126. 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 

127. 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 

128. 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 

129. 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 

130. 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 

131. CJM 1997 (vol 49 pp. 916)
 Brylinski, Ranee

Quantization of the $4$dimensional nilpotent orbit of SL(3,$\mathbb{R}$)
We give a new geometric model for the quantization
of the 4dimensional conical (nilpotent) adjoint orbit
$O_\mathbb{R}$ of SL$(3,\mathbb{R})$. The space of quantization is the space of
holomorphic functions on $\mathbb{C}^2 \{ 0 \}$ which are square integrable
with respect to a signed measure defined by a Meijer $G$function.
We construct the quantization out a nonflat Kaehler structure on
$\mathbb{C}^2  \{ 0 \}$ (the universal cover of $O_\mathbb{R}$ ) with Kaehler potential
$\rho=z^4$.
Categories:81S10, 32C17, 22E70 

132. CJM 1997 (vol 49 pp. 820)
 Robart, Thierry

Sur l'intÃ©grabilitÃ© des sousalgÃ¨bres de Lie en dimension infinie
Une des questions fondamentales de la th\'eorie des groupes de
Lie de dimension infinie concerne l'int\'egrabilit\'e des
sousalg\`ebres de Lie topologiques $\cal H$ de l'alg\`ebre
de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie
au sens de Milnor. Par contraste avec ce qui se passe en
th\'eorie classique il peut exister des sousalg\`ebres de Lie
ferm\'ees $\cal H$ de $\cal G$ nonint\'egrables en un
sousgroupe de Lie. C'est le cas des alg\`ebres de Lie de champs
de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne
d\'efinissent pas un feuilletage de Stefan. Heureusement cette
``imperfection" de la th\'eorie n'est pas partag\'ee par tous les
groupes de Lie int\'eressants. C'est ce que montre cet article
en exhibant une tr\`es large classe de groupes de Lie de
dimension infinie exempte de cette imperfection. Cela permet de
traiter compl\`etement le second probl\`eme fondamental de
Sophus Lie pour les groupes de jauge de la
physiquemath\'ematique et les groupes formels de
diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.
Categories:22E65, 58h05, 17B65 

133. CJM 1997 (vol 49 pp. 736)
134. CJM 1997 (vol 49 pp. 417)
 Boe, Brian D.; Fu, Joseph H. G.

Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the KazhdanLusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 

135. CJM 1997 (vol 49 pp. 133)
 Reeder, Mark

Exterior powers of the adjoint representation
Exterior powers of the adjoint representation of a complex semisimple Lie
algebra are decomposed into irreducible representations, to varying
degrees of satisfaction.
Keywords:Lie algebras, adjoint representation, exterior algebra Categories:20G05, 20C30, 22E10, 22E60 
