Expand all Collapse all | Results 26 - 41 of 41 |
26. CJM 2006 (vol 58 pp. 1203)
Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de Harish-Chandra Dans un travail ant\'erieur, nous
avions montr\'e que l'induite parabolique (normalis\'ee) d'une
repr\'esentation irr\'eductible cuspidale $\sigma $ d'un
sous-groupe de Levi $M$ d'un groupe $p$-adique contient un
sous-quotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre
\'egal au rang parabolique de $M$. L'objet de cet article est
d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de
Langlands.
Categories:11F70, 11F80, 22E50 |
27. CJM 2006 (vol 58 pp. 1095)
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$ The Casselman--Shalika method is a way to compute explicit
formulas for periods of irreducible unramified representations of
$p$-adic groups that are associated to unique models (i.e.,
multiplicity-free induced representations). We apply this method
to the case of the Shalika model of $GL_n$, which is known to
distinguish lifts from odd orthogonal groups. In the course of our
proof, we further develop a variant of the method, that was
introduced by Y. Hironaka, and in effect reduce many such problems
to straightforward calculations on the group.
Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs Categories:22E50, 11F70, 11F85 |
28. CJM 2006 (vol 58 pp. 344)
Reducibility for $SU_n$ and Generic Elliptic Representations We study reducibility of representations
parabolically induced from discrete series
representations of $SU_n(F)$ for $F$ a $p$-adic field of
characteristic zero. We use the approach of studying the relation
between $R$-groups when a reductive subgroup of a quasi-split group
and the full group have the same derived group. We use restriction to
show the quotient of $R$-groups is in natural bijection with a group
of characters. Applying this to $SU_n(F)\subset U_n(F)$ we show the
$R$ group for $SU_n$ is the semidirect product of an $R$-group for
$U_n(F)$ and this group of characters. We derive results on
non-abelian $R$-groups and generic elliptic representations as well.
Categories:22E50, 22E35 |
29. CJM 2005 (vol 57 pp. 159)
Duality and Supports of Induced Representations for Orthogonal Groups In this paper, we construct a duality for $p$-adic orthogonal groups.
Category:22E50 |
30. 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 |
31. 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 |
32. 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 |
33. 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 |
34. 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 |
35. 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 |
36. 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 |
37. 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 |
38. 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 |
39. CJM 1999 (vol 51 pp. 130)
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 |
40. CJM 1999 (vol 51 pp. 164)
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 |
41. CJM 1998 (vol 50 pp. 1105)
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 |