101. CJM 2001 (vol 53 pp. 195)
 Mokler, Claus

On the Steinberg Map and Steinberg CrossSection for a Symmetrizable Indefinite KacMoody Group
Let $G$ be a symmetrizable indefinite KacMoody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2nl}}$ be the characters of the
fundamental irreducible representations of $G$, defined as convergent
series on a certain part $G^{\tralg} \subseteq G$. Following
Steinberg in the classical case and Br\"uchert in the affine case, we
define the Steinberg map $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2nl}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,nl} \to C$. We investigate the local behaviour of
$\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1)
\bigr)$, and we show that there exists a neighborhood of $(0,\dots,0)
\times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular
analytical map, satisfying a certain functional identity. This
identity has its origin in an action of the center of $G$ on~$C$.
Categories:22E65, 17B65 

102. CJM 2000 (vol 52 pp. 1192)
 Herb, Rebecca A.

Orbital Integrals on $p$Adic Lie Algebras
Let $G$ be a connected reductive $p$adic group and let $\frakg$ be its
Lie algebra. Let $\calO$ be any $G$orbit in $\frakg$. Then the orbital
integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution
on $\frakg $, and HarishChandra proved that its Fourier transform $\hat
\mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular
semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of
$\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give
a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$
sufficiently large. In the case that $\calO$ is a regular semisimple orbit,
the formula is already known by work of Waldspurger. In the case that
$\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at
infinity is already known because of its homogeneity properties. The
general case combines aspects of these two extreme cases. The formula
for $\hat\mu _{\calO}$ at infinity can be used to formulate a ``theory
of the constant term'' for the space of distributions spanned by the
Fourier transforms of orbital integrals. It can also be used to show
that the Fourier transforms of orbital integrals are ``linearly
independent at infinity.''
Categories:22E30, 22E45 

103. CJM 2000 (vol 52 pp. 1101)
 Zhang, Yuanli

Discrete Series of Classical Groups
Let $G_n$ be the split classical groups $\Sp(2n)$, $\SO(2n+1)$ and
$\SO(2n)$ defined over a $p$adic field F or the quasisplit
classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a
quadratic extension $E/F$. We prove the selfduality of unitary
supercuspidal data of standard Levi subgroups of $G_n(F)$ which
give discrete series representations of $G_n(F)$.
Category:22E35 

104. CJM 2000 (vol 52 pp. 804)
105. CJM 2000 (vol 52 pp. 539)
106. CJM 2000 (vol 52 pp. 449)
107. CJM 2000 (vol 52 pp. 412)
 Varopoulos, N. Th.

Geometric and Potential Theoretic Results on Lie Groups
The main new results in this paper are contained in the geometric
Theorems 1 and~2 of Section~0.1 below and they are related to
previous results of M.~Gromov and of myself (\cf\
\cite{1},~\cite{2}). These results are used to prove some general
potential theoretic estimates on Lie groups (\cf\ Section~0.3) that
are related to my previous work in the area (\cf\
\cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos
(\cf\ \cite{5},~\cite{21}).
Categories:22E30, 43A80, 60J60, 60J65 

108. CJM 2000 (vol 52 pp. 438)
 Wallach, N. R.; Willenbring, J.

On Some $q$Analogs of a Theorem of KostantRallis
In the first part of this paper generalizations of Hesselink's
$q$analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the KostantRallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$analog of the
KostantRallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.
Categories:22E47, 20G05 

109. CJM 2000 (vol 52 pp. 306)
 Cunningham, Clifton

Characters of DepthZero, Supercuspidal Representations of the Rank2 Symplectic Group
This paper expresses the character of certain depthzero
supercuspidal representations of the rank2 symplectic group as the
Fourier transform of a finite linear combination of regular
elliptic orbital integralsan expression which is ideally suited
for the study of the stability of those characters. Building on
work of F.~Murnaghan, our proof involves Lusztig's Generalised
Springer Correspondence in a fundamental way, and also makes use of
some results on elliptic orbital integrals proved elsewhere by the
author using MoyPrasad filtrations of $p$adic Lie algebras. Two
applications of the main result are considered toward the end of
the paper.
Categories:22E50, 22E35 

110. CJM 1999 (vol 51 pp. 1135)
 Arthur, James

Endoscopic $L$Functions and a Combinatorial Identity
The trace formula contains terms on the spectral side that are
constructed from unramified automorphic $L$functions. We shall
establish an identify that relates these terms with corresponding
terms attached to endoscopic groups of $G$. In the process, we
shall show that the $L$functions of $G$ that come from automorphic
representations of endoscopic groups have meromorphic continuation.
Categories:22E45, 22E46 

111. CJM 1999 (vol 51 pp. 1307)
 Johnson, Norman W.; Weiss, Asia Ivić

Quadratic Integers and Coxeter Groups
Matrices whose entries belong to certain rings of algebraic
integers can be associated with discrete groups of transformations
of inversive $n$space or hyperbolic $(n+1)$space
$\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups,
generated by reflections, or certain subgroups whose generators
include direct isometries of $\mbox{H}^{n+1}$. We show how linear
fractional transformations over rings of rational and (real or
imaginary) quadratic integers are related to the symmetry groups of
regular tilings of the hyperbolic plane or 3space. New light is
shed on the properties of the rational modular group $\PSL_2
(\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it
i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.
Categories:11F06, 20F55, 20G20, 20H10, 22E40 

112. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
squareintegrable $\Gamma$automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 

113. CJM 1999 (vol 51 pp. 835)
 Kim, Henry H.

LanglandsShahidi Method and Poles of Automorphic $L$Functions: Application to Exterior Square $L$Functions
In this paper we use LanglandsShahidi method and the result of
Langlands which says that non selfconjugate maximal parabolic
subgroups do not contribute to the residual spectrum, to prove the
holomorphy of several \emph{completed} automorphic $L$functions on the
whole complex plane which appear in constant terms of the Eisenstein
series. They include the exterior square $L$functions of $\GL_n$, $n$
odd, the RankinSelberg $L$functions of $\GL_n\times \GL_m$, $n\ne m$,
and $L$functions $L(s,\sigma,r)$, where $\sigma$ is a generic
cuspidal representation of $\SO_{10}$ and $r$ is the halfspin
representation of $\GSpin(10, \mathbb{C})$. The main part is
proving the holomorphy and nonvanishing of the local normalized
intertwining operators by reducing them to natural conjectures in
harmonic analysis, such as standard module conjecture.
Categories:11F, 22E 

114. CJM 1999 (vol 51 pp. 816)
 Hall, Brian C.

A New Form of the SegalBargmann Transform for Lie Groups of Compact Type
I consider a twoparameter family $B_{s,t}$ of unitary transforms
mapping an $L^{2}$space over a Lie group of compact type onto a
holomorphic $L^{2}$space over the complexified group. These were
studied using infinitedimensional analysis in joint work with
B.~Driver, but are treated here by finitedimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
SegalBargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.
Categories:22E30, 81S30, 58G11 

115. CJM 1999 (vol 51 pp. 636)
116. CJM 1999 (vol 51 pp. 266)
 Deitmar, Anton; Hoffman, Werner

Spectral Estimates for Towers of Noncompact Quotients
We prove a uniform upper estimate on the number of cuspidal
eigenvalues of the $\Ga$automorphic Laplacian below a given bound
when $\Ga$ varies in a family of congruence subgroups of a given
reductive linear algebraic group. Each $\Ga$ in the family is assumed
to contain a principal congruence subgroup whose index in $\Ga$ does
not exceed a fixed number. The bound we prove depends linearly on the
covolume of $\Ga$ and is deduced from the analogous result about the
cutoff Laplacian. The proof generalizes the heatkernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.
Categories:11F72, 58G25, 22E40 

117. CJM 1999 (vol 51 pp. 164)
 Tan, Victor

Poles of Siegel Eisenstein Series on $U(n,n)$
Let $U(n,n)$ be the rank $n$ quasisplit unitary group over a
number field. We show that the normalized Siegel Eisenstein series
of $U(n,n)$ has at most simple poles at the integers or half
integers in certain strip of the complex plane.
Categories:11F70, 11F27, 22E50 

118. CJM 1999 (vol 51 pp. 130)
 Savin, Gordan; Gan, Wee Teck

The Dual Pair $G_2 \times \PU_3 (D)$ ($p$Adic Case)
We study the correspondence of representations arising by
restricting the minimal representation of the linear group of type
$E_7$ and relative rank $4$. The main tool is computations of the
Jacquet modules of the minimal representation with respect to
maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.
Categories:22E35, 22E50, 11F70 

119. CJM 1998 (vol 50 pp. 1090)
120. CJM 1998 (vol 50 pp. 1105)
 Roberts, Brooks

Tempered representations and the theta correspondence
Let $V$ be an even dimensional nondegenerate symmetric bilinear
space over a nonarchimedean local field $F$ of characteristic zero,
and let $n$ be a nonnegative integer. Suppose that $\sigma \in
\Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$
correspond under the theta correspondence. Assuming that $\sigma$
is tempered, we investigate the problem of determining the
Langlands quotient data for $\pi$.
Categories:11F27, 22E50 

121. CJM 1998 (vol 50 pp. 972)
 Brüchert, Gerd

Trace class elements and crosssections in KacMoody groups
Let $G$ be an affine KacMoody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$
its fundamental irreducible representations and $\chi_0, \dots,
\chi_r, \chi_{\delta}$ their characters. We determine the set of all
group elements $x$ such that all $\pi_i(x)$ act as trace class
operators, \ie, such that $\chi_i(x)$ exists, then prove that the
$\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r,
\chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$.
In a second part, following Steinberg, we define a crosssection $C$
for the potential regular classes in $G$. We prove that the
restriction $\chi_C$ behaves well algebraically. Moreover, we obtain
an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a
functional identity for $\chi_C$ which shows that $\chi_C$ is
quasihomogeneous.
Categories:22E65, 17B67 

122. CJM 1998 (vol 50 pp. 356)
 Gross, Leonard

Some norms on universal enveloping algebras
The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$
supports some norms and seminorms that have arisen naturally in the
context of heat kernel analysis on Lie groups. These norms and seminorms
are investigated here from an algebraic viewpoint. It is shown
that the norms corresponding to heat kernels on the associated Lie
groups decompose as product norms under the natural isomorphism
$U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak
g_2)$. The seminorms corresponding to Green's functions are
examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It
is also shown that the algebraic dual space $U'$ is spanned by its
finite rank elements if and only if $\frak g$ is nilpotent.
Categories:17B35, 16S30, 22E30 

123. CJM 1998 (vol 50 pp. 74)
 Flicker, Yuval Z.

Elementary proof of the fundamental lemma for a unitary group
The fundamental lemma in the theory of automorphic forms is proven
for the (quasisplit) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
Categories:22E35, 11F70, 11F85, 11S37 

124. CJM 1997 (vol 49 pp. 1117)
 Hu, Zhiguo

The von Neumann algebra $\VN(G)$ of a locally compact group and quotients of its subspaces
Let $\VN(G)$ be the von Neumann algebra of a locally
compact group $G$. We denote by $\mu$ the initial
ordinal with $\abs{\mu}$ equal to the smallest cardinality
of an open basis at the unit of $G$ and $X= \{\alpha;
\alpha < \mu \}$. We show that if $G$ is nondiscrete
then there exist an isometric $*$isomorphism $\kappa$
of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear
mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that
$\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and
$\pi$ have certain additional properties. Let $\UCB
(\hat{G})$ be the $C^{*}$algebra generated by
operators in $\VN(G)$ with compact support and
$F(\hat{G})$ the space of all $T \in \VN(G)$ such that
all topologically invariant means on $\VN(G)$ attain the
same value at $T$. The construction of the mapping $\pi$
leads to the conclusion that the quotient space $\UCB
(\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has
$l^{\infty}(X)$ as a continuous linear image if $G$ is
nondiscrete. When $G$ is further assumed to be
nonmetrizable, it is shown that $\UCB(\hat{G})/F
(\hat{G})\cap \UCB(\hat{G})$ contains a linear
isomorphic copy of $l^{\infty}(X)$. Similar results are
also obtained for other quotient spaces.
Categories:22D25, 43A22, 43A30, 22D15, 43A07, 47D35 

125. CJM 1997 (vol 49 pp. 1224)
 Ørsted, Bent; Zhang, Genkai

Tensor products of analytic continuations of holomorphic discrete series
We give the irreducible decomposition
of the tensor product of an analytic continuation of
the holomorphic discrete
series of $\SU(2, 2)$ with its conjugate.
Keywords:Holomorphic discrete series, tensor product, spherical function, ClebschGordan coefficient, Plancherel theorem Categories:22E45, 43A85, 32M15, 33A65 
