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1. CJM Online first
Symmetric Genuine Spherical Whittaker Functions on $\overline{GSp_{2n}(F)}$ Let $F$ be a p-adic field of odd residual characteristic. Let
$\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general
symplectic group and the symplectic group attached to the $2n$
dimensional symplectic space over $F$. Let $\sigma$ be a genuine,
possibly reducible, unramified principal series representation of
$\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas for a spanning
set for the space of Spherical Whittaker functions attached to
$\sigma$. For odd $n$, and generically for even $n$, this spanning set
is a basis. The significant property of this set is that each of its
elements is unchanged under the action of the Weyl group of
$\overline{Sp_{2n}(F)}$.
If $n$ is odd then each element in the set has an equivariant property
that generalizes a uniqueness result of Gelbart, Howe and
Piatetski-Shapiro. Using this symmetric set, we
construct a family of reducible genuine unramified principal series
representations which have more then one generic constituent. This
family contains all the reducible genuine unramified principal series
representations induced from a unitary data and exists only for $n$
even.
Keywords:metaplectic group, Casselman Shalika Formula Category:11F85 |
2. CJM Online first
Expression d'un facteur epsilon de paire par une formule intÃ©grale Let $E/F$ be a quadratic extension of $p$-adic fields and
let $d$, $m$ be nonnegative integers of distinct parities. Fix
admissible irreducible tempered representations $\pi$ and $\sigma$ of
$GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and
$\sigma$ are conjugate-dual. That is to say $\pi\simeq \pi^{\vee,c}$
and $\sigma\simeq \sigma^{\vee,c}$ where $c$ is the non trivial
$F$-automorphism of $E$. This implies, we can extend $\pi$ to an
unitary representation $\tilde{\pi}$ of a nonconnected group
$GL_d(E)\rtimes \{1,\theta\}$. Define $\tilde{\sigma}$ the same
way. We state and prove an integral formula for
$\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of
$\tilde{\pi}$ and $\tilde{\sigma}$. This formula is related to the
local Gan-Gross-Prasad conjecture for unitary groups.
Keywords:epsilon factor, twisted groups Categories:22E50, 11F85 |
3. 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 |
4. CJM 2009 (vol 61 pp. 674)
A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$ We give a constructive proof, in the special case of ${\rm GL}_3$, of
a theorem of Ash and Stevens which compares overconvergent cohomology
to classical cohomology. Namely, we show that every ordinary
classical Hecke-eigenclass can be lifted uniquely to a rigid analytic
eigenclass. Our basic method builds on the ideas of M. Greenberg; we
first form an arbitrary lift of the classical eigenclass to a
distribution-valued cochain. Then, by appropriately iterating the
$U_p$-operator, we produce a cocycle whose image in cohomology is the
desired eigenclass. The constructive nature of this proof makes it
possible to perform computer computations to approximate these
interesting overconvergent eigenclasses.
Categories:11F75, 11F85 |
5. 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 |
6. 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 |