j,i}=c
\cdot T_{j,j} R_{j,i}$. We will need the following fact due to
Casselman and Wallach (see \cite{casselman}).
%***************************  Proposition 1.9 *****************************************
\begin{proposition} Let $G$ be a real reductive group. Let $K$ be its maximal
compact subgroup. Let $\xi: X \str Y$ be a morphism of two
admissible Banach $G$- modules of finite length which has  a dense
image. Then $\xi$ induces an epimorphism on the spaces of smooth
vectors.
%More precisely,
%if $f\in Y$ is a smooth vector  then there exists a smooth vector $g\in X$
%such that $\xi(g)=f$ and the $K$- types entering into the decomposition of $g$
%are the same as those of $f$.
\end{proposition}

In our situation we will need the following more precise form of
Proposition 1.9. In fact it can be proved in much more general
context, but we do not need it and do not have precise reference.

\begin{lemma}
Let $G= GL(n, \RR)$. Let $K=O(n)$ be the maximal compact subgroup.
Let $X$ and $Y$ be $G$- modules of continuous sections of some
finite dimensional $G$- equivariant vector bundles over the
Grassmannians (or any other partial flag manifolds). Let $\xi: X
\str Y$ be a morphism of these $G$- modules. Then if $f\in Y$ is a
smooth vector  then there exists a smooth vector $g\in X$ such
that $\xi(g)=f$ and the $K$- types entering into the decomposition
of $g$ are the same as those of $f$.
\end{lemma}

We postpone the proof of this lemma till the end of the section.
Now let us continue the proof of Theorem 1.2. We have given an
interpretation of $T_{j,j}$ as an intertwining operator of two
$GL(n,\RR)$- modules; they satisfy the conditions of Proposition
1.10 , since they are induced from characters of parabolic
subgroups (see \cite{wallach}). Hence by Proposition 1.10 there
exists a $C^\infty$- smooth function $g$ on the Grassmannian
$Gr_{j,n}$ such that $f= T_{j,j} (g)$ and with the same $K$- types
as $f$.
 Next there exists an interpretation of the Radon
transform as an intertwining operator of some admissible
$GL(n,\RR)$- modules of finite length (it was given in
\cite{gelfand-graev-rosu}). Hence Proposition 1.10 implies the
statement. \qed

{\bf Proof} of Lemma 1.10. Let $f\in Y$ be as in the statement of
Proposition 1.10. By Proposition 1.9 we can choose a smooth vector
$h\in X$ such that $\xi (h)=f$. Then $h$ is just a smooth section
of the corresponding vector bundle over the Grassmannian (or the
partial flags manifold). Let us consider the image $g$ under the
orthogonal projection of $h$ to the closure with respect to the
$L_2$- metric of the span in $X$ of those $K$- irreducible
subspaces which have the same $K$- types as those entering into
the decomposition of $f$. Clearly $\xi(g)=\xi(h)=f$. To finish the
proof it remains to show that $g$ is also a smooth section. Indeed
since the orthogonal projection commutes with the action of $K$,
$K$- smooth vectors go to $K$- smooth, hence $g$ is $K$- smooth
vector. But since $K$ acts transitively on the Grassmannians (and
in fact on all partial flag manifolds) every $K$- smooth section
of every $K$- equivariant bundle must be smooth in the usual
sense. \qed

%***************   Section 2   ******************************************************
%************************************************************************************
%************************************************************************************
\section{Non- Archimedean analogue of the cosine transform.}
In this section we study non- Archimedean analogue of the cosine
transform. More precisely we study a non-Archimedean  analogue of
the intertwining operator $T_{i,i}$ (in the notation of the
previous section). We show that it has an irreducible image.

Now let us introduce the necessary notation. Let $F$ be a non-
Archimedean local field. In this section we will denote by
$Gr_{i,n}$ the Grassmannian of $i$- dimensional subspaces in
$F^n$. Denote by $L$ the line bundle over the Grassmannian
$\gr_{i,n}$ whose fiber over a subspace $E\in \gr_{i,n}$ is the
space of Lebesgue measures on $E$ (which is denoted by $|\wedge ^i
E^*|$). Clearly $L$ is $GL(n, F)$- equivariant line bundle over
$\gr_{i,n}$. Let $M$ denote the line bundle over the Grassmannian
$\gr_{n-i,n}$ whose fiber over $H \in  \gr_{n-i,n}$ is the space
of Lebesgue measures on the quotient space $F^n /H$ denoted by
$|\wedge ^{i} (F^n /H)^*|$.
 Let $|\omega|$ denote the line bundle
of densities over $\gr_{n-i,n}$. Let $N:=M \otimes |\omega|$.
Define an intertwining operator $T$ from the space of continuous
sections of $N$ to the space of continuous sections of $L$
$$T:\Gamma (\gr_{n-i,n},N) \str \Gamma (\gr_{i,n}, L) $$
as follows. For $E\in \gr_{i,n}$ and $f\in \Gamma (\gr_{n-i,n},N)$ set
$$(Tf)(E)=\int _{H\in \gr_{n-i,n}} pr^*_{E,H} (f(H)) ,$$
where $pr_{E,H}$ denotes the natural map $E\str F^n /H$ and
$pr^*_{E,H}$ is the induced map $|\wedge ^{i} (F^n /H)^*| \str |\wedge ^i E^*|$.
Clearly $T$ is a non-trivial operator commuting with the action of $GL(n,F)$.
Recall that an irreducible $GL(n,F)$- module is called {\itshape unramified}
if it has a non-zero vector invariant with respect to maximal compact
subgroup of $GL(n,F)$.
%**************************** Theorem 2.1 *************************************************
\begin{theorem}
The operator $T$ has an irreducible image. Moreover its image is
an unramified $GL(n,F)$- module.
\end{theorem}
\begin{remark}
It can be shown that the representation of $GL(n,F)$ in
$\Gamma(Gr_{i,n}, L)$ is irreducible for $i=0, 1, n-1, n$, and for
$2\leq i \leq n-2$ it has length two. This follows from the results of
Zelevinsky \cite{zelevinsky} (see also the discussion below).
\end{remark}
The proof of this theorem is an application of the results of the
paper by Zelevinsky \cite{zelevinsky}. Let us remind the necessary
facts following the notation of \cite {zelevinsky}.

We will denote for brevity the group $GL(n,F)$ by $G_n$. By $Irr(G_n)$
we will denote the set of isomorphism classes of irreducible
representations of $G_n$, and by $Rep(G_n)$ we will denote the set of isomorphism
classes of all smooth representations of $G_n$.
 Let $\alpha =(n_1, \dots , n_r)$ be an ordered partition
of $n$. Let $G_{\alpha}$ be the subgroup $G_{n_1} \times \dots \times G_{n_r}$
of $G_n$  consisting of block- diagonal matrices. Let $P_{\alpha}$
denote the subgroup of $G_n$ consisting of block-upper-diagonal matrices.
Then $P_{\alpha}$ is a parabolic subgroup with the Levi factor isomorphic to $G_{\alpha}$.
For $\rho _i  \in Rep(G_{n_i}) , \, i= 1, \dots ,r$, let $\rho_1 \otimes \dots \otimes \rho _r \in
Rep(G_{\alpha})$ be their (exterior) tensor product. This representation can be extended
(trivially) to the representation of $P_{\alpha}$. Define
$$ \rho_1 \times \dots \times \rho _r :=
Ind _{P_{\alpha}} ^{G_n}( \rho_1 \otimes \dots \otimes \rho _r), $$
where the induction is normalized.

Let ${\cal C}$ be the set of equivalence classes of irreducible cuspidal
representations of $G_n, \, n=1,2, \dots$. Note that a (one- dimensional) character
of $F^*$ can be considered as a cuspidal representation of $G_1=F^*$.
Let us call a segment in ${\cal C}$ any subset of ${\cal C}$ of the form
$\Delta =\{\rho, \nu \rho , \dots , \nu ^k \rho =\rho '\}$, where $\nu$ is the
character $\nu (g)= |det(g)|$. We will write it also as $\Delta = [\rho , \rho ']$.
To each segment $\Delta$ one can
associate the irreducible representation $<\Delta>$ which can be defined
as the unique irreducible submodule of $\rho \times \nu \rho \times \dots \times \nu ^k \rho$.

Let $\Delta _1 =[\rho _1, \rho _1 ']$ and $\Delta _2=[\rho _2, \rho _2 '] $
be two segments in ${\cal C}$. The segments
$\Delta _1$ and $\Delta _2$ are called {\itshape linked} if $\Delta _1 \cup \Delta _2$
is also a segment and $\Delta _1 \not \subset \Delta _2, \,
\Delta _2 \not \subset \Delta _1$. The segments
$\Delta _1$ and $\Delta _2$ are called {\itshape juxtaposed} if
they are linked and $\Delta _1 \cap \Delta _2 =\emptyset$.
We say that $\Delta _1$ {\itshape precedes} $\Delta _2$
if they are linked and $\rho _2 =\nu ^k \rho _1$ for $k>0$.

The following theorem is one of the main results of \cite{zelevinsky} (Theorem 6.1).
% **************************************  Proposition 2.3 ********************************
\begin{proposition}
(a) Let $\Delta _1, \dots , \Delta _r$ be segments in ${\cal C}$. Suppose that
for each pair of indices $i<j$, $\Delta _i$ does not precede $\Delta _j$.
Then the representation $<\Delta _1> \times \dots \times <\Delta _r>$
has a unique irreducible submodule; denote it by $<\Delta _1, \dots , \Delta _r>$.

(b) The representations $<\Delta _1, \dots , \Delta _r>$ and $<\Delta '_1, \dots , \Delta '_s>$
are isomorphic iff the sequences
$<\Delta _1>, \dots , <\Delta _r>$ and $<\Delta '_1>, \dots , <\Delta '_s>$
are equal up to rearrangement.

(c) Any irreducible representation of $G_n$ is isomorphic to some representation of the form
$<\Delta _1, \dots , \Delta _r>$.
\end{proposition}

We will also need the following fact (\cite{zelevinsky}, Theorem 4.2).
\begin{proposition}
Let $\Delta _1, \dots , \Delta _r$ be segments in ${\cal C}$.
Then the representation $<\Delta _1> \times \dots \times <\Delta _r>$ is irreducible
iff for all $i \ne j$ the segments $ \Delta _i$ and $\Delta _j$ are not linked.
\end{proposition}

Now let us introduce few more notation. Let $\Delta $ and $\Delta '$
be linked segments. Set
$$\Delta ^\cup = \Delta \cup \Delta ', \, \Delta ^\cap =\Delta \cap \Delta '.$$
By definition $\Delta ^\cup$ is a segment. $\Delta ^\cap$ is a
segment iff $\Delta $ and $\Delta '$ are not juxtaposed; otherwise
$\Delta ^\cap =\emptyset$. It was shown in \cite{zelevinsky},
Section 4, that $\omega := <\Delta ^\cup> \times <\Delta ^\cap>$
is irreducible (and hence by Proposition 2.3 it is isomorphic to
$<\Delta ^\cup , \Delta ^\cap >$). (Here the term $\Delta ^\cap$
should be ignored if it is empty.) The next result will be also
used (\cite{zelevinsky}, Proposition 4.6).
\begin{proposition}
Suppose $\Delta '$ precedes $\Delta$ and set $\pi =<\Delta> \times <\Delta '>$.
Then $\pi$ has a unique irreducible submodule $\omega _0= <\Delta, \Delta '>$.
Moreover $\pi / \omega _0 \simeq \omega =<\Delta ^\cup> \times <\Delta ^\cap>$.
\end{proposition}

Let us describe in terms of segments the representation dual to the given one.
For each segment $\Delta$ in ${\cal C}$ let  $\Delta  ^* := \{ \rho ^* | \rho \in \Delta \}$, where
$ \rho ^*$ is the representation dual to $\rho$. Clearly $\Delta ^*$ is also a segment.
The next result was proved in \cite{zelevinsky}, Theorem 7.10.
%*********************************Proposition 2.6 **********************************
\begin{proposition}
For each segments $\Delta _1, \dots , \Delta _r$ in ${\cal C}$
$$ <\Delta _1, \dots , \Delta _r>^* =<\Delta _1 ^*, \dots , \Delta _r ^*>.$$
\end{proposition}

Now let us return to our situation.

{\bf Proof } of Theorem 2.1. We use the notation of the beginning
of this section. Recall that we study the intertwining operator
$$ T: \Gamma (Gr_{n-i,i}, N) \str \Gamma (Gr_{i,n}, L) .$$
For brevity, we will denote by $\kappa$ the number
$\frac{n-1}{2}$. It is easy to see that the representation of
$GL(n,F)$ is isomorphic to $\Delta _1 \times \Delta _1'$, where
$$\Delta _1 =\underbrace {
( \kappa -i, \kappa -(i-1), \dots , \kappa -1)}_{i \mbox { times
}},$$
$$ \Delta _1 ' = \underbrace {
( -\kappa , -\kappa +1, \dots , \kappa -i)}_{n-i \mbox { times
}}.$$ We see that if $i= 1\mbox { or } n-1$ then the segments
$\Delta _1$ and $\Delta _1 '$ are not linked (since one of them is
contained in the other). Hence by Proposition 2.4 the
representation $\Delta _1 \times \Delta _1'$ is irreducible. Hence
in the cases $i=1, \, n-1$ Theorem 2.1 is proved.

Now assume that $ 2\leq i \leq n-2$. Then the segments $\Delta _1$ and $\Delta _1'$
 are linked and $\Delta _1'$ precedes $\Delta _1$.
Hence by Proposition 2.5 $<\Delta _1> \times <\Delta _1'>$ has a unique
irreducible submodule isomorphic to $<\Delta _1, \Delta_1'>$,
and the quotient module is irreducible and isomorphic to
$<\Delta _1 ^\cup , \Delta _1 ^\cap >$, where
$$\Delta _1 ^\cup =
\underbrace {(-\kappa, -\kappa+1, \dots , \kappa -1)}_{ n-1 \mbox{
times }},$$
$$\Delta _1 ^\cap =(\kappa-i).$$

Now let us describe $\Gamma (Gr_{n-i,n}, N)$. Since $N=M \otimes
|\omega|$ then $\Gamma (Gr_{n-i,n}, N) ^* =\Gamma (Gr_{n-i,n},
M^*)$. It is easy to see that the representation of $GL (n,F)$ in
$\Gamma (Gr_{n-i,n}, M^*)$ is equal to $<\Delta _2> \times <\Delta
_2'>$, where $$\Delta _2 = \underbrace{( -\kappa+i, -\kappa+i+1 ,
\dots , \kappa)}_{n-i \mbox{ times }},$$
$$\Delta _2' =
\underbrace{(-\kappa +1,-\kappa +2, \dots , -\kappa +i)}_{i \mbox{
times }}.$$ Clearly $\Delta _2'$ precedes $\Delta _2$. Hence by
Proposition 2.5 the unique irreducible submodule of $<\Delta _2>
\times <\Delta _2'>$ is isomorphic to $<\Delta _2, \Delta _2'>$,
and the quotient module is irreducible and isomorphic to $<\Delta
_2 ^\cup, \Delta _2 ^\cap >$, where
$$\Delta _2 ^\cup =
\underbrace {(-\kappa+1,-\kappa+2, \dots , \kappa)}_{n-1 \mbox{
times }},$$
$$\Delta _2 ^\cap =(-\kappa +i).$$
Dualising and using Proposition 2.6 we get that the $GL(n,F)$-
module $\Gamma (Gr_{n-i,n}, N)$ has a unique irreducible submodule
isomorphic to $<\Delta _3 ^\cup , \Delta _3 ^\cap >$ with
$$\Delta _3 ^\cup
=\underbrace {(-\kappa, -\kappa +1, \dots , \kappa -1)}_{n-1 \mbox
{ times }},$$
$$\Delta _3 ^\cap =(\kappa -i);$$
and the quotient module is irreducible and is isomorphic to
$<\Delta _3 ,\Delta _3'>$ with
$$\Delta _3= \underbrace{(-\kappa , -\kappa +1,
\dots , \kappa -i)}_{n-i \mbox{ times }},$$
$$\Delta _3' =\underbrace{(\kappa -i , \kappa -i +1,
\dots , \kappa -1)}_{i \mbox{ times }}.$$ Comparing these
computations with computations for $\Gamma(Gr_{i,n}, L)$ and using
Proposition 2.3 (b) we conclude that the image of any non-zero
intertwining operator from $\Gamma(Gr_{n-i,n}, N)$ to
$\Gamma(Gr_{i,n}, L)$ must have irreducible image.

It is easy to see that this image is unramified.
\qed










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\end{document}

math0111031.extracted_bib
\begin{thebibliography}{9}
\bibitem{alesker1}
 Alesker, Semyon;
 On P. McMullen's conjecture on translation invariant valuations.
Adv. Math. 155 (2000), no. 2, 239--263.

\bibitem{alesker}
Alesker, Semyon; Description of translation invariant valuations
with the solution of P. McMullen's conjecture. Geom. Funct. Anal.
11 (2001), no. 2, 244--272.

%\bibitem{bernstein-zelevinsky}
%Bernstein, I. N.; Zelevinsky, A. V.;
 %Induced representations of reductive p-adic groups. I.
%Ann. Sci. \'Ecole Norm. Sup. (4) 10 (1977), no. 4, 441--472.

\bibitem{casselman}
Casselman, William; Canonical extensions of Harish-Chandra modules
to representations of $G$. Canad. J. Math. 41 (1989), no. 3,
385--438.

\bibitem{gelfand-graev-rosu}
Gelfand, I. M.; Graev, M. I.; Ro\c su, R.;
 The problem of integral geometry and intertwining operators for a pair of real Grassmannian
manifolds. J. Operator Theory 12 (1984), no. 2, 359--383.

\bibitem{goodey-howard1}
 Goodey, Paul; Howard, Ralph;
 Processes of flats induced by higher-dimensional processes. Adv. Math. 80 (1990), no. 1, 92--109.

\bibitem{goodey-howard2}
Goodey, Paul R.; Howard, Ralph; Processes of flats induced by higher-dimensional processes. II.
Integral geometry and tomography (Arcata, CA,
1989), 111--119, Contemp. Math., 113, Amer. Math. Soc., Providence, RI, 1990.

\bibitem{goodey-howard-reeder}
 Goodey, Paul; Howard, Ralph; Reeder, Mark;
 Processes of flats induced by higher-dimensional processes. III. Geom. Dedicata 61 (1996), no. 3,
257--269.

\bibitem{goodey-zhang}
 Goodey, Paul; Zhang, Gaoyong;
 Inequalities between projection functions of convex bodies.
Amer. J. Math. 120 (1998), no. 2, 345--367.

\bibitem{grinberg}
Grinberg, Eric L.;
 Radon transforms on higher Grassmannians.
J. Differential Geom. 24 (1986), no. 1, 53--68.

\bibitem{groemer}
Groemer, H.;
Geometric applications of Fourier series and spherical harmonics.
Encyclopedia of Mathematics and its Applications, 61.
Cambridge University Press, Cambridge, 1996.

\bibitem{hadwiger}
Hadwiger, H.; Vorlesungen \"uber Inhalt, Oberfl\"ache und Isoperimetrie.
(German) Springer-Verlag, Berlin-G\"ottingen-Heidelberg 1957.

\bibitem{howe-lee}
Howe, Roger; Lee, Soo Teck;
Degenerate principal series representations
of $ GL_n(\CC)$ and ${\rm GL}\sb n(\RR)$.
J. Funct. Anal. 166 (1999), no. 2, 244--309.

\bibitem{klain1}
Klain, Daniel A.;
 A short proof of Hadwiger's characterization theorem. Mathematika 42 (1995), no. 2, 329--339.

\bibitem{klain2}
Klain, Daniel A.;
 Even valuations on convex bodies. Trans. Amer. Math. Soc. 352 (2000), no. 1, 71--93.

\bibitem{matheron1}
Matheron, G.;
 Un th\'eor\`eme d'unicit\'e pour les hyperplans poissoniens. (French)
J. Appl. Probability 11 (1974), 184--189.

\bibitem{matheron2}
Matheron, G.;
 Random sets and integral geometry.
Wiley Series in Probability and Mathematical
Statistics. Wiley, New York-London-Sydney, 1975.

\bibitem{mcmullensurvey}
McMullen, Peter;
 Valuations and dissections.
Handbook of convex geometry, Vol. A, B, 933--988, North-Holland, Amsterdam, 1993.

\bibitem{mcmullen-schneider}
McMullen, Peter; Schneider, Rolf ;
Valuations on convex bodies. Convexity and its applications, 170--247, Birkh�user, Basel, 1983.

\bibitem{schneider70}
Schneider, Rolf; \"Uber eine Integralgleichung in der Theorie der konvexen K\"orper.
 (German) Math. Nachr. 44 1970 55--75.

\bibitem{schneider-weil}
 Schneider, Rolf; Weil, Wolfgang; Integralgeometrie. (German)
[Integral geometry] Teubner Skripten zur Mathematischen
Stochastik. [Teubner Texts on Mathematical Stochastics] B. G.
Teubner, Stuttgart, 1992.

\bibitem{sugiura}
 Sugiura, Mitsuo;
 Representations of compact groups realized by spherical functions on symmetric spaces.
Proc. Japan Acad. 38 1962 111--113.

\bibitem{takeuchi}
Takeuchi, Masaru; Modern spherical functions.
Translated from the 1975 Japanese original by Toshinobu Nagura. Translations of Mathematical
Monographs, 135. American Mathematical Society, Providence, RI, 1994.

\bibitem{wallach}
 Wallach, Nolan R.; Real reductive groups. I, II.
  Pure and Applied Mathematics, 132. Academic Press, Inc., Boston, MA,
  1988, 1992.

\bibitem{zhelobenko}
 Zhelobenko, D. P.; Kompaktnye gruppy Li i ikh predstavleniya.
 (Russian) [Compact Lie groups and their representations] Izdat.
``Nauka'', Moscow, 1970.

\bibitem{zelevinsky}
Zelevinsky, A.; Induced representations of $p$- adic reductive groups, II;
Irreducible representations of $GL(n)$, Ann. Sci. \'Ecole Norm. Sup. 3 (1980), 165-210.

\end{thebibliography}