\boldmath Cartan determinants and Shapovalov forms
\sc Jonathan Brundan and Alexander Kleshchev
We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of $p$-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.
math0111306.tex
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\begin{document}
\title[Cartan determinants]{\boldmath
Cartan determinants and
Shapovalov forms}
\author{\sc Jonathan Brundan and Alexander Kleshchev}
\address
{Department of Mathematics\\ University of Oregon\\
Eugene\\ OR~97403, USA}
\email{brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu}

\thanks{Second author
partially supported by the NSF (grant no. DMS-9900134).}
\begin{abstract}
We compute the determinant of the Gram matrix of the Shapovalov form on weight
spaces of the basic representation of an affine
Kac-Moody algebra of ADE type (possibly twisted).
As a consequence, we obtain explicit formulae for the determinants
of the Cartan matrices of $p$-blocks of the symmetric group
and its double cover, and of the associated Hecke algebras at roots of
unity.
\end{abstract}
\maketitle

\section{Introduction}

Let $\mathfrak g$ be an affine Kac-Moody algebra
of type $X_N^{(r)}$ as in the table:
\vspace{2mm}
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|}
\hline
X_N^{(r)}&A_{\ell}^{(1)}&D_{\ell}^{(1)}&E_{\ell}^{(1)}&
A_{2\ell-1}^{(2)}&A_{2\ell}^{(2)}&D_{\ell+1}^{(2)}&E_6^{(2)}&D_4^{(3)}\\
\hline
\ell&\geq 1&\geq 4&6,7\hbox{ or }8&\geq 3&\geq 1&\geq 2&4&2\\
\hline
k&0&0&0&\ell-1&\ell&1&2&1\\
\hline
\alpha&\ell+1&4&9-\ell&2&1&2&1&1\\
\hline
\beta&1&1&1&\ell&2\ell+1&2&3&2\\
\hline
\end{array}
$$
\vspace{1mm}

\noindent
We are interested here in the basic representation $V = V(\Lambda_0)$
of $\mathfrak g$, see \cite{Kac}.
Let $|0\rangle$ be a vacuum vector and define the
{lattice}
$V_\Z := U_\Z |0 \rangle$
in $V$, where $U_\Z$ is the $\Z$-subalgebra of 
the universal enveloping algebra
of $\mathfrak g$ generated by the divided powers
$$
e_i^n / n!, \quad f_i^n / n!\qquad(i = 0,1,\dots,\ell,\ n \geq 1)
$$
in the Chevalley generators.
Let $(.\,,.)_S$ denote the Shapovalov form, 
the unique Hermitian form on $V$ satisfying
$(|0\rangle, |0\rangle)_S = 1$ and $(e_i v, v')_S = (v, f_i v')_S$
for $i = 0,\dots,\ell$ and all $v,v' \in V$.
Its restriction to $V_\Z$ gives a symmetric bilinear form
$$
(.\,,.)_S:V_\Z \times V_\Z \rightarrow \Z.
$$ 
Our Main Theorem gives an explicit formula for the determinant of the Gram matrix of this form on each weight space of $V_\Z$.

To state the result precisely, 
recall the description of the weights of $V$ \cite[$\S$12.6]{Kac}:
every weight is of the form $w\Lambda_0 -d \delta$
for some $w$ in the Weyl group $W$ associated to $\mathfrak g$
and some integer $d \geq 0$.
Also
let $\mathscr P(d)$ denote the set of all partitions
$\la = (\la_1 \geq \la_2 \geq \dots)$ of $d$.
Given $\la \in \mathscr P(d)$, we can gather together its equal parts
to represent it as $\la  = 
(1^{r_1} 2^{r_2} \dots)$.
Also recall the number $r\in\{1,2,3\}$ which comes from the type $X_N^{(r)}$. 
Then:

\vspace{2mm}
\noindent
{\bf Main Theorem.} {\em The determinant of the Gram matrix of the 
Shapovalov form on the 
$(w\Lambda_0 - d \delta)$-weight space of $V_\Z$ is
$\alpha^{a(d)}\beta^{b(d)}$ where
$a(d) = \sum_{\la \in \mathscr P(d)} a_\la$,
$b(d) = \sum_{\la \in \mathscr P(d)} b_\la$ and
for $\la = (1^{r_1} 2^{r_2} \dots)$, 
\begin{align*}
a_\la &=
\prod_{i\text{ with }r|i} \binom{\ell+r_i-1}{r_i}\cdot
\prod_{i\text{ with }r \nmid i} \binom{k+r_i-1}{r_i}\cdot\sum_{i\text{ with }r|i} \frac{r_i}{\ell},\\
b_\la
&=
\prod_{i\text{ with }r|i} \binom{\ell+r_i-1}{r_i}\cdot
\prod_{i\text{ with }r\nmid i} \binom{k+r_i-1}{r_i}\cdot\sum_{i\text{ with }r\nmid i} \frac{r_i}{k},
\end{align*}
$\ell,k,\alpha,\beta$ being as in the above table.
The generating functions
$a(q) = \sum_{d \geq 0} a(d) q^d$ and
$b(q) = \sum_{d \geq 0} b(d) q^d$ are given by the formulae
\begin{align*}
a(q)&=  T(q^r)P(q)^k P(q^r)^{\ell-k},\\
b(q) &= (T(q)-T(q^r))P(q)^k P(q^r)^{\ell-k}
\end{align*}
where $P(q) = \prod_{i \geq 1} \frac{1}{1-q^i}$ is the generating function
for the number of partitions of $d$ and 
$T(q) = \sum_{i \geq 1} \frac{q^i}{1-q^i}$ is the generating function for
the number of divisors of $d$.
}

\vspace{2mm}

In \cite{CKK}, De Concini, Kac and Kazhdan constructed
the basic representation over $\Z$ (at least in the untwisted cases)
using an integral version of the vertex operator construction of \cite{KF}.
They showed in particular that the basic representation
remains irreducible on reduction modulo $p$ if and only if
$p \nmid \det X_N$, where $X_N$ is the Cartan matrix of the underlying
finite root system; this also follows immediately from our Main Theorem
on noting that $\det X_N = \alpha \beta^{r-1}$.

Our interest in the theorem comes instead from modular representation
theory.
Suppose now that $\mathfrak g$ is of type $A_\ell^{(1)}$ and set $p = 
(\ell+1)$.
Let $FS_n$ denote the group algebra of the symmetric group over a field
$F$ of characteristic $p$ (assuming in this case that $p$ is prime), 
and let $H_n$ denote the 
Iwahori-Hecke algebra associated to $S_n$
over an arbitrary field but at a primitive $p$th root of $1$
(this case making sense for arbitrary $p \geq 2$).
By \cite{Ariki, G}, there is an isomorphism between 
the basic representation $V_\Z$ of $\mathfrak g$ and the 
direct sum $K = \bigoplus_{n \geq 0} K_n$ of the Grothendieck groups $K_n$  
of finitely generated projective $FS_n$- (resp. $H_n$-) modules for all $n$.
Under the isomorphism, the weight spaces of $V_\Z$ are in 1--1 correspondence
with the block components 
of $K$, a weight space of the form $w \Lambda - d \delta$
corresponding to a block of $p$-weight $d$ (see e.g. \cite[$\S$5.3]{Mat} 
for the
definition of the $p$-weight of a block).
Moreover, according to \cite[Theorem 14.2]{G}, the Shapovalov form  
corresponds to the usual Cartan pairing
$([P], [Q]) = \dim \hom(P,Q)$ between projective modules $P,Q$.
Thus the theorem has the following immediate corollary:

\vspace{2mm}
\noindent
{\bf Corollary 1.} {\em 
Let $B$ be a block of $p$-weight $d$ of either the
group algebra
$FS_n$ of the symmetric group 
over a field of prime characteristic $p$, or
the Hecke algebra $H_n$ over an arbitrary field but
at a primitive $p$th root of unity, in which case $p\geq 2$ is an arbitrary 
integer.
Then the determinant of the Cartan matrix of $B$ is
$p^{N(d)}$ where $$
N(d) = 
\sum_{\lambda = (1^{r_1} 2^{r_2} \dots) \in \mathscr P(d)} 
\frac{r_1+r_2+\dots}{p-1}
\binom{p-2+r_1}{r_1}
\binom{p-2+r_2}{r_2}
\dots.
$$
The generating function $N(q) = \sum_{d \geq 0} N(d) q^d$
equals $T(q) P(q)^{p-1}$.
}
\vspace{2mm}

It is a classical result
of Brauer that the determinant of the Cartan matrix
of a block of $FS_n$ is a power of $p$ (see \cite[84.17]{CR}).
Donkin \cite{Donkin} has proved similarly
that the determinant of the Cartan matrix of a block of $H_n$
{\em divides} a power of $p$. The corollary shows in particular
that the determinant is exactly a power of $p$, even in those
cases where $p$ is not prime, as had been conjectured by Mathas.
We remark that in the case of blocks of $FS_n$, but not of $H_n$,
the explicit generating function given in the
corollary has also recently been obtained by Bessenrodt and Olsson \cite{BO}
using methods from block theory.

Finally suppose that $\mathfrak g$ is of type $A_{2\ell}^{(2)}$
and set $p = (2\ell+1)$.
In this case, the Main Theorem can be reinterpreted as a
computation of Cartan determinants of the $p$-blocks of
the double covers $\widehat S_n$ of the symmetric group.
Following \cite[$\S$9-c]{BKcrys} for notation,
let $S(n)$ be the twisted group algebra of $S_n$ over an algebraically 
closed field 
$F$ of characteristic $p$ (assuming $p$ is an odd prime in this case), 
and let $W(n)$ be the Hecke-Clifford
superalgebra over an algebraically closed field of
characteristic different from $2$ 
at a primitive $p$th root of unity (for arbitrary odd $p \geq 3$). 
By \cite[7.16, 8.13, 9.9]{BKcrys},
there is an isomorphism between the basic representation $V_\Z$
and the direct sum $K = \bigoplus_{n \geq 0} K_n$ of the Grothendieck
groups of finitely generated projective $S(n)$- (resp. $W(n)$-)
supermodules, under which a weight space of the form
$w \Lambda - d \delta$ maps to a superblock of $p$-bar weight $d$ (see
\cite[$\S$9-a]{BKcrys} for the definition of $p$-bar weight of a superblock),
and the Shapovalov form corresponds to the Cartan pairing on 
projective supermodules (see \cite[$\S$7-c]{BKcrys}). So:

\vspace{2mm}
\noindent
{\bf Corollary 2.} {\em 
Let $B$ be a superblock of $p$-bar weight $d$ of either 
$S(n)$ in odd characteristic $p$, or $W(n)$ at a primitive $p$th root of unity,
in which case $p \geq 3$ is an arbitrary odd integer.
Then the determinant of the Cartan matrix of $B$ is
$p^{N(d)}$ where $$
N(d) = 
\sum_{\lambda = (1^{r_1} 2^{r_2} \dots) \in \mathscr P(d)} 
\frac{2r_1+2r_3+2r_5+\dots}{p-1}
\binom{\frac{p-3}{2}+r_1}{r_1}
\binom{\frac{p-3}{2}+r_2}{r_2}
\binom{\frac{p-3}{2}+r_3}{r_3}\dots.
$$
The generating function $N(q) = \sum_{d \geq 0} N(d) q^d$ equals
$(T(q)-T(q^2))P(q)^{(p-1)/2}$.
}
\vspace{2mm}

It is more natural from the point of view of finite group theory
to ask for the Cartan determinant
of a block $B$ of the twisted group algebra 
$S(n)$ in the usual ungraded sense.
According to Humphreys' classification \cite{Hum}, see also 
\cite[9.16]{BKcrys}, we can associated to $B$ its
$p$-bar weight $d$ and a type $\eps \in \{\Mtype, \Qtype\}$.
In case $\eps = \Mtype$, $B$ coincides with a superblock of $p$-bar weight $d$
and it is immediate that its Cartan determinant is as in Corollary 2.
But in the cases when $\eps = \Qtype$ and $d > 0$, the Cartan matrix
of $B$ has twice as many rows and columns 
as the Cartan matrix of the corresponding superblock. 
Nevertheless, we believe the Cartan determinant is the same, based on
explicit computations for small $d$.
In other words, we conjecture that
Cartan determinants of $p$-blocks of $S(n)$
depend only on the $p$-bar weight $d$, not on the type $\eps$, of the block.

\section{The affine algebras}\label{s1}

We begin by recalling the construction of the affine Lie algebras 
from \cite[Chapter 8]{Kac}.
Let $X_N^{(r)}$ be an affine Dynkin diagram of ADE type as in the introduction,
and let $X_N$ be the underlying finite Dynkin diagram.
We use the same numbering of Dynkin diagrams as \cite[$\S$4.8]{Kac}
with two exceptions:
in the case $X_N^{(r)} = E_6^{(2)}$ we will number the vertices of the
finite Dynkin diagram $X_N = E_6$ by
$$
{\begin{picture}(81, 50)%
\put(5, 35){\circle{4}}%
\put(7, 35){\line(1, 0){15}}%
\put(24, 35){\circle{4}}%
\put(26, 35){\line(1, 0){15}}%
\put(43, 35){\circle{4}}%
\put(45, 35){\line(1, 0){15}}%
\put(62, 35){\circle{4}}%
\put(64, 35){\line(1, 0){15}}%
\put(81, 35){\circle{4}}%
\put(43, 33){\line(0, -11){15}}%
\put(43, 16){\circle{4}}%
\put(5, 41){\makebox(0, 0)[b]{$_1$}}%
\put(24, 41){\makebox(0, 0)[b]{$_2$}}%
\put(43, 41){\makebox(0, 0)[b]{$_3$}}%
\put(62, 41){\makebox(0, 0)[b]{$_5$}}%
\put(81, 41){\makebox(0, 0)[b]{$_6$}}%
\put(43, 11){\makebox(0, 0)[t]{$_4$}}%
\end{picture}}
$$
and in the case 
$X_N^{(r)} = A_{2\ell}^{(2)}$ we will number the vertices
of the finite Dynkin diagram $X_N = A_{2\ell}$
by
$$
{\begin{picture}(126, 25)%
\put(5, 5){\circle{4}}%
\put(7, 5){\line(1, 0){15}}%
\put(24, 5){\circle{4}}%
\put(26, 5){\line(1, 0){1}}%
\put(29, 5){\line(1, 0){1}}%
\put(32, 5){\line(1, 0){1}}%
\put(35, 5){\line(1, 0){1}}%
\put(38, 5){\line(1, 0){1}}%
\put(41, 5){\line(1, 0){1}}%
\put(44, 5){\line(1, 0){1}}%
\put(47, 5){\line(1, 0){1}}%
\put(50, 5){\line(1, 0){1}}%
\put(53, 5){\line(1, 0){1}}%
\put(56, 5){\circle{4}}%
\put(58, 5){\line(1, 0){15}}%
\put(75, 5){\circle{4}}%
\put(77, 5){\line(1, 0){1}}%
\put(80, 5){\line(1, 0){1}}%
\put(83, 5){\line(1, 0){1}}%
\put(86, 5){\line(1, 0){1}}%
\put(89, 5){\line(1, 0){1}}%
\put(92, 5){\line(1, 0){1}}%
\put(95, 5){\line(1, 0){1}}%
\put(98, 5){\line(1, 0){1}}%
\put(101, 5){\line(1, 0){1}}%
\put(104, 5){\line(1, 0){1}}%
\put(107, 5){\circle{4}}%
\put(109, 5){\line(1, 0){15}}%
\put(126, 5){\circle{4}}%
\put(5, 11){\makebox(0, 0)[b]{$_{\ell-1}$}}%
\put(24, 11){\makebox(0, 0)[b]{$_{\ell - 2}$}}%
\put(56, 11){\makebox(0, 0)[b]{$_{0}$}}%
\put(75, 11){\makebox(0, 0)[b]{$_{\ell+1}$}}%
\put(107, 11){\makebox(0, 0)[b]{$_{2\ell-1}$}}%
\put(126, 11){\makebox(0, 0)[b]{$_{2\ell}$}}%
\end{picture}}
$$

Let $Q'$ denote the root lattice of type $X_N$, with simple roots
$\alpha_i'$ and invariant
bilinear form $(.|.)'$ normalized so that each
$(\alpha_i'| \alpha_i')' = 2$.
Let $\mu:Q' \rightarrow Q'$ be a graph automorphism of order $r$,
as in e.g. \cite[$\S$7.9]{Kac}.
Let 
$$
\eps:Q' \times Q' \rightarrow \{\pm 1\}
$$ 
be an asymmetry function
as in \cite[$\S$7.8]{Kac} chosen so that 
$\eps(\mu(\alpha'), \mu(\beta')) = \eps(\alpha',\beta')$.
In case $X_N^{(r)} = A_{2\ell}^{(2)}$ this is not possible so we instead require here that
$\eps(\mu(\alpha'), \mu(\beta')) = \eps(\beta',\alpha')$.
Let $\mathfrak h' = \C \otimes_\Z Q'$ viewed as an abelian Lie algebra,
and extend $\mu$ and $(.|.)'$ linearly to $\mathfrak h'$.
Then we can construct the finite dimensional simple Lie algebra $\mathfrak g'$
of type $X_N$ as the vector space
$$
\mathfrak g' = \mathfrak h' \oplus \bigoplus_{\alpha' \text{ a root}} \C 
E_{\alpha'}
$$
viewed as a Lie algebra so that
$\mathfrak h'$ is abelian and
\begin{align*}
[\alpha', E_{\beta'}] &= (\alpha'|\beta')' E_{\beta'},
\qquad[E_{\alpha'}, E_{-\alpha'}] = -\alpha',\\
[E_{\alpha'},E_{\beta'}] &= \left\{
\begin{array}{ll}
\eps(\alpha',\beta') E_{\alpha'+\beta'}&\hbox{if $\alpha'+\beta'$ is a root,}\\
0&\hbox{otherwise.}
\end{array}\right.
\end{align*}
The invariant form on $\mathfrak h'$ extends to $\mathfrak g'$
by $(\mathfrak h'|E_{\alpha'})' = 0$ and $(E_{\alpha'}|E_{\beta'})' = 
- \delta_{\alpha',-\beta'}$ for all roots $\alpha',\beta'$.

Let $a_i, a_i^\vee\:(i = 0,\dots,\ell)$ be the numerical labels 
on the
Dynkin diagram $X_N^{(r)}$ and its dual as in \cite[$\S$4.8]{Kac}.
We note especially that $a_0 = 1$ if
$X_N^{(r)} \neq A_{2\ell}^{(2)}$ and
$a_0 = 2$ if $X_N^{(r)} = A_{2\ell}^{(2)}$.
It will also be convenient to define 
\begin{align*}
c_i &= 
\left\{
\begin{array}{ll}
2&\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$ and $i = 0$,}\\
1&\hbox{otherwise;}
\end{array}
\right.\\
d_i &= c_i a_i^\vee a_i^{-1} \in \{1,r\}
\end{align*}
for $i = 0,1,\dots,\ell$.
Let $m = a_0 r$ and fix a primitive $m$th root of unity
$\omega \in \C$. %(e.g. $\omega = e^{2\pi i / m}$).
In all types other than $A_{2\ell}^{(2)}$, 
let $\eta:Q' \rightarrow \C^\times$ denote the constant function with
$\eta(\alpha') = 1$ for all $\alpha' \in Q'$;
in type $A_{2\ell}^{(2)}$, define $\eta$ instead
by the rules
$$
\eta(0) = 1,
\quad
\eta(\alpha'+\beta') = \eta(\alpha') \eta(\beta') (-1)^{(\alpha'|\beta')'},
\qquad
\eta(\alpha_j') = \left\{
\begin{array}{ll}
1&j \neq 0, \ell+1,\\
\omega
&
j = 0, \ell+1.
\end{array}\right.
$$
Now extend $\mu$ from $\mathfrak h'$ to $\mathfrak g'$
by declaring that
$\mu(E_{\alpha'}) = \eta(\alpha') E_{\mu(\alpha')}$ for all roots $\alpha' 
\in Q'$.
The order of the resulting
automorphism $\mu$ of $\mathfrak g'$ is equal to $m$ in all cases.

Decompose 
$$
\mathfrak g' = \bigoplus_{n \in \Z / m} \mathfrak g'_n
\qquad\hbox{where}\qquad
\mathfrak g'_n = \{X \in \mathfrak g'\:|\:\mu(X) = \omega^n X\}.
$$
Also write
$\mathfrak h'_n = \mathfrak h' \cap \mathfrak g'_n$.
Introduce the infinite dimensional Lie algebras
\begin{align*}
{\mathfrak g} &= \bigoplus_{n \in \Z} \mathfrak g'_n \otimes t^n
\oplus \C c \oplus \C d \subseteq \mathfrak g' \otimes \C[t,t^{-1}] \oplus \C c \oplus \C d,\\
{\mathfrak h} &= \mathfrak h'_0 \otimes 1
\oplus \C c \oplus \C d \subset {\mathfrak g},\\
\mathfrak t &= \mathfrak t^+ \oplus \C c \oplus \mathfrak t^- \subset {\mathfrak g}
\quad\hbox{where}\quad
\mathfrak t^{\pm} = \bigoplus_{\pm n > 0} \mathfrak h'_n \otimes t^n.
\end{align*}
Multiplication is defined by the rules
\begin{align*}
& [d, X\otimes t^n ] = n X \otimes t^n,
\qquad
[c, {\mathfrak g}] = 0,\\
& [X\otimes t^n , Y \otimes t^k ] =
[X,Y]\otimes t^{n+k} + \delta_{n,-k} n\frac{(X|Y)'}{m} c.
\end{align*}
Then ${\mathfrak{g}}$ is the affine Lie algebra of type
$X_N^{(r)}$ with canonical central element $c$ and scaling element $d$,
and
${\mathfrak{h}}$ is a Cartan subalgebra.
As a matter of notation, we will write
\begin{equation*}\label{xndef}
X(n) := 
\sum_{j=0}^{m-1} \omega^{-nj} \mu^j(X) \otimes t^n \in \mathfrak g'_n \otimes t^n
\end{equation*}
for $X \in \mathfrak g'$ and $n \in \Z$.
The normalized invariant form on $\mathfrak g$ will be denoted
$(.|.)$, and is defined by
$$
(X \otimes t^n| Y \otimes t^k)
= 
\delta_{n,-k} (X|Y)' / r
$$
for all $X \in \mathfrak g_n, Y \in \mathfrak g_k$.

In order to write down a choice of Chevalley generators for
${\mathfrak{g}}$,
let $\ell$ denote the number of $\mu$-orbits on the simple roots in $Q'$.
Let 
$$
\eps = 
\left\{
\begin{array}{ll}
0 &\hbox{if $X_N^{(r)} \neq A_{2\ell}^{(2)}$},\\
\ell &\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$},
\end{array}
\right.
$$ 
and set
$$
I = \{0,1,\dots,\ell\} - \{\eps\}.
$$
Then, the $\alpha_i'$ for
$i \in I$ give a set
of representatives for the $\mu$-orbits on the simple roots.
Define
$$
-\alpha_\eps' = \left\{
\begin{array}{ll}
\hbox{the longest root in $Q'$}&\hbox{if $r = 1$ or $X_N^{(r)} = A_{2\ell}^{(2)}$,}\\
\alpha_1'+\dots+\alpha_{2\ell-2}'&\hbox{if $X_N^{(r)}  = A_{2\ell-1}^{(2)}$,}\\
\alpha_1'+\dots+\alpha_\ell'&\hbox{if $X_N^{(r)}  = D_{\ell+1}^{(2)}$,}\\
\alpha_2'+\alpha_3'+\alpha_4'&\hbox{if $X_N^{(r)} = D_4^{(3)}$,}\\
\alpha_1'+2\alpha_2' + 2\alpha_3' + \alpha_4' + \alpha_5' + \alpha_6'
&\hbox{if $X_N^{(r)} = E_6^{(2)}$}.\\
\end{array}
\right.
$$
For $i = 0,1,\dots,\ell$, write
$$
e_i(n) = \frac{\sqrt{c_i}}{a_0 d_i} E_{\alpha_i'}(n)
\quad\hbox{and}\quad
f_i(n) = -\frac{\sqrt{c_i}}{a_0 d_i} E_{-\alpha_i'}(n).
$$
The Chevalley generators of ${\mathfrak g}$ are
$e_0 = e_0(1), e_i = e_i(0)$ and $f_0 = f_0(-1), f_i = f_i(0)$
for $i = 1,\dots,\ell$, as is proved in \cite[$\S$8.7]{Kac}
(taking $s_0 = 1, s_1=\dots=s_\ell = 0$).
We also define 
$$
h_i = [e_i, f_i] = 
\delta_{i,0} c + \frac{c_i}{a_0 d_i} \alpha_i'(0).
$$


Next let $Q 
\subset {\mathfrak h}^*$ denote the root lattice associated
to ${\mathfrak g}$. So following \cite[$\S$6.2]{Kac},
$$
Q = \bigoplus_{i=0}^\ell \Z {\alpha}_i 
\oplus \Z \Lambda_0
$$ where
$\alpha_0,\dots,\alpha_\ell$ are the simple roots
corresponding to $h_0,\dots,h_\ell$ and $\Lambda_0$ is the zeroth fundamental
dominant weight, i.e.
\begin{align*}
\langle h_i, \alpha_j \rangle 
&= \hbox{the $ij$-entry of the Cartan matrix of type $X_{N}^{(r)}$},\\
\langle h_i, \Lambda_0 \rangle &= 
\langle d, \alpha_i \rangle = \delta_{i,0},\\
\langle d, \Lambda_0 \rangle &= 0,
\end{align*}
for $i,j = 0,\dots,\ell$.
Also as in \cite[$\S$6.2]{Kac}, we 
have the normalized invariant form $(.|.)$ on $\mathfrak h^*$
and the element $\delta = \sum_{i=0}^\ell a_i \alpha_i \in Q$.

To conclude, we explain the relationship between the
form $(.|.)'$ on $Q'$ and the form $(.|.)$ on $Q$.
Introduce the new symmetric bilinear form $(.|.)_\mu$ on $Q'$ 
defined by
\begin{equation*}\label{newform}
(\alpha'|\beta')_\mu = (\alpha'|\sum_{j=0}^{r-1} \mu^j(\beta'))'
\end{equation*}
for all $\alpha',\beta' \in Q'$. 
There is an orthogonal decomposition
$$
\mathfrak h^* = \ocirc{\mathfrak h}{\!\!^*} \oplus
(\C \delta + \C \Lambda_0)
$$
where $\ocirc{\mathfrak h}{\!\!^*} = \bigoplus_{i=1}^\ell \C \alpha_i$,
see \cite[$\S$6.2]{Kac}.
As in {\em loc. cit.} we write $-:\mathfrak h^* \rightarrow
\ocirc{\mathfrak h}{\!\!^*}$ for the orthogonal projection, in particular
$\overline{Q}$ denotes the orthogonal projection of $Q$ onto
$\ocirc{\mathfrak h}{\!\!^*}$.
Define a $\Z$-linear map 
\begin{equation}\label{atend}
\iota:Q' \rightarrow \overline{Q}
\end{equation}
by 
$\iota(\mu^j(\alpha_i')) = 
\overline{\alpha}_i$
for each $i \in I$ and $j \geq 0$.
The kernel of $\iota$ is the space
\begin{equation}
\label{EM}
M' = \{\alpha' - \mu(\alpha')\:|\:\alpha' \in Q'\}
\end{equation}
which is precisely the radical of the bilinear form $(.|.)_\mu$.
Moreover, $\iota$ induces an isometry between $Q' / M'$
and $\overline{Q}$
with respect to the forms induced by $(.|.)_\mu$ and $(.|.)$ 
respectively.

\section{The basic representation}

Next we recall the construction of the basic representation $V = V(\Lambda_0)$
of $\mathfrak g$, following Lepowsky \cite{Lep}.
Let $Z = \langle -1, \omega \rangle \subset \C^\times$ be the multiplicative group generated by $-1$ and $\omega$.
Form the central extension
$$
1 \longrightarrow Z \longrightarrow \widehat Q \stackrel{\pi}{\longrightarrow} 
Q' \longrightarrow 1,
$$
namely,
$\widehat Q = \{e^{\alpha'}_x\:|\:\alpha' \in Q', x \in Z\}$
with multiplication
$$
e^{\alpha'}_x e^{\beta'}_y = 
\left\{
\begin{array}{ll}
e^{\alpha'+\beta'}_{xy \eps(\alpha',\beta')}&\hbox{if $X_N^{(r)} \neq A_{2\ell}^{(2)}$, $D_4^{(3)}$,}\\
e^{\alpha'+\beta'}_{xy \eps(\alpha',\beta')(-\omega)^{-(\alpha'|\mu(\beta'))'}}
&\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$ or $D_4^{(3)}$,}
\end{array}\right.
$$
for $\alpha',\beta' \in Q',\ x,y \in Z$.
The map $\pi:\widehat Q \rightarrow Q'$ here 
is defined by $\pi(e^{\alpha'}_x) = \alpha'$.
Let $\widehat M = \pi^{-1}(M')$, where $M'$ is as in (\ref{EM}).
There is a well-defined 
multiplicative character $\tau:\widehat M \rightarrow \C^\times$
defined in \cite[Proposition~6.1]{Lep} by
$$
\tau(e^{\alpha' - \mu(\alpha')}_x)
= (-1)^{(\alpha'|\alpha')'/2} x \eta(\alpha') \eps(\alpha',\mu(\alpha'))
\omega^{-a_0^2(\alpha'|\alpha')_\mu/2}.
$$
So we can form the induced $\widehat Q$-module
$\C [\widehat Q] \otimes_{\C [\widehat M]} \tau.$
We note the useful formula %for all $\alpha' \in Q'$:
\begin{equation*}
\label{EUseful}
e^{\alpha'}_1 \otimes \tau = 
\eta(\alpha') \omega^{a_0(\alpha'|\alpha')_\mu/2}
e^{\mu(\alpha')}_1 \otimes \tau \qquad (\alpha' \in Q').
\end{equation*}

View the symmetric algebra $S(\mathfrak t^-)$ as a $\mathfrak t$-module
in the unique way so that $c$ acts as $1$, elements of $\mathfrak t^-$ act 
by multiplication, and elements of $\mathfrak t^+$ annihilate $1$.
It is $\Z$-graded by declaring that $$
\deg(h \otimes t^{-n}) = \frac{n}{a_0}
$$
for each $h \in \mathfrak h_{-n}', n \geq 1$.
Let
$$
V = S(\mathfrak t^-) \otimes \C [\widehat Q] \otimes_{\C [\widehat M]} \tau.
$$
Let $\mathfrak t$ 
act on $S(\mathfrak t^-)$ as given 
and trivially on $\C[\widehat Q] \otimes_{\C[\widehat M]} \tau$, 
let $h \otimes t^0$ for $h \in \mathfrak h_0'$ act by
$$
(h \otimes t^0) (f \otimes e_x^{\alpha'} \otimes \tau) = (h|\alpha')' 
f \otimes e_x^{\alpha'}\otimes \tau,
$$
and let $d$ act by
$$
d(f \otimes e_x^{\alpha'} \otimes \tau) = -a_0\left(\deg(f) 
+ (\alpha'|\alpha')_\mu/2\right)f \otimes e_x^{\alpha'} \otimes \tau.
$$
We have now defined the action of ${\mathfrak h} + \mathfrak t$ on $V$.
To extend the action to all of ${\mathfrak g}$, 
let $\alpha' \in Q'$ be a root.
As in \cite[(4.8)]{Lep}, let
$$
\sigma(\alpha')
=
\left\{
\begin{array}{ll}
1&\hbox{$r=1$,}\\
\sqrt{2}^{(\alpha'|\mu(\alpha'))'}&\hbox{if $X_N^{(r)} = A_{2\ell-1}^{(2)}, 
D_{\ell+1}^{(2)}$ or $E_6^{(2)}$,}\\
(1-\omega^{-1})^{(\alpha'|\mu(\alpha'))'}&\hbox{if $X_N^{(r)} = D_4^{(3)}$,}\\
2(1+\omega)^{(\alpha'|\mu(\alpha'))'}&\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$}.
\end{array}
\right.
$$
Also define
$$
P_{\alpha'}(z) = 
\exp\left(\sum_{n \geq 1} \frac{\alpha'(- n) z^n}{n}\right),
\qquad
Q_{\alpha'}(z) = 
\exp\left(-\sum_{n \geq 1} \frac{\alpha'(n) z^n}{n}\right),
$$
viewed as elements of $\End(V)[[z^{\pm 1}]]$.
Let
$$
E_{\alpha'}(z) = \sigma(\alpha')P_{\alpha'}(z) Q_{\alpha'}(z^{-1})
e_1^{\alpha'}
z^{a_0\alpha'} z^{a_0(\alpha'|\alpha')_\mu/2-1}.
$$
Here, $z^{a_0\alpha'}$ denotes the operator with
$$
z^{a_0\alpha'} (f \otimes e_x^{\beta'} \otimes \tau) 
= z^{(a_0\alpha'|\beta')_\mu} f \otimes e_x^{\beta'} \otimes \tau
$$ 
for each $f \in S(\mathfrak t^-)$ and $\beta \in Q'$,  and
$$
e_1^{\alpha'}(f \otimes e_x^{\beta'}\otimes\tau) =
f\otimes (e_1^{\alpha'} e_x^{\beta'}) \otimes \tau.
$$
Expanding $E_{\alpha'}(z)$ in powers of $z$ we get the required action of
$E_{\alpha'}(n) 
\in {\mathfrak g}$  on $V$ for each root $\alpha' \in Q'$ 
and each $n \in \Z$:
$$
E_{\alpha'}(z) = \sum_{n \in \Z} E_{\alpha'}(n) z^{-n-1}.
$$
For a proof that this is a well-defined irreducible representation of
${\mathfrak g}$ in case $r = 1$ see \cite[$\S$14.8]{Kac};
the general case is due to Lepowsky \cite{Lep}.


Let $\C[\overline Q]$ denote the group algebra of $\overline Q$, 
with natural basis
$e^{\alpha}$ for $\alpha \in \overline{Q}$
and multiplication
$e^{\alpha} e^{\beta} = e^{\alpha+\beta}$.
Note $\C[\widehat Q] \otimes_{\C[\widehat M]} \tau$ has
a basis given by the elements $e^{\alpha'}_1 \otimes \tau$ for
all $\alpha' \in \sum_{i \in I} \Z \alpha_i'$.
For such an $\alpha'$, let
$$
\iota(e_1^{\alpha'} \otimes \tau) = 
\left\{
\begin{array}{ll}
e^{\iota(\alpha')}&\hbox{if $X_N^{(r)} \neq A_{2\ell}^{(2)}$, $D_4^{(3)}$,}\\
(-\omega)^{(\alpha'|\mu(\alpha'))'/2}e^{\iota(\alpha')}
&\hbox{if $X_N^{(r)} = D_4^{(3)}$,}\\
\left(\frac{1-\omega}{\sqrt 2} \right)^{(\alpha'|\mu(\alpha'))'}
e^{\iota(\alpha')}
&\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$,}
\end{array}\right.
$$
recalling the map $\iota:Q' \rightarrow \overline{Q}$ defined in (\ref{atend}).
Extending linearly, we obtain a vector space isomorphism
$\iota:\C [\widehat Q] \otimes_{\C [\widehat M]} \tau
 \rightarrow \C[\overline{Q}]$.
For $i = 0,1,\dots,\ell$, we 
define functions $\sigma_i^{\pm}:\overline{Q} \rightarrow \C^\times$ 
by the equation
\begin{equation*}\label{sdef}
\sigma^{\pm}_i(\alpha) e^{\alpha \pm \overline{\alpha}_i}
=
\pm \frac{\sqrt{c_i}}{a_0 d_i}\sigma(\alpha_i') 
\iota(e_1^{\pm \alpha_i'} \iota^{-1}(e^{\alpha}))
\end{equation*}
for all $\alpha \in \overline{Q}$.
The choice of the renormalization map $\iota$ above ensures:

\vspace{1mm}
\begin{Lemma}\label{sigmal}
For all $i = 0,1,\dots,\ell$ and $\alpha \in \overline{Q}$, 
$\sigma_i^{\pm}(\alpha) \in \{\pm 1\}$.
Moreover, for $i \in I$, we have that $\sigma_i^- = - \sigma_i^+$,
and $\sigma_i^+:\overline{Q} \rightarrow \{\pm 1\}$
is a group homomorphism such that
$\sigma_i^+(\overline{\alpha}_j) = \eps(\alpha_i', \alpha_j')$
for each $j \in I$.
\end{Lemma}
\vspace{1mm}

\iffalse
I could not find a nice description of the
signs $\sigma_\eps^{\pm}$ in the twisted cases.
In all types other than $A_{2\ell}^{(2)}$ I did check that
$\sigma_\eps^+ = - \sigma_\eps^-$; in type $A_{2\ell}^{(2)}$
one has instead that $\sigma_\eps^+ = \sigma_\eps^-$.
Here are the descriptions I got if you want to check them:

$r = 1$. $\sigma_0^+$ is a group homomorphism defined by
$\sigma_0^+(\overline{\alpha_j}) = \eps(\alpha_0', \alpha_j')$.

$r = 2$. Write $\alpha_0' = p + \mu(q)$ with $p, q \in \sum_{i=1}^\ell
\Z\alpha_i'$.
Then 
$\sigma_0^+(\iota(\alpha')) = \eps(\alpha_0',\alpha')
\eps(\alpha'+p, q+\mu(q)) (-1)^{(q| q+\mu(q))'/2}$ for $\alpha' \in \sum_{i=1}^\ell \Z \alpha_i'$.

$r=3$. 
For $\alpha' \in \Z \alpha_1' + \Z \alpha_2'$,
$\sigma_0^+(\iota(\alpha')) = \eps(\alpha_0', \alpha')
\eps(\alpha', \alpha_3'+\alpha_4') \eps(\alpha_1', \alpha_3')$.

$r=4$.
For $\alpha' \in \Z \alpha_0'+\dots+\Z\alpha_{\ell-1}'$,
$\sigma_\ell^+(\iota(\alpha')) = (-1)^{\ell-1} \eps(\alpha_\ell', \alpha')
\eps(\alpha'+p, \alpha_\ell')$ where $p = \alpha_0'+\dots+\alpha_{\ell-1}'$.
\fi

Now we can rewrite the construction of the basic representation $V$
in terms of the Chevalley generators.
We will identify 
$$
V = S(\mathfrak t^-) \otimes \C[\widehat{Q}]\otimes_{\C[\widehat{M}]} \tau 
= S(\mathfrak t^-) \otimes \C[\overline{Q}]
$$
via the map $\id\otimes\iota$.
Then,
the actions of $h_i$ for $i = 0,\dots,\ell$ and of $d$ are as
\begin{align*}
h_i (f \otimes e^{\alpha}) &= (\delta_{i,0} + \langle h_i, \alpha \rangle) f \otimes e^{\alpha},\\
d(f \otimes e^{\alpha})&= - a_0\left(\deg(f) +  
(\alpha|\alpha)/2\right) f \otimes e^{\alpha}
\end{align*}
for all $\alpha \in \overline{Q}$.
In particular, we note from this that
\begin{equation}
\label{EWeight}
\wt(f \otimes e^{\alpha}) = 
\Lambda_0 + \alpha  - \left(\deg(f) +
(\alpha|\alpha)/2
\right)\delta
\end{equation}
for each homogeneous $f \in S(\mathfrak t^-)$ and $\alpha \in \overline{Q}$. 
This shows that $1 \otimes e^{0}$ is a highest weight vector
in $V$ of highest weight $\Lambda_0$ (cf. \cite[Lemma 12.6]{Kac}), 
identifying $V$ with the irreducible
highest weight module $V(\Lambda_0)$.
Finally, for $i = 0,\dots,\ell$,
\begin{align}
e_i(z) &= \sum_{n \in \Z} e_i(n) \otimes z^{-n-1} =
P_{\alpha_i'}(z) Q_{\alpha_i'}(z^{-1}) e^{\overline{\alpha}_i} z^{a_0 {\alpha}_i} 
z^{{a_0}({\alpha}_i|{\alpha}_i)/2 - 1} s^+_i,\label{v1}\\\label{v2}
f_i(z) &= \sum_{n \in \Z} f_i(n) \otimes z^{-n-1} =
P_{-\alpha_i'}(z) Q_{-\alpha_i'}(z^{-1}) e^{-\overline{\alpha}_i} z^{-a_0{\alpha}_i} 
z^{{a_0}({\alpha}_i|{\alpha}_i)/2 - 1} s^-_i,
\end{align}
where
\begin{align*}
z^{\pm a_0{\alpha}_i} (f \otimes e^{\beta})
&= z^{(\pm a_0{\alpha}_i|\beta)} f \otimes e^{\beta},\\
s^{\pm}_i(f \otimes e^{\beta}) &= \sigma^{\pm}_i(\beta) f \otimes e^{\beta},\\
e^{\pm\overline{\alpha}_i}(f \otimes e^{\beta}) &= 
f \otimes e^{\beta \pm \overline{\alpha}_i}.
\end{align*}
The following lemma will be needed later on:

\vspace{1mm}
\begin{Lemma}\label{bigprod}
For $i_1,\dots,i_s \in \{0,\dots,\ell\}$,
roots $\beta_1',\dots,\beta_t' \in Q'$
and 
$\gamma\in \overline{Q}$, we have that
\begin{eqnarray*}
& & 
\hspace {-1.3 cm}e_{i_1}(z_1)e_{i_2}(z_2)
\dots e_{i_s}(z_s) P_{\beta_1'}(w_1) \dots P_{\beta_{t}'}(w_t)
\otimes e^\gamma = \\
& & 
\pm 
%\left(
\prod_{1\leq u \leq s} 
z_u^{\frac{a_0}{2}(\alpha_{i_u}|\alpha_{i_u}) - 1 +
a_0 (\alpha_{i_u}|\gamma)}
%\right)
%\qquad\qquad\qquad\qquad\qquad
\\
%\:\quad\qquad
& &
\times 
%\left(
\prod_{1 \leq u  < v \leq s}
z_u^{a_0(\alpha_{i_u}|\alpha_{i_v})}
\prod_{k\in \Z/m}\left(1 - \omega^{-k} 
\frac{z_v}{z_u}\right)^{(\mu^k(\alpha_{i_u}')|\alpha_{i_v}')'}
%\right)
\\
%\qquad
& &
\times 
%\left(
%\prod_{\stackrel{\hbox{$_{1 \leq u \leq s}$}}{1 \leq v \leq t}}
\prod_{1\leq u \leq s}\prod_{1 \leq v \leq t}
\prod_{k \in \Z / m}
\left(1 - \omega^{-k} \frac{w_v}{z_u}\right)^{(\mu^k(\alpha_{i_u}')|\beta_v')'}
%\right)
\\
& &
\times P_{\alpha_{i_1}'}(z_1) \dots P_{\alpha_{i_s}'}(z_s)
P_{\beta_1'}(w_1) \dots P_{\beta_t'}(w_t) \otimes 
e^{\gamma+\overline{\alpha}_{i_1} + \dots +\overline{\alpha}_{i_s}}.\quad
\end{eqnarray*}
A similar formula holds for $f_{i_1}(z_1) \dots f_{i_s}(z_s)
P_{\beta_1'}(w_1) \dots P_{\beta_{s}'}(w_s)
\otimes e^\gamma$, replacing 
$\alpha_{i_u}$ by $- \alpha_{i_u}$,
$\alpha_{i_u}'$ by $- \alpha_{i_u}'$, and $\bar\alpha_{i_u}$ by $- \bar\alpha_{i_u}$
everywhere.
\end{Lemma}

\begin{proof}
This follows from the following commutation relation
obtained in \cite[3.4]{Lep}: for $\alpha',\beta' \in Q'$,
\begin{equation*}\label{useful}
Q_{\alpha'}(z^{-1}) P_{\beta'}(w)
= 
P_{\beta'}(w)Q_{\alpha'}(z^{-1}) 
\prod_{k\in \Z / m} 
(1 - \omega^{-k} \frac{w}{z})^{(\mu^k(\alpha')|\beta')'},
\end{equation*}
which is a consequence of the Campbell-Hausdorff formula,
cf. \cite[(14.8.12)]{Kac}.
\end{proof}

\section{The Integral form}

As in the introduction, 
let $U_\Z$ denote the $\Z$-subalgebra of the universal enveloping
algebra of ${\mathfrak g}$ generated by the elements
$e_i^r / r!, f_i^r / r!$ for $i = 0,\dots,\ell$ and $r \geq 0$, and
let 
$$
V_\Z := U_\Z (1 \otimes e^0) \subset V.
$$ 
In this section, we will give an explicit description
of $V_\Z$.

To start with,
let $\tau:\mathfrak g \rightarrow \mathfrak g$ be the antilinear Chevalley 
antiautomorphism, so 
$$
\tau(d) = d,\quad 
\tau(e_i(n)) = f_i(-n),\quad \tau(f_i(n)) = e_i(-n)
$$
for each $i = 0,\dots,\ell$ and $n \in \Z$, cf. \cite[$\S\S$7.6, 8.3]{Kac}. 
The {\em Shapovalov form}\, $(.|.)_S$ on $V$ is the unique Hermitian form
such that
$$
(1 \otimes e^{0}, 1 \otimes e^{0})_S = 1\quad \text{and} \quad
(x v, w)_S = (v, \tau(x) w)_S
$$ 
for all $v,w \in V$, $x \in \mathfrak g$.
The restriction of $\tau$ to $\mathfrak t$ gives the antilinear Chevalley
antiautomorphism of $\mathfrak t$, and we can also 
consider the Shapovalov form on $S(\mathfrak t^-)$,
satisfying
$(1,1)_S = 1$ and $(x f, g)_S = (f, \tau(x) g)_S$
for all $f,g \in S(\mathfrak t^-)$, $x \in \mathfrak t$.

\vspace{1mm}
\begin{Lemma}\label{one} For all $f,g \in S(\mathfrak t^-)$ and
$\alpha,\beta \in \overline{Q}$,
$(f \otimes e^{\alpha}, g \otimes e^{\beta})_S
= (f,g)_S.$
\end{Lemma}

\begin{proof}
Since different weight spaces are orthogonal and in view of (\ref{EWeight}), 
this reduces to checking that
$(1 \otimes e^{\alpha}, 1 \otimes
e^{\alpha})_S = 1$ for all $\alpha \in \overline{Q}$.
Proceeding by induction, we may assume that
there is some $\beta \in \overline{Q}$ and $i \in I$ such that
$(1 \otimes e^{\beta},1 \otimes e^{\beta})_S = 1$
and 
either $\alpha = \beta + \overline{\alpha}_i$
or $\alpha = \beta - \overline{\alpha}_i$.

Suppose that $\alpha = \beta + \overline{\alpha}_i$.
Letting $n = - a_0(\alpha_i|\beta) - a_0(\alpha_i|\alpha_i)/2$,
one checks easily using (\ref{v1}), (\ref{v2}) that
\begin{equation}\label{inpf}
e_i(n) (1 \otimes e^{\beta}) = \sigma_i^+(\alpha)( 1 \otimes e^{\alpha}),\qquad
f_i(-n) (1 \otimes e^{\alpha}) = \sigma_i^-(\beta) (1 \otimes e^{\beta}).
\end{equation}
Hence, 
\begin{align*}
(1 \otimes e^{\alpha}, 1 \otimes e^{\alpha})_S
&=
\sigma^+_i(\beta)(e_i(n)(1 \otimes e^{\beta}),1 \otimes e^{\alpha})_S\\
&=
\sigma^+_i(\beta)(1 \otimes e^{\beta}, f_i(-n)(1 \otimes e^{\alpha}))_S\\
&=
\sigma_i^-(\alpha) \sigma_i^+(\beta) (1 \otimes e^{\beta}, 1 \otimes 
e^{\beta})_S = 1,
\end{align*}
since $\sigma_i^-(\alpha) = \sigma_i^+(\beta)$ by Lemma~\ref{sigmal}.

A similar argument in the case that $\alpha = \beta - \overline{\alpha}_i$
completes the proof.
\end{proof}


\begin{Lemma}\label{two}
For all $i =0,1,\dots,\ell$ and $n \in \Z$, the elements
$e_i(n)$ and $f_i(n)$ belong to $U_\Z$.
\end{Lemma}

\begin{proof}
Suppose that $e_i(n) \neq 0$.
Then, $\wt(e_i(n)) = \overline{\alpha}_i + \frac{n}{a_0} \delta$ is a real
root, hence is conjugate under the Weyl group $W$ associated to
$\mathfrak g$ to some simple root $\alpha_j$.
So we can find simple reflections $s_{i_1},\dots,s_{i_t} \in W$
such that $\overline{\alpha}_i + \frac{n}{a_0} \delta
= s_{i_1} \dots s_{i_t} \alpha_j$.
Let $r^\ad_i$ be the automorphism of $\mathfrak g$
defined by
$r_i^\ad = \exp (\ad f_i) \exp (-\ad e_i) \exp (\ad f_i)$, for $i = 0,1,\dots,\ell$.
Since real root spaces of $\mathfrak g$ are one dimensional,
$$
r_{i_1}^\ad \dots r_{i_t}^{\ad} e_j = c e_i(n)
$$
for some non-zero scalar $c$. 
Now, $\tau(\exp(\ad y) (x))=\exp(-\ad\tau(y))(\tau(x))$, whence by an $SL_2$-calculation we have $r_i^\ad (x) = \tau(r_i^\ad(\tau (x)))$ for all $x \in 
\mathfrak g$, we also get that
$$
r_{i_1}^\ad \dots r_{i_t}^{\ad} f_j = c f_i(-n).
$$
But the $r_i^\ad$ preserve the normalized invariant form on $\mathfrak g$,
so
$$
a_j (a_j^{\vee})^{-1} = (e_j| f_j) = (c e_i(n)|c f_i(-n)) = c^2
a_i (a_i^{\vee})^{-1}.
$$
Clearly, $\alpha_i$ and $\alpha_j$ are roots of the same length, i.e.
$a_j (a_j^{\vee})^{-1} = 
a_i (a_i^{\vee})^{-1}$, so this gives that $c = \pm 1$.
Finally, the action of $r^\ad_i$ on $\mathfrak g$ 
leaves $U_\Z \cap \mathfrak g$ invariant, so
$$
e_i(n) = \pm r^\ad_{i_1} \dots r^\ad_{i_s} e_j \in U_\Z,
$$
and similarly $f_i(n) \in U_\Z$ too.
\end{proof}

For $n \geq 1$ and $i = 0,1,\dots,\ell$, define
$$
y_{nd_i}^{(i)} = \frac{\alpha_i'(-a_0n d_i)}{a_0 nd_i},
\qquad
x_{nd_i}^{(i)} = \sum_{k_1+2k_2+\dots = n}
\frac{{y_{d_i}^{(i)}}^{k_1}}{k_1!}
\frac{{y_{2d_i}^{(i)}}^{k_2}}{k_2!}
\frac{{y_{3d_i}^{(i)}}^{k_3}}{k_3!}\cdots.
$$
Observe that
\begin{equation}
\label{EYX}
P_{\alpha_i'}(z) = \exp\left(\sum_{n \geq 1} y_{nd_i}^{(i)} z^{a_0 n d_i}
\right)
= 1 + \sum_{n \geq 1} x_{nd_i}^{(i)} z^{a_0 n d_i}.
\end{equation}
The $y_{nd_i}^{(i)}$ for $n \geq 1$ and $i \in I$
give a basis
for $\mathfrak t^-$. So
$S(\mathfrak t^-)$ is equal to the
free polynomial algebra 
$$
B := \C[y_{nd_i}^{(i)}\:|\:n \geq 1, i \in I].
$$
Since the $x_{nd_i}^{(i)}$ are related to the $y_{nd_i}^{(i)}$ 
in a unitriangular way,
we obtain a $\Z$-form
$$
B_\Z := \Z[x_{nd_i}^{(i)}\:|\:n \geq 1, i \in I]
\subset B
$$
for $B$.
As $\al_\eps'$ is an integral linear combination of the $\al_i'$ with $i\in I$,
it follows from (\ref{EYX}) that the elements $x_{nd_\eps}^{(\eps)}$
also belong to the lattice $B_\Z$.
The $\Z$-grading on $B_\Z$ induced by the grading on $S(\mathfrak t^-)$
is determined by $\deg(y_{n}^{(i)}) = \deg(x_{n}^{(i)}) = n$.

The following theorem (or rather its $q$-analogue) for the non-twisted 
case has been proved in \cite{CJ}. Our argument for the general case is 
similar.

\vspace{1mm}
\begin{Theorem}\label{zform}
$V_\Z = B_\Z \otimes_{\Z} \Z[\overline{Q}]$.
\end{Theorem}

\begin{proof}
Let us first show that
$B_\Z \otimes_{\Z} \Z[\overline{Q}] \subseteq V_\Z$.
Fix $i_1,\dots,i_s \in I$, and let
$$
M(i_1,\dots,i_s) = \{(n_1,\dots,n_s)\:|\:n_1 \geq \dots \geq n_s \geq 0
\hbox{ and }
d_{i_j} | n_j\hbox{ for all $j=1,\dots,s$}\}.
$$
Denote by $>$ the dominance ordering on partitions belonging
to $M(i_1,\dots,i_s)$.
We will show that
$x_{n_1}^{(i_1)} \dots x_{n_s}^{(i_s)} \otimes e^\beta \in V_\Z$
for all $(n_1,\dots,n_s) \in M(i_1,\dots,i_s)$ and each
$\beta \in \overline{Q}$.
Clearly every monomial in $B_\Z$ is of the form
$x_{n_1}^{(i_1)} \dots x_{n_s}^{(i_s)}$ for some choice
of $i_1,\dots,i_s$ and $(n_1,\dots,n_s) \in M(i_1,\dots,i_s)$,
so this is good enough.

To start with, each $e_i(n), f_i(-n) \in U_\Z$ by Lemma~\ref{two}.
So an obvious inductive argument using (\ref{inpf})
gives that $1 \otimes e^\gamma \in V_\Z$
for each $\gamma \in \overline{Q}$.
Hence, letting $\gamma = \beta - \bar\alpha_{i_1} - \dots - \bar\alpha_{i_s}$,
Lemma~\ref{two} implies that 
all coefficients of $e_{i_1}(z_1) \dots e_{i_s}(z_s) \otimes e^\gamma$
belong to $V_\Z$.
Applying Lemma~\ref{bigprod}, we deduce that all the coefficients of
$$
X := \left(\prod_{1 \leq u  < v \leq s}
\prod_{k\in \Z/m}
\left(1 - \omega^{-k} \frac{z_v}{z_u}\right)^{(\mu^k(\alpha_{i_u}')|\alpha_{i_v}')'}
\right)
P_{\alpha_{i_1}'}(z_1) \dots P_{\alpha_{i_s}'}(z_s) \otimes 
e^{\beta}
$$
belong to $V_\Z$.
One checks that in all cases,
$$
\prod_{k\in \Z/m}
\left(1 - \omega^{-k} \frac{z_v}{z_u}\right)^{(\mu^k(\alpha_{i_u}')|\alpha_{i_v}')'}
= 1 + (*),
$$
where ($*$) is a $\Z$-linear combination of $\left(\frac{z_v}{z_u} \right)^p$
for $p \geq 1$.
It follows that the $z_1^{a_0n_1} \dots z_s^{a_0 n_s}$-coefficient
of $X$ equals
$$
x_{n_1}^{(i_1)} \dots x_{n_s}^{(i_s)}\otimes e^\beta + (**),
$$
where ($**$) is a $\Z$-linear combination
of $x_{n_1'}^{(i_1)} \dots x_{n_s'}^{(i_s)}\otimes e^\beta$ for
$(n_1',\dots,n_s') > (n_1,\dots,n_s)$.
Using downward induction on this ordering,
we deduce that $x_{n_1}^{(i_1)} \dots x_{n_s}^{(i_s)} \in V_\Z$.

Finally, we prove that $B_\Z \otimes_{\Z} \Z[\overline{Q}] \supseteq V_\Z$.
As the high weight vector $1\otimes e^0$ belongs to 
$B_\Z \otimes_{\Z} \Z[\overline{Q}] \supseteq V_\Z$, it 
suffices to show that $B_\Z \otimes_{\Z} \Z[\overline{Q}]$ is invariant 
under each of the
operators $f_i(n)^s / s!$ for $n \in \Z, s \geq 1$
and $i = 0,1,\dots,\ell$. Fix $i\in\{0,1,\dots,\ell\}$
and consider
$$
Y := f_i(z_1)\dots f_i(z_s) P_{\alpha_{i_1}'}(w_1) \dots P_{\alpha_{i_t}'}(w_t)
\otimes e^{\gamma+s \overline{\alpha}_i}
$$
for $i_1,\dots,i_t \in I$ and $\gamma \in \overline{Q}$.
Applying Lemma~\ref{bigprod} and simplifying, 
\begin{eqnarray*}
Y & = & 
\pm (z_1 \dots z_s)^{\frac{a_0}{2}(\alpha_{i}|\alpha_{i}) - 1 -
a_0 (\alpha_{i}|\gamma + s \overline{\alpha}_i)} \\
& & 
\times 
%\left(
\prod_{1 \leq u  < v \leq s}
\prod_{k\in \Z/m}
\left(z_u - \omega^{-k} z_v\right)^{(\mu^k(\alpha_{i}')|\alpha_{i}')'}
%\right)
\\
& & \times 
%\left(
\prod_{\stackrel{\hbox{$_{1 \leq u \leq s}$}}{1 \leq v \leq t}}
\prod_{k \in \Z / m}
\left(1 - \omega^{-k} \frac{w_v}{z_u}\right)^
{-(\mu^k(\alpha_{i}')|\alpha_{i_v}')'}
%\right)
\\
& &
\times P_{-\alpha_{i}'}(z_1) \dots P_{-\alpha_{i}'}(z_s)
P_{\alpha_{i_1}'}(w_1) \dots P_{\alpha_{i_t}'}(w_t) \otimes 
e^{\gamma}.
\end{eqnarray*}
Certainly, each coefficient of 
$P_{-\alpha_{i}'}(z_1) \dots P_{-\alpha_{i}'}(z_s)P_{\alpha_{i_1}'}(w_1) \dots P_{\alpha_{i_t}'}(w_t) \otimes 
e^{\gamma}$ belongs to $V_\Z$.
One checks that
$$
\prod_{k\in \Z/m}
\left(z_u - \omega^{-k} z_v\right)^{(\mu^k(\alpha_{i}')|\alpha_{i}')'}
= \left\{
\begin{array}{ll}
(z_u - z_v)^2 \frac{(z_u+z_v)^2}{z_u^2+z_v^2}&\hbox{if $X_N^{(r)} = A_{2\ell}^{(2)}$ and $i = 0$},\\
(z_u^{a_0 d_i} - z_v^{a_0 d_i})^2&\hbox{otherwise.}
\end{array}
\right.
$$
Hence in all cases, $Y$ 
looks like $\prod_{1 \leq u < v \leq s} (z_u - z_v)^2$ times an expression that is symmetric in $z_1,\dots,z_s$.
Hence, by \cite[Lemma 2.5(ii)]{CJ}, the coefficient of $(z_1\dots z_s)^{-n-1}$
in $Y$ is divisible by $s!$. Hence,
all coefficients of 
$(f_i(n)^s / s!) P_{\alpha_{i_1}'}(w_1) \dots P_{\alpha_{i_s}'}(w_s)
\otimes e^{\gamma - s \overline{\alpha}_i}$
belong to $V_\Z$, which completes the proof.
\end{proof}

\section{The determinant}

Fix now some $d \geq 0$.
Lemma~\ref{one} and Theorem~\ref{zform} reduce the problem of computing the
determinant of the Shapovalov form on the $(w \Lambda_0 - d \delta)$ weight
space of $V_\Z$ for any $w \in W$ 
to the problem of computing the determinant of
the Shapovalov form on the degree $d$ component of $B_\Z$.
To tackle the latter question, observe
$$
B = \bigotimes_{i \in I}
\C[y_{d_i}^{(i)},y_{2d_i}^{(i)}, \dots],
\qquad
B_\Z = \bigotimes_{i \in I}
\Z[x_{d_i}^{(i)},x_{2d_i}^{(i)}, \dots].
$$
So according to \cite[Corollary 2.1]{CKK}, there is a well-defined Hermitian 
form on $B$ determined by the rules
$$
(1,1)_K = 1,
\qquad
(n y_{nd_i}^{(i)} f, g)_K = (f, \frac{\partial}{\partial y_{nd_i}^{(i)}} g)_K
$$
for all $i \in I, n \geq 1$ and $f,g \in B$.
Moreover, there is a homogeneous 
basis for the lattice $B_\Z$ (given by Schur polynomials)
that is orthonormal with respect to the form $(.\,,.)_K$.
In particular, the determinant of the form $(.\,,.)_K$ on the degree $d$
 component of $B_\Z$ is equal to $1$.
Our strategy will therefore be to relate the Shapovalov form $(.\,,.)_S$ to
the form $(.\,,.)_K$.

Define $I(n) = \{i \in I\:|\:d_i | n\}$.
Introduce the matrices $A^{(n)} = (a_{i,j}^{(n)})_{i,j \in I(n)}$ 
with
$$
a_{i,j}^{(n)} = \frac{1}{d_i} 
(\alpha_i'|\sum_{k=0}^{r-1}\omega^{a_0nk} \mu^k(\alpha_j'))'.
$$
Recall $\alpha,\beta$ from the introduction. One verifies:

\vspace{1mm}
\begin{Lemma}\label{f0}
For any $n \geq 0$, we have 
$
\det A^{(n)} = \left\{
\begin{array}{ll}
\alpha&\hbox{if $r \mid n$,}\\
\beta&\hbox{if $r \nmid n$,}
\end{array}
\right.
$
\end{Lemma}
\vspace{1mm}

The significance of the matrices $A^{(n)}$ is that for $i \in I(n)$, 
the element
$\tau(n y_{n}^{(i)} / d_i) = \alpha_i'(a_0 n) / a_0 d_i
$
acts on $B$ as the operator
$\sum_{j \in I{(n)}} a^{(n)}_{i,j} 
\frac{\partial}{\partial y_{n}^{(j)}}.$
We set
$$
z_n^{(i)} = \sum_{j \in I(n)} a_{i,j}^{(n)} y_n^{(j)}.
$$
For a partition $\lambda = (\lambda_1 \geq \dots \geq \lambda_h > 0)$
we let $I(\lambda) = I(\lambda_1) \times \dots \times I(\lambda_h)$.
Given $\bi = (i_1,\dots,i_h) \in I(\lambda)$, let
$$
x_\lambda^{(\bi)} = x_{\lambda_1}^{(i_1)}
\dots x_{\lambda_h}^{(i_h)},\qquad
y_\lambda^{(\bi)} = y_{\lambda_1}^{(i_1)}
\dots x_{\lambda_h}^{(i_h)},\qquad
z_\lambda^{(\bi)} = z_{\lambda_1}^{(i_1)}
\dots x_{\lambda_h}^{(i_h)},
$$
all elements of $B$ of degree $|\lambda|$.

\vspace{1mm}
\begin{Lemma}\label{f2}
For $\bi \in I(\lambda)$ and any $f \in B$, we have 
$(y_\lambda^{(\bi)}, f)_S
= (z_\lambda^{(\bi)}, f)_K$.
\end{Lemma}

\begin{proof}
Proceed by induction on the number of non-zero parts of $\lambda$, starting 
induction from the obvious fact that $(1, f)_S=(1, f)_K$.
For the induction step, note that for $i \in I(n)$,
\begin{align*}
(y_{n}^{(i)} y_\lambda^{(\bi)}, f)_S
&=
(\frac{d_i}{n} y_\lambda^{(\bi)},
\sum_{j\in I(n)} {a_{i,j}^{(n)}}
\frac{\partial}{\partial y_{n}^{(j)} }f)_S\\
&=
(\frac{d_i}{n} z_\lambda^{(\bi)},
\sum_{j\in I(n)} {a_{i,j}^{(n)}}
\frac{\partial}{\partial y_{n}^{(j)} }f)_K
=
(z_{n}^{(i)} z_\lambda^{(\bi)}, f)_K.
\end{align*}
\end{proof}


Let
$$
\Omega(\lambda) = \{\bi \in I(\lambda)\:|\:
i_j \leq i_{j+1}\text{ whenever }\lambda_j = \lambda_{j+1}\}.
$$
Then
$\{x_{\lambda}^{(\bi)}\:|\:\lambda \in \mathscr P(d), \bi \in \Omega(\lambda)\}$, 
$\{y_{\lambda}^{(\bi)}\:|\:\lambda \in \mathscr P(d), \bi \in 
\Omega(\lambda)\}$
and
$\{z_{\lambda}^{(\bi)}\:|\:\lambda \in \mathscr P(d), 
\bi \in \Omega(\lambda)\}$ 
give three different bases for
the degree $d$ component of $B$.
Consider the transition matrices
$P = (p_{\lambda,\mu}^{\bi,\bj})$
and 
$Q = (q_{\lambda,\mu}^{\bi,\bj})$
where $\lambda,\mu \in \mathscr P(d)$, $\bi \in \Omega(\lambda), 
\bj \in \Omega(\mu)$
defined from
$$
x_\lambda^{(\bi)} = \sum_{\mu \in \mathscr P(d),\ \bj \in \Omega(\mu)}
p_{\lambda,\mu}^{\bi,\bj} y_\mu^{(\bj)},
\qquad
z_\lambda^{(\bi)} = \sum_{\mu \in \mathscr P(d),\ \bj \in \Omega(\mu)}
q_{\lambda,\mu}^{\bi,\bj} y_\mu^{(\bj)}.
$$

\begin{Lemma}\label{f1} %Let $P$ and $Q$ be the transition matrices
%defined above.
%\begin{itemize}
%\item[(i)]
%The matrix $P$ is block unitriangular, i.e.
%$p_{\la,\la}^{\bi,\bj} = \delta_{\bi,\bj}$ and
%$p_{\la,\mu}^{\bi,\bj} = 0$ if $\mu\not\leq \la$ in the usual dominance
%ordering on partitions.
%\item[(ii)] 
The matrix $Q$ is block diagonal, i.e.
$q_{\la,\mu}^{\bi,\bj} = 0$ for $\la \neq \mu$.
Moreover, the determinant of the $\la$-block 
$Q_\la = (q^{\bi,\bj}_{\la,\la})_{\bi,\bj \in \Omega(\la)}$ of $Q$
is $\alpha^{a_\lambda}\beta^{b_\lambda}$,
notation as in the introduction.
%\end{itemize}
\end{Lemma}

\begin{proof}
%(i) This follows since by definition
%$x_n^{(i)} = y_n^{(i)} + \hbox{(lower terms)}.$
%
%(ii) 
It is immediate from the definition that
$q_{\la,\mu}^{\bi,\bj} = 0$ for $\la \neq \mu$.
So consider the $\la$-block $Q_\la$ of $Q$.
Represent $\la = (\lambda_1 \geq \dots \geq \lambda_h > 0)$
instead as $(1^{r_1} 2^{r_2} \dots s^{r_s})$.
%Let $G$ be the naturally embedded parabolic subgroup
%$\Sym_{r_s}\times \dots \times \Sym_{r_2} \times \Sym_{r_1}$ of the
%symmetric group $\Sym_{h}$.
%Note $G$ acts naturally on $I(\lambda)$,
%by $g \cdot (i_1,\dots,i_h) = (i_{g^{-1}1}, \dots, i_{g^{-1}h}).$
%For $\bi\in I(\lambda)$, we let $G_{\bi}$ be the stabilizer in 
%$G$ of $\bi$, and
%choose a set $G^{\bi}$ of representatives for the cosets $G / G_\bi$.
By definition,
$$
z_\la^{(\bi)} = \sum_{\bj \in I(\lambda)}
a^{(\lambda_1)}_{i_1,j_1} \dots a^{(\lambda_h)}_{i_h,j_h} y_{\la}^{(\bj)}.
$$
%Hence, since $\Omega(\lambda)$ is a set of representatives for the $G$-orbits
%on $I(\lambda)$,
%we see that
%$$
%q_{\lambda,\lambda}^{\bi,\bj}
%= \sum_{g \in {G^\bj}} 
%a^{(\lambda_1)}_{i_1,j_{g^{-1}1}} \dots a^{(\lambda_h)}_{i_h,j_{g^{-1}h}}.
%$$
Thus $Q_\lambda$ is the matrix
$S^{r_1} (A^{(1)}) \otimes \dots \otimes S^{r_s}(A^{(s)})$,
a tensor product of symmetric powers of the matrices $A^{(n)}$.
Now, note that for an $n \times n$ matrix $A$,
$$
\det S^m(A) = (\det A)^{\binom{n+m-1}{n}},
$$
while for an $n \times n$ matrix $B$ and an $m \times m$ matrix $C$,
$$
\det(B \otimes C) = (\det B)^m (\det C)^n.
$$
These are both proved by reducing to the case that the matrices are diagonal.
Combining the formulae with Lemma~\ref{f0}, one computes
$\det Q_\lambda = \alpha^{a_\lambda} \beta^{b_\lambda}$.
\end{proof}

Now we can prove the main theorem:

\vspace{1mm}
\begin{Theorem}
The determinant of the restriction of the Shapovalov form to the
degree $d$ part of $B_\Z$ is
$\prod_{\lambda \in \mathscr P(d)}\alpha^{a_\lambda} \beta^{b_\lambda}$.
\end{Theorem}

\begin{proof}
Consider the matrices
$M = (m_{\la,\mu}^{\bi,\bj})$
and
$N = (n_{\la,\mu}^{\bi,\bj})$
for $\la,\mu \in \mathscr P(d)$, $\bi \in \Omega(\la)$, $\bj \in \Omega(\mu)$
defined from
$$
m_{\la,\mu}^{\bi,\bj} = (x_\la^{(\bi)}, x_\mu^{(\bj)})_S,
\qquad
n_{\la,\mu}^{\bi,\bj} = (x_\la^{(\bi)}, x_\mu^{(\bj)})_K.
$$
Recalling the transition matrices $P$ and $Q$ introduced above,
Lemma~\ref{f2} gives at once that
$M = P Q P^{-1} N.$
On the other hand, $N$ has determinant $1$,
since as we observed above 
the degree $d$ component of 
$B_\Z$ admits an  orthonormal basis with respect to the 
contravariant form $(.\,,.)_K$ (see \cite[Corollary 2.1]{CKK}).
So we can compute $\det M$ at once using Lemma~\ref{f1}.
\end{proof}

We finally indicate how to deduce the generating functions
$a(q)$ and $b(q)$ stated in the introduction.
By definition, $a_\lambda$ is  $$
\frac{h}{\ell}
\times \left(
\begin{array}{l}
\hbox{the number of ways of coloring}\\
\hbox{the parts $\la_i \equiv 0 \:(r)$ with $\ell$ colors}
\end{array}
\right)
\times
\left(
\begin{array}{l}
\hbox{the number of ways of coloring}\\
\hbox{the parts $\la_i \not\equiv 0 \:(r)$ with $k$ colors}
\end{array}
\right)
$$
where $h$ is the number of $\la_i \equiv 0\:(r)$.
Consider
$$
G(q,t,u) = 
\left(\prod_{n \geq 1} \frac{1}{1-q^{nr}t}\right)^{\ell}
\left(\prod_{n \geq 1} \frac{1-q^{nr} u}{1-q^{n}u}
\right)^k.
$$
The coefficient of $q^d t^h u^i$ is equal to the number of 
partitions of $d$ with $h$ parts divisible by $r$ colored by $\ell$
different colors and with $i$ parts not divisible by $r$ colored
by $k$ different colors.
Hence the generating function $a(q)$ 
for $a(d) = \sum_{\lambda \in\mathscr P(d)} a_\lambda$ is equal to 
$\frac{1}{\ell} \frac{d}{dt} G(q,t,u)|_{t=u=1}$.
Similarly, $b(q) = \frac{1}{k} \frac{d}{du} G(q,t,u)|_{t=u=1}$.


\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi

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\parskip -0.5mm

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\end{thebibliography}
\end{document}


math0111306.extracted_bib
\begin{thebibliography}{BKS}
\parskip -0.5mm

\bibitem{Ariki} S. Ariki,
On the decomposition numbers of the Hecke algebra of type $G(m,1,n)$,
{\em J. Math. Kyoto Univ.} {\bf 36} (1996), 789--808.

\bibitem{BO}
C. Bessenrodt and J. Olsson,
A note on Cartan matrices for symmetric groups,
preprint, 2001.

%\bibitem{BKProj}
%J. Brundan and A. Kleshchev,
%Projective representations of the symmetric group via Sergeev
%duality, to appear in {\em Math. Z.}.

\bibitem{BKcrys}
J. Brundan and A. Kleshchev,
Hecke-Clifford superalgebras, crystals of type $A_{2\ell}^{(2)}$
and modular branching rules for $\widehat S_n$,
{\em Representation Theory} {\bf 5} (2001), 317--403.

\bibitem{CJ}
V. Chari and N. Jing,
Realization of level one representations of
$U_q(\widehat{\mathfrak g})$ at a root of unity,
{\em Duke Math. J.} {\bf 108} (2001), 183--197.

\bibitem{CKK}
C. De Concini, V. Kac and D. Kazhdan,
``Boson-Fermion correspondence over $\Z$'',
in:
{Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988)}, 
{\em Adv. Ser. Math. Phys.}
{\bf 7} (1989), 
124--137.

\bibitem{CR}
C. Curtis and I. Reiner, 
{\em Representation theory of finite groups and associative algebras},
Wiley, 1988.

\bibitem{Donkin}
S. Donkin, Representations of Hecke algebras and characters of symmetric
groups, preprint, 2001.

\bibitem{KF}
I. Frenkel and V. Kac,
Basic representations of affine Lie algebras and dual resonance modules,
{\em Invent. Math.} {\bf 62} (1980), 23--66.

%\bibitem{Garland}
%H. Garland, The arithmetic theory of loop algebras,
%{\em J. Algebra} {\bf 53} (1978), 480--551.

\bibitem{G}
I. Grojnowski, Affine $\widehat{\mathfrak{sl}}_p$ controls the modular
representation theory of the symmetric group and related Hecke algebras,
preprint, 1999.

\bibitem{Hum}
J.~F. Humphreys, Blocks of projective representations of the 
symmetric groups, {\em J. London Math. Soc.} {\bf 33} (1986), 441--452.

%\bibitem{Jing}
%N. Jing,
%Twisted vertex representations of quantum affine algebras,
%{\em Invent. Math.} {\bf 102} (1990), 663--690.

\bibitem{Kac}
V. Kac,
{\em Infinite dimensional Lie algebras},
Cambridge University Press, third edition, 1995.

\bibitem{Lep}
J. Lepowsky,
{Calculus of twisted vertex operators},
{\em Proc. Natl. Acad. Sci. USA} {\bf 82} (1985), 8295--8299.

%\bibitem{Mac}
%I. G. Macdonald, {\em Symmetric functions and Hall polynomials},
%Oxford Mathematical Monographs, second edition, OUP, 1995.

\bibitem{Mat}
A. Mathas,
{\em Iwahori-Hecke algebras and Schur algebras of the symmetric
group}, University Lecture Series {\bf 15},
Amer. Math. Soc., 1999.

%\bibitem{Mitz}
%D. Mitzman,
%Integral bases for affine Lie algebras and their universal enveloping
%algebras,
%{\em Contemp. Math.} {\bf 40} (1985), 1--159.

\end{thebibliography}