Expand all Collapse all | Results 76 - 100 of 121 |
76. CJM 2004 (vol 56 pp. 168)
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 |
77. CJM 2003 (vol 55 pp. 1155)
The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$ |
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 invariants Categories:17B20, 17B45, 22E47 |
78. CJM 2003 (vol 55 pp. 1121)
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 |
79. CJM 2003 (vol 55 pp. 1080)
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 |
80. CJM 2003 (vol 55 pp. 969)
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 |
81. CJM 2003 (vol 55 pp. 353)
Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$-Adic Simple Algebras |
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 |
82. CJM 2002 (vol 54 pp. 1100)
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, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 |
83. CJM 2002 (vol 54 pp. 795)
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 graphs Categories:22D05, 20B07, 20B27, 05C25 |
84. CJM 2002 (vol 54 pp. 828)
Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$ |
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 |
85. CJM 2002 (vol 54 pp. 769)
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 |
86. CJM 2002 (vol 54 pp. 263)
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 |
87. CJM 2002 (vol 54 pp. 92)
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 |
88. CJM 2001 (vol 53 pp. 1141)
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 paires Category:22E50 |
89. CJM 2001 (vol 53 pp. 675)
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 |
90. CJM 2001 (vol 53 pp. 244)
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 |
91. CJM 2001 (vol 53 pp. 278)
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 manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 |
92. CJM 2001 (vol 53 pp. 195)
On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ 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 $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,n-l} \to C$. 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 |
93. CJM 2000 (vol 52 pp. 1192)
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 Harish-Chandra 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 |
94. CJM 2000 (vol 52 pp. 1101)
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 quasi-split
classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a
quadratic extension $E/F$. We prove the self-duality of unitary
supercuspidal data of standard Levi subgroups of $G_n(F)$ which
give discrete series representations of $G_n(F)$.
Category:22E35 |
95. CJM 2000 (vol 52 pp. 804)
The Distributions in the Invariant Trace Formula Are Supported on Characters J.~Arthur put the trace formula in invariant form for all connected
reductive groups and certain disconnected ones. However his work was
written so as to apply to the general disconnected case, modulo two
missing ingredients. This paper supplies one of those missing
ingredients, namely an argument in Galois cohomology of a kind first
used by D.~Kazhdan in the connected case.
Categories:22E50, 11S37, 10D40 |
96. CJM 2000 (vol 52 pp. 449)
An Intertwining Result for $p$-adic Groups For a reductive $p$-adic group $G$, we compute the supports of the Hecke
algebras for the $K$-types for $G$ lying in a certain frequently-occurring
class. When $G$ is classical, we compute the intertwining between any
two such $K$-types.
Categories:22E50, 20G05 |
97. CJM 2000 (vol 52 pp. 539)
On Square-Integrable Representations of Classical $p$-adic Groups In this paper, we use Jacquet module methods to study the problem
of classifying discrete series for the classical $p$-adic groups
$\Sp(2n,F)$ and $\SO(2n+1,F)$.
Category:22E50 |
98. CJM 2000 (vol 52 pp. 412)
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 |
99. CJM 2000 (vol 52 pp. 438)
On Some $q$-Analogs of a Theorem of Kostant-Rallis 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 Kostant-Rallis 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
Kostant-Rallis 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 |
100. CJM 2000 (vol 52 pp. 306)
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 |