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76. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

77. CJM 2005 (vol 57 pp. 616)

Muić, Goran
 Reducibility of Generalized Principal Series In this paper we describe reducibility of non-unitary generalized principal series for classical $p$-adic groups in terms of the classification of discrete series due to M\oe glin and Tadi\'c. Categories:22E35, and, 50, 11F70

78. CJM 2005 (vol 57 pp. 535)

Kim, Henry H.
 On Local $L$-Functions and Normalized Intertwining Operators In this paper we make explicit all $L$-functions in the Langlands--Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$-functions. Categories:11F70, 22E55

79. CJM 2005 (vol 57 pp. 159)

Jantzen, Chris
 Duality and Supports of Induced Representations for Orthogonal Groups In this paper, we construct a duality for $p$-adic orthogonal groups. Category:22E50

80. CJM 2005 (vol 57 pp. 17)

Bédos, Erik; Conti, Roberto; Tuset, Lars
 On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories. Keywords:quantum group, amenabilityCategories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32

81. CJM 2004 (vol 56 pp. 945)

Helminck, Aloysius G.; Schwarz, Gerald W.
 Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$ and $K:=G^\theta$ be the fixed point groups. We have an action $(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson \cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$. Categories:20G15, 20G20, 22E15, 22E46

82. CJM 2004 (vol 56 pp. 963)

Ishiwata, Satoshi
 A Berry-Esseen Type Theorem on Nilpotent Covering Graphs We prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel. Categories:22E25, 60J15, 58G32

83. CJM 2004 (vol 56 pp. 883)

Tandra, Haryono; Moran, William
 Kirillov Theory for a Class of Discrete Nilpotent Groups This paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups $G$. $G$ is assumed to be discrete, countable and $\pi$-radicable, with $\pi$ containing the primes less than or equal to the nilpotence class of $G$. In addition, it is assumed that all of the characters of $G$ have idempotent absolute value. Such groups are shown to be plentiful. Category:22D10

84. CJM 2004 (vol 56 pp. 293)

Khomenko, Oleksandr; Mazorchuk, Volodymyr
 Structure of modules induced from simple modules with minimal annihilator We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\Oo$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals. Keywords:parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categoriesCategories:17B10, 22E47

85. CJM 2004 (vol 56 pp. 168)

Pogge, James Todd
 On a Certain Residual Spectrum of $\Sp_8$ Let $G=\Sp_{2n}$ be the symplectic group defined over a number field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of $G(\mathbb{A})$ acting on the Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A}) \bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A}) \bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a cuspidal automorphic representation $\pi$ taken modulo conjugacy (Here we normalize $\pi$ so that the action of the maximal split torus in the center of $G$ at the archimedean places is trivial.) and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$ is a space of residues of Eisenstein series associated to $(M,\pi)$. In this paper, we will completely determine the space $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when $M\simeq\GL_2\times\GL_2$. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\GL_n$. Categories:11F70, 22E55

86. CJM 2003 (vol 55 pp. 1155)

Đoković, Dragomir Ž.; Litvinov, Michael
 The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K = \mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x = axb^{-1}$, and so does its identity component $K^0 = \SO_p ({\bf C}) \times \SO_q ({\bf C})$. A $K$-orbit (or $K^0$-orbit) in $M_{p,q}$ is said to be nilpotent if its closure contains the zero matrix. The closure, $\overline{\mathcal{O}}$, of a nilpotent $K$-orbit (resp.\ $K^0$-orbit) ${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some nilpotent $K$-orbits (resp.\ $K^0$-orbits) of smaller dimensions. The description of the closure of nilpotent $K$-orbits has been known for some time, but not so for the nilpotent $K^0$-orbits. A conjecture describing the closure of nilpotent $K^0$-orbits was proposed in \cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ and determination of the basic relative invariants of these spaces. The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra $\mathfrak{so} (p,q)$ under the adjoint action of the identity component of the real orthogonal group $\mathrm{O}(p,q)$. Keywords:orthogonal $ab$-diagrams, prehomogeneous vector spaces, relative invariantsCategories:17B20, 17B45, 22E47

87. CJM 2003 (vol 55 pp. 1121)

Bettaïeb, Karem
 Classification des reprÃ©sentations tempÃ©rÃ©es d'un groupe $p$-adique Soit $G$ le groupe des points d\'efinis sur un corps $p$-adique d'un groupe r\'eductif connexe. A l'aide des caract\eres virtuels supertemp\'er\'es de $G$, on prouve (conjectures de Clozel) que toute repr\'esentation irr\'eductible temp\'er\'ee de $G$ est irr\'eductiblement induite d'une essentielle d'un sous-groupe de L\'evi de~$G$. Category:22E

88. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
 Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$. Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

89. CJM 2003 (vol 55 pp. 969)

Glöckner, Helge
 Lie Groups of Measurable Mappings We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$. Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25

90. CJM 2003 (vol 55 pp. 353)

Silberger, Allan J.; Zink, Ernst-Wilhelm
 Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$-Adic Simple Algebras Let $F$ be a $p$-adic local field and let $A_i^\times$ be the unit group of a central simple $F$-algebra $A_i$ of reduced degree $n>1$ ($i=1,2$). Let $\mathcal{R}^2 (A_i^\times)$ denote the set of irreducible discrete series representations of $A_i^\times$. The Abstract Matching Theorem'' asserts the existence of a bijection, the Jacquet-Langlands'' map, $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map $\mathcal{J} \mathcal{L}$, but only for level zero'' representations. We prove that the restriction $\mathcal{J} \mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to \mathcal{R}_0^2 (A_2^\times)$ is a bijection of level zero discrete series (Proposition~3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra $A_i$ and is invariant under $\mathcal{J} \mathcal{L}_{A_2,A_1}$ (Theorem~4.1). Categories:22E50, 11R39

91. CJM 2002 (vol 54 pp. 1100)

Wood, Peter J.
 The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete. Keywords:locally compact group, Fourier algebra, operator space, projectiveCategories:13D03, 18G25, 43A95, 46L07, 22D99

92. CJM 2002 (vol 54 pp. 795)

Möller, Rögnvaldur G.
 Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results. Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphsCategories:22D05, 20B07, 20B27, 05C25

93. CJM 2002 (vol 54 pp. 828)

Moriyama, Tomonori
 Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$ Let $\pi$ be an irreducible generalized principal series representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi$ to the representation induced from an irreducible admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function. Categories:22E45, 11F70

94. CJM 2002 (vol 54 pp. 769)

Miyazaki, Takuya
 Nilpotent Orbits and Whittaker Functions for Derived Functor Modules of $\Sp(2,\mathbb{R})$ We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi$ of a unipotent subgroup, for the non-tempered cohomological representation of $G = \Sp (2,\mathbb{R})$. Through an explicit calculation of a holonomic system which characterizes these functions we observe that their existence is determined by the including relation between the real nilpotent coadjoint $G$-orbit of $\psi$ in $\mathfrak{g}_{\mathbb{R}}^\ast$ and the asymptotic support of the cohomological representation. Categories:22E46, 22E30

95. CJM 2002 (vol 54 pp. 263)

Chaudouard, Pierre-Henri
 IntÃ©grales orbitales pondÃ©rÃ©es sur les algÃ¨bres de Lie : le cas $p$-adique Soit $G$ un groupe rÃ©ductif connexe dÃ©fini sur un corps $p$-adique $F$ et $\ggo$ son algÃ¨bre de Lie. Les intÃ©grales orbitales pondÃ©rÃ©es sur $\ggo(F)$ sont des distributions $J_M(X,f)$---$f$ est une fonction test---indexÃ©es par les sous-groupes de LÃ©vi $M$ de $G$ et les Ã©lÃ©ments semi-simples rÃ©guliers $X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales composantes du cÃ´tÃ© gÃ©omÃ©trique des formules des traces locale et globale d'Arthur. Si $M=G$, on retrouve les intÃ©grales orbitales invariantes qui, vues comme fonction de $X$, sont bornÃ©es sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un rÃ©sultat bien connu de Harish-Chandra. Si $M \subsetneq G$, les intÃ©grales orbitales pondÃ©rÃ©es explosent au voisinage des Ã©lÃ©ments singuliers. Nous construisons dans cet article de nouvelles intÃ©grales orbitales pondÃ©rÃ©es $J_M^b(X,f)$, Ã©gales Ã  $J_M(X,f)$ Ã  un terme correctif prÃ¨s, qui tout en conservant les principales propriÃ©tÃ©s des prÃ©cÃ©dentes (comportement par conjugaison, dÃ©veloppement en germes, {\it etc.}) restent bornÃ©es quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons Ã©galement que les intÃ©grales orbitales pondÃ©rÃ©es globales, associÃ©es Ã  des Ã©lÃ©ments semi-simples rÃ©guliers, se dÃ©composent en produits de ces nouvelles intÃ©grales locales. Categories:22E35, 11F70

96. CJM 2002 (vol 54 pp. 92)

Mezo, Paul
 Comparisons of General Linear Groups and their Metaplectic Coverings I We prepare for a comparison of global trace formulas of general linear groups and their metaplectic coverings. In particular, we generalize the local metaplectic correspondence of Flicker and Kazhdan and describe the terms expected to appear in the invariant trace formulas of the above covering groups. The conjectural trace formulas are then placed into a form suitable for comparison. Categories:11F70, 11F72, 22E50

97. CJM 2001 (vol 53 pp. 1141)

Bushnell, Colin J.; Henniart, Guy
 Sur le comportement, par torsion, des facteurs epsilon de paires Soient $F$ un corps commutatif localement compact non archim\'edien et $\psi$ un caract\ere additif non trivial de $F$. Soient $n$ et $n'$ deux entiers distincts, sup\'erieurs \a $1$. Soient $\pi$ et $\pi'$ des repr\'esentations irr\'eductibles supercuspidales de $\GL_n(F)$, $\GL_{n'}(F)$ respectivement. Nous prouvons qu'il existe un \'el\'ement $c= c(\pi,\pi',\psi)$ de $F^\times$ tel que pour tout quasicaract\ere mod\'er\'e $\chi$ de $F^\times$ on ait $\varepsilon(\chi\pi\times \pi',s,\psi) = \chi(c)^{-1}\varepsilon(\pi\times\pi',s,\psi)$. Nous examinons aussi certains cas o\u $n=n'$, $\pi'=\pi^\vee$. Les r\'esultats obtenus forment une \'etape vers une d\'emonstration de la conjecture de Langlands pour $F$, qui ne fasse pas appel \a la g\'eom\'etrie des vari\'et\'es modulaires, de Shimura ou de Drinfeld. Let $F$ be a non-Archimedean local field, and $\psi$ a non-trivial additive character of $F$. Let $n$ and $n'$ be distinct positive integers. Let $\pi$, $\pi'$ be irreducible supercuspidal representations of $\GL_n(F)$, $\GL_{n'}(F)$ respectively. We prove that there is $c= c(\pi,\pi',\psi)\in F^\times$ such that for every tame quasicharacter $\chi$ of $F^\times$ we have $\varepsilon(\chi\pi\times \pi',s,\psi) = \chi(c)^{-1}\varepsilon(\pi\times\pi',s,\psi)$. We also treat some cases where $n=n'$ and $\pi'=\pi^\vee$. These results are steps towards a proof of the Langlands conjecture for $F$, which would not use the geometry of modular---Shimura or Drinfeld---varieties. Keywords:corps local, correspondance de Langlands locale, facteurs epsilon de pairesCategory:22E50

98. CJM 2001 (vol 53 pp. 675)

Ban, Dubravka
 Jacquet Modules of Parabolically Induced Representations and Weyl Groups The representation parabolically induced from an irreducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations. Category:22E50

99. CJM 2001 (vol 53 pp. 278)

Helminck, G. F.; van de Leur, J. W.
 Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\KP$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau$-functions is given. Keywords:KP hierarchy, Darboux transformation, Grassmann manifoldCategories:22E65, 22E70, 35Q53, 35Q58, 58B25

100. CJM 2001 (vol 53 pp. 244)

Goldberg, David; Shahidi, Freydoon
 On the Tempered Spectrum of Quasi-Split Classical Groups II We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension $E/F$ of $p$-adic fields of characteristic zero. We study the case where the Levi component $M \simeq \GL_n (E) \times U_m (F)$, with $n \equiv m$ $(\mod 2)$. This, along with earlier work, determines the poles of the local Rankin-Selberg product $L$-function $L(s, \tau' \times \tau)$, with $\tau'$ an irreducible unitary supercuspidal representation of $\GL_n (E)$ and $\tau$ a generic irreducible unitary supercuspidal representation of $U_m (F)$. The results are interpreted using the theory of twisted endoscopy. Categories:22E50, 11S70
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