Expand all Collapse all | Results 1 - 18 of 18 |
1. CJM Online first
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes Suppose that $\tilde{G}$ is a connected reductive group
defined over a field $k$,
and
$\Gamma$ is a finite group acting via $k$-automorphisms
of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition.
Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$
is reductive.
We axiomatize the main features of the relationship between this
fixed-point group and the pair $(\tilde{G},\Gamma)$,
and consider any group $G$ satisfying the axioms.
If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we
can consider their duals $\tilde{G}^*$ and $G^*$.
We show the existence of and give an explicit formula for a natural map
from the set of semisimple stable conjugacy classes in $G^*(k)$
to the analogous set for $\tilde{G}^*(k)$.
If $k$ is finite, then our groups are automatically quasisplit,
and our result specializes to give a map
of semisimple conjugacy classes.
Since such classes parametrize packets of irreducible representations
of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.
Keywords:reductive group, lifting, conjugacy class, representation, Lusztig series Categories:20G15, 20G40, 20C33, 22E35 |
2. CJM 2013 (vol 66 pp. 566)
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms |
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence Categories:22E50, 11F70, 22E55, 22E35 |
3. CJM 2011 (vol 63 pp. 1137)
Distribution Algebras on p-adic Groups and Lie Algebras When $F$ is a $p$-adic field, and $G={\mathbb
G}(F)$ is the group of $F$-rational points of a connected algebraic
$F$-group, the complex vector space ${\mathcal H}(G)$ of compactly
supported locally constant distributions on $G$ has a natural
convolution product that makes it into a ${\mathbb C}$-algebra
(without an identity) called the Hecke algebra. The Hecke algebra is a
partial analogue for $p$-adic groups of the enveloping algebra of a
Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of
an identity element, and the process $G \mapsto \mathcal{H}(G)$
is not a functor. Bernstein introduced an enlargement
$\mathcal{H}\,\hat{\,}(G)$
of $\mathcal{H}(G)$. The algebra
$\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left
essentially compact. We show that the process $G \mapsto
\mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow
H$ is a morphism of $p$-adic groups, let $F(\tau) \colon
\mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be
the morphism of $\mathbb{C}$-algebras. We identify the kernel of
$F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$-adic
Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra,
$\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the
natural projection, we show that $F(\tau)$ maps $G$-invariant distributions
on $\mathcal{G}$ to $N_G (\mathfrak{m})$-invariant distributions on
$\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant
essentially compact distributions on $\mathfrak{g}$ associated with a
$G$-invariant non-degenerate symmetric bilinear form on ${\mathfrak g}$
and in the case of $SL(2)$ show how certain members of the family can
be moved to the group.
Keywords:distribution algebra, p-adic group Categories:22E50, 22E35 |
4. CJM 2010 (vol 62 pp. 1340)
Holomorphie des opÃ©rateurs d'entrelacement normalisÃ©s Ã l'aide des paramÃ¨tres d'Arthur
In this paper we prove holomorphy for certain intertwining operators
arising from the theory of Eisenstein series. To do that we need to
normalize using the Langlands--Shahidi's normalization arising from
the twisted endoscopy and the associated representations of the
general linear group.
Categories:22E50, 22E35 |
5. CJM 2009 (vol 62 pp. 94)
The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series Let $G$ be a connected semisimple split group over a $p$-adic field.
We establish the explicit link between principal nilpotent
orbits and the irreducible constituents of principal series
in terms of $L$-group objects.
Keywords:Langlands correspondence, nilpotent orbits, principal series Categories:22E50, 22E35 |
6. CJM 2009 (vol 61 pp. 427)
On Reducibility and Unitarizability for Classical $p$-Adic Groups, Some General Results The aim of this paper is to prove two general results on parabolic
induction of classical $p$-adic groups (actually, one of them holds also
in the archimedean case), and to obtain from them some consequences about
irreducible unitarizable representations. One of these consequences is a
reduction of the unitarizability problem for these groups. This
reduction is similar to the reduction of the unitarizability problem
to the case of real infinitesimal
character for real reductive groups.
Categories:22E50, 22E35 |
7. CJM 2008 (vol 60 pp. 1306)
Theta Lifts of Tempered Representations for Dual Pairs $(\Sp_{2n}, O(V))$ This paper is the continuation of our previous work on the explicit
determination of the structure of theta lifts for dual pairs
$(\Sp_{2n}, O(V))$ over a non-archimedean field $F$ of characteristic
different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the
dimension of the space $V$ (over $F$) is even. We determine the
structure of theta lifts of tempered representations in terms of theta
lifts of representations in discrete series.
Categories:22E35, 22E50, 11F70 |
8. CJM 2007 (vol 59 pp. 1050)
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$ Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicity-free by proving an appropriate theorem on invariant
distributions; this then proves a multiplicity-one theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.
Categories:22E50, 22E35, 11S37 |
9. CJM 2007 (vol 59 pp. 148)
On Certain Classes of Unitary Representations for Split Classical Groups In this paper we prove the unitarity of duals of tempered
representations supported on minimal parabolic subgroups for split
classical $p$-adic groups. We also construct a family of unitary
spherical representations for real and complex classical groups
Categories:22E35, 22E50, 11F70 |
10. CJM 2006 (vol 58 pp. 897)
Distributions invariantes sur les groupes rÃ©ductifs quasi-dÃ©ployÃ©s Soit $F$ un corps local non archim\'edien, et $G$ le groupe des
$F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$.
Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes
par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de
Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact
biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout
d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$
sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de
dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la
r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise
ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions
sur $G$, que cet espace est engendr\'e d'une part par certaines
int\'egrales orbitales semi-simples, d'autre part par les int\'egrales
orbitales unipotentes, en montrant tout d'abord des r\'esultats
analogues sur les groupes finis.
Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions Categories:22E35, 20G40 |
11. 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 |
12. CJM 2005 (vol 57 pp. 648)
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, $K$-types Categories:20G25, 22E35, 20H25 |
13. CJM 2005 (vol 57 pp. 616)
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 |
14. 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 |
15. 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 |
16. 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 |
17. 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 |
18. CJM 1998 (vol 50 pp. 74)
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 |