Expand all Collapse all | Results 1 - 25 of 36 |
1. CJM Online first
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$ Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi
subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage
in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic
representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$,
we construct (under a certain
technical assumption, which is always satisfied when $n=2$) an
automorphic representation $\pi$
of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the
representations $\pi_1,\dots,\pi_k$. This is
the global analogue of the metaplectic tensor product
defined by P. Mezo in the sense that locally at each place $v$,
$\pi_v$ is equivalent to the local metaplectic tensor product of
$\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all
of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then
the metaplectic tensor product is cuspidal (resp. square-integrable
modulo center). We also show that (both
locally and globally) the metaplectic tensor product behaves in the
expected way under the action of a Weyl group element, and show the
compatibility with parabolic inductions.
Keywords:automorphic forms, representations of covering groups Category:11F70 |
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 2012 (vol 64 pp. 497)
Le lemme fondamental pondÃ©rÃ© pour le groupe mÃ©taplectique Dans cet article, on Ã©nonce une variante du lemme fondamental
pondÃ©rÃ© d'Arthur pour le groupe mÃ©taplectique de Weil, qui sera un
ingrÃ©dient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractÃ©ristique rÃ©siduelle suffisamment
grande, on en donne une dÃ©monstration Ã l'aide de la mÃ©thode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondÃ©rÃ© non standard sur les algÃ¨bres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend Ã ce que cette condition soit
ultÃ©rieurement vÃ©rifiÃ©e.
Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula Categories:11F70, 11F27, 22E50 |
4. CJM 2011 (vol 65 pp. 22)
Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters We prove a non-vanishing result for families of
$\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip,
as one factor runs over twists by Hecke characters. As an
application, we simplify the proof, due to Luo, Rudnick, and Sarnak,
of the best known bounds towards the Generalized Ramanujan Conjecture
at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is
the regularization of the units in residue classes by the use of an
Arakelov ray class group.
Keywords:non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture Categories:11F70, 11M41 |
5. CJM 2011 (vol 63 pp. 1083)
Decomposition of Splitting Invariants in Split Real Groups For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor Categories:11F70, 22E47, 11S37, 11F72, 17B22 |
6. CJM 2011 (vol 63 pp. 591)
Rank One Reducibility for Metaplectic Groups via Theta Correspondence We calculate reducibility for the representations of
metaplectic groups induced from cuspidal representations of
maximal parabolic subgroups via theta correspondence, in terms of the
analogous representations of the odd orthogonal groups. We also
describe the lifts of all relevant subquotients.
Categories:22E50, 11F70 |
7. CJM 2010 (vol 62 pp. 914)
Reducibility of the Principal Series for Sp^{~}_{2}(F) over a p-adic Field Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with
entries in a nondyadic $p$-adic field $F$. We further let $\widetilde{G}_n$ be
the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times}
\mid |z|=1\}$ defined using the Leray cocycle. In this paper, we aim to
demonstrate the complete list of reducibility points of the genuine
principal series of ${\widetilde{G}_2}$. In most cases, we will use
some techniques developed by TadiÄ that analyze the Jacquet
modules with respect to all of the parabolics containing a fixed
Borel. The exceptional cases involve representations induced from
unitary characters $\chi$ with $\chi^2=1$. Because such
representations $\pi$ are unitary, to show the irreducibility of
$\pi$, it suffices to show that
$\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this
by examining the poles of certain intertwining operators associated to
simple roots. Then some results of Shahidi and Ban decompose arbitrary
intertwining operators into a composition of operators corresponding
to the simple roots of ${\widetilde{G}_2}$. We will then be able to
show that all such operators have poles at principal series
representations induced from quadratic characters and therefore such
operators do not extend to operators in
$\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.
Categories:22E50, 11F70 |
8. CJM 2010 (vol 62 pp. 563)
Whittaker Functions on Real Semisimple Lie Groups of Rank Two We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.
Categories:11F70, 22E30 |
9. CJM 2009 (vol 61 pp. 1383)
Integral Representation for $U_{3} \times \GL_{2}$ Gelbart and Piatetskii-Shapiro constructed
various integral
representations of Rankin--Sel\-berg type for groups $G \times
\GL_{n}$,
where $G$
is of split rank $n$. Here we show that their method
can equally well be applied
to the product $U_{3} \times \GL_{2}$, where $U_{3}$
denotes the quasisplit
unitary group in three variables. As an application, we describe which
cuspidal automorphic representations of $U_{3}$ occur
in the Siegel induced
residual spectrum of the quasisplit $U_{4}$.
Categories:11F70, 11F67 |
10. CJM 2009 (vol 61 pp. 779)
Residual Spectra of Split Classical Groups and their Inner Forms This paper is concerned with the residual spectrum of the
hermitian quaternionic classical groups $G_n'$ and $H_n'$ defined
as algebraic groups for a quaternion algebra over an algebraic
number field. Groups $G_n'$ and
$H_n'$ are not quasi-split. They are inner forms of the split
groups $\SO_{4n}$ and $\Sp_{4n}$. Hence, the parts of the residual
spectrum of $G_n'$ and $H_n'$ obtained in this paper are compared
to the corresponding parts for the split groups $\SO_{4n}$ and
$\Sp_{4n}$.
Categories:11F70, 22E55 |
11. CJM 2009 (vol 61 pp. 617)
Square Integrable Representations and the Standard Module Conjecture for General Spin Groups In this paper we study square integrable representations and
$L$-functions for quasisplit general spin groups over a $p$-adic
field. In the first part, the holomorphy of $L$-functions in a half
plane is proved by using a variant form of Casselman's square
integrability criterion and the Langlands--Shahidi method. The
remaining part focuses on the proof of the standard module
conjecture. We generalize Mui\'c's idea via the Langlands--Shahidi method
towards a proof of the conjecture. It is used in the work of M. Asgari
and F. Shahidi on generic transfer for general spin groups.
Categories:11F70, 11F85 |
12. CJM 2009 (vol 61 pp. 395)
$L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications Let $\Pi$ be a generic cuspidal automorphic representation of
$\GSp(2)$ defined over a totally real algebraic number field $\gfk$
whose archimedean type is either a (limit of) large discrete series
representation or a certain principal series representation. Through
explicit computation of archimedean local zeta integrals, we prove the
functional equation of tensor product $L$-functions $L(s,\Pi \times
\sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$
of $\GL(2)$. We also give an application to the spinor $L$-function
of $\Pi$.
Categories:11F70, 11F41, 11F46 |
13. CJM 2009 (vol 61 pp. 373)
An Infinite Order Whittaker Function In this paper we construct a flat smooth section of an induced space
$I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function
is not of finite order.
An asymptotic method of classical analysis is used.
Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15 |
14. 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 |
15. CJM 2008 (vol 60 pp. 790)
Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons |
Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons Soit $F_0$ un corps local non archim\'edien de caract\'eristique
nulle et de ca\-rac\-t\'eristique r\'esiduelle impaire.
J. Rogawski a montr\'e l'existence du changement de base entre le
groupe unitaire en trois variables $U(2,1)(F_{0})$, d\'efini
relativement \`a une extension quadratique $F$ de $F_{0}$, et le
groupe lin\'eaire $GL(3,F)$. Par ailleurs, nous
avons d\'ecrit les repr\'esentations supercuspidales irr\'eductibles
de $U(2,1)(F_{0})$ comme induites \`a partir d'un sous-groupe compact
ouvert de $U(2,1)(F_{0})$, description analogue \`a celle des
repr\'esentations admissibles irr\'eductibles de $GL(3,F)$ obtenue
par C. Bushnell et P. Kutzko. A partir de ces
descriptions, nous construisons explicitement le changement de base
des repr\'esentations tr\`es cuspidales de $U(2,1)(F_{0})$.
Categories:22E50, 11F70 |
16. CJM 2008 (vol 60 pp. 412)
Quelques calculs de traces compactes et leurs transform{Ã©es de Satake On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique
des traces tordues compactes d'un ensemble de repr{\'e}sentations
explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est
un corps $p$-adique. Ces calculs r\'esolve en particulier une
question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.
Categories:22E50, 11F70 |
17. CJM 2007 (vol 59 pp. 1323)
On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67 |
18. 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 |
19. 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 |
20. 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 |
21. CJM 2006 (vol 58 pp. 643)
Centralizers and Twisted Centralizers: Application to Intertwining Operators ABSTRACT
The equality of the centralizer and twisted centralizer is proved
based on a case-by-case analysis when the unipotent radical of a
maximal parabolic subgroup is abelian.
Then this result is used to determine the poles of intertwining operators.
Category:11F70 |
22. 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 |
23. CJM 2005 (vol 57 pp. 535)
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 |
24. 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 |
25. 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 |