26. CJM 2006 (vol 58 pp. 1203)
 Heiermann, Volker

Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de HarishChandra
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
sousgroupe de Levi $M$ d'un groupe $p$adique contient un
sousquotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de HarishChandra 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)
 Sakellaridis, Yiannis

A CasselmanShalika Formula for the Shalika Model of $\operatorname{GL}_n$
The CasselmanShalika 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.,
multiplicityfree 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:CasselmanShalika, periods, Shalika model, spherical functions, Gelfand pairs Categories:22E50, 11F70, 11F85 

28. CJM 2006 (vol 58 pp. 344)
 Goldberg, David

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 quasisplit 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
nonabelian $R$groups and generic elliptic representations as well.
Categories:22E50, 22E35 

29. CJM 2005 (vol 57 pp. 159)
30. CJM 2003 (vol 55 pp. 353)
 Silberger, Allan J.; Zink, ErnstWilhelm

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 ``JacquetLanglands'' 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)
 Mezo, Paul

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)
 Bushnell, Colin J.; Henniart, Guy

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 nonArchimedean local field, and $\psi$ a nontrivial
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 modularShimura or Drinfeldvarieties.
Keywords:corps local, correspondance de Langlands locale, facteurs epsilon de paires Category:22E50 

33. CJM 2001 (vol 53 pp. 675)
34. CJM 2001 (vol 53 pp. 244)
 Goldberg, David; Shahidi, Freydoon

On the Tempered Spectrum of QuasiSplit Classical Groups II
We determine the poles of the standard intertwining operators for a
maximal parabolic subgroup of the quasisplit 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 RankinSelberg 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)
36. CJM 2000 (vol 52 pp. 449)
37. CJM 2000 (vol 52 pp. 539)
38. CJM 2000 (vol 52 pp. 306)
 Cunningham, Clifton

Characters of DepthZero, Supercuspidal Representations of the Rank2 Symplectic Group
This paper expresses the character of certain depthzero
supercuspidal representations of the rank2 symplectic group as the
Fourier transform of a finite linear combination of regular
elliptic orbital integralsan 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 MoyPrasad 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)
 Savin, Gordan; Gan, Wee Teck

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)
 Tan, Victor

Poles of Siegel Eisenstein Series on $U(n,n)$
Let $U(n,n)$ be the rank $n$ quasisplit 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)
 Roberts, Brooks

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 
