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101. CJM 2000 (vol 52 pp. 306)

Cunningham, Clifton
Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group
This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals---an 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 Moy-Prasad filtrations of $p$-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

Categories:22E50, 22E35

102. 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

103. 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 3-space. 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

104. CJM 1999 (vol 51 pp. 952)

Deitmar, Anton; Hoffmann, Werner
On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rank-one 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 square-integrable $\Gamma$-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 DeGeorge-Wallach and Delorme.

Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus
Categories:11F72, 22E30, 22E40, 43A85, 58G25

105. CJM 1999 (vol 51 pp. 835)

Kim, Henry H.
Langlands-Shahidi Method and Poles of Automorphic $L$-Functions: Application to Exterior Square $L$-Functions
In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate 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 Rankin-Selberg $L$-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 half-spin representation of $\GSpin(10, \mathbb{C})$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

Categories:11F, 22E

106. CJM 1999 (vol 51 pp. 816)

Hall, Brian C.
A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type
I consider a two-parameter 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 infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s \rightarrow \infty$ and $s \rightarrow t/2$.

Categories:22E30, 81S30, 58G11

107. CJM 1999 (vol 51 pp. 636)

Paul, Annegret
First Occurrence for the Dual Pairs $\bigl(U(p,q),U(r,s)\bigr)$
We prove a conjecture of Kudla and Rallis about the first occurrence in the theta correspondence, for dual pairs of the form $\bigl(U(p,q),U(r,s)\bigr)$ and most representations.

Category:22E46

108. 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 cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$.

Categories:11F72, 58G25, 22E40

109. CJM 1999 (vol 51 pp. 164)

Tan, Victor
Poles of Siegel Eisenstein Series on $U(n,n)$
Let $U(n,n)$ be the rank $n$ quasi-split 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

110. 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

111. CJM 1998 (vol 50 pp. 1090)

Lohoué, Noël; Mustapha, Sami
Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes
Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$ and let $\nabla$ be the associated gradient. In this paper we investigate the boundness of the Riesz transform $\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with exponential volume growth and on certain homogenous spaces.

Categories:22E30, 35H05, 43A80, 43A85

112. 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

113. CJM 1998 (vol 50 pp. 972)

Brüchert, Gerd
Trace class elements and cross-sections in Kac-Moody groups
Let $G$ be an affine Kac-Moody 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 cross-section $C$ 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 quasi-homogeneous.

Categories:22E65, 17B67

114. 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

115. 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 (quasi-split) 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

116. 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 non-metrizable, 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

117. 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, Clebsch-Gordan coefficient, Plancherel theorem
Categories:22E45, 43A85, 32M15, 33A65

118. 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 4-dimensional 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 non-flat Kaehler structure on $\mathbb{C}^2 - \{ 0 \}$ (the universal cover of $O_\mathbb{R}$ ) with Kaehler potential $\rho=|z|^4$.

Categories:81S10, 32C17, 22E70

119. CJM 1997 (vol 49 pp. 820)

Robart, Thierry
Sur l'intégrabilité des sous-algèbres 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 sous-alg\`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 sous-alg\`ebres de Lie ferm\'ees $\cal H$ de $\cal G$ non-int\'egrables en un sous-groupe 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 physique-math\'ematique et les groupes formels de diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.

Categories:22E65, 58h05, 17B65

120. CJM 1997 (vol 49 pp. 736)

Fendler, Gero
Dilations of one parameter Semigroups of positive Contractions on $L^{\lowercase {p}}$ spaces
It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an $L^p$-space, for $1
Categories:47D03, 22D12, 43A22

121. 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 Kazhdan-Lusztig 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

122. 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
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