26. CJM 2011 (vol 63 pp. 1107)
 Liu, Baiying

Genericity of Representations of pAdic $Sp_{2n}$ and Local Langlands Parameters
Let $G$ be the $F$rational points of the symplectic group $Sp_{2n}$,
where $F$ is a nonArchimedean local field
of characteristic
$0$. Cogdell, Kim, PiatetskiShapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$case of a
conjecture of
GrossPrasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.
Keywords:generic representations, local Langlands parameters Categories:22E50, 11S37 

27. CJM 2011 (vol 63 pp. 591)
28. CJM 2011 (vol 63 pp. 327)
 Jantzen, Chris

Discrete Series for $p$adic $SO(2n)$ and Restrictions of Representations of $O(2n)$
In this paper we give a classification of discrete series for
$SO(2n,F)$, $F$ $p$adic, similar to that of
MÅglinTadiÄ for
the other classical groups. This is obtained by taking the
MÅglinTadiÄ classification for $O(2n,F)$ and studying how the
representations restrict to $SO(2n,F)$. We then extend this to an
analysis of how admissible representations of $O(2n,F)$ restrict.
Category:22E50 

29. CJM 2010 (vol 62 pp. 1340)
30. CJM 2010 (vol 62 pp. 1310)
 Lee, KyuHwan

IwahoriHecke Algebras of $SL_2$ over $2$Dimensional Local Fields
In this paper we construct an analogue of IwahoriHecke algebras of $\operatorname{SL}_2$ over $2$dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is welldefined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider IwahoriMatsumoto type relations.
Categories:22E50, 20G25 

31. CJM 2010 (vol 62 pp. 914)
 Zorn, Christian

Reducibility of the Principal Series for Sp^{~}_{2}(F) over a padic 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 

32. CJM 2010 (vol 62 pp. 1116)
 Jin, Yongyang; Zhang, Genkai

Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups
Let $\mathbb G$ be a steptwo nilpotent group of Htype with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$Laplacian operator $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.
Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, Htype groups Categories:35H30, 26D10, 22E25 

33. CJM 2010 (vol 62 pp. 563)
34. CJM 2009 (vol 62 pp. 52)
 Deng, Shaoqiang

An Algebraic Approach to Weakly Symmetric Finsler Spaces
In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous RiemannFinsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the nonRiemannian reversible weakly
symmetric Finsler spaces we find are nonBerwaldian and with vanishing
Scurvature. This means that reversible nonBerwaldian Finsler spaces
with vanishing Scurvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, Scurvature Categories:53C60, 58B20, 22E46, 22E60 

35. CJM 2009 (vol 62 pp. 94)
36. CJM 2009 (vol 61 pp. 1375)
 Spallone, Steven

Stable Discrete Series Characters at Singular Elements
Write $\Theta^E$ for the stable discrete series character associated
with an irreducible finitedimensional representation $E$ of a connected
real reductive group $G$. Let $M$ be the centralizer of the split
component of a maximal torus $T$, and denote by $\Phi_M(\gm,\Theta^E)$
Arthur's extension of $ D_M^G(\gm)^{\lfrac 12}
\Theta^E(\gm)$ to $T(\R)$. In this paper we give a simple
explicit expression for
$\Phi_M(\gm,\Theta^E)$ when $\gm$ is elliptic in $G$. We do not assume $\gm$ is regular.
Category:22E47 

37. CJM 2009 (vol 61 pp. 1407)
 Will, Pierre

Traces, CrossRatios and 2Generator Subgroups of $\SU(2,1)$
In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 

38. CJM 2009 (vol 61 pp. 1325)
 Nien, Chufeng

Uniqueness of Shalika Models
Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$adic field,
and $D$ a quaternion division algebra over $\CF$. This paper proves
uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$,
and reobtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for
any $n\in \BN$.
Keywords:Shalika models, linear models, uniqueness, multiplicity free Category:22E50 

39. CJM 2009 (vol 61 pp. 961)
 Bernon, Florent

Transfert des intÃ©grales orbitales pour les algÃ¨bres de Lie classiques
Dans cet article, on consid\`ere un groupe semisimple $\rmG$ classique
r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un
sousgroupe de Cartan compact. On d\'efinit une famille de
sousalg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang
que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$.
Soit $\g'$ une telle sousalg\`ebre de Lie. On construit alors une
application de transfert des int\'egrales orbitales de $\g'$ dans
l'espace des int\'egrales orbitales de $\g$. On montre que cette
application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur
simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$.
Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de
rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette
application de transfert.
On utilise cette application de transfert pour obtenir une formule de
r\'eduction de l'int\'egrale de Cauchy HarishChandra pour les paires
duales d'alg\`ebres de Lie r\'eductives $\bigl( \Ug(p,q),\Ug(r,s)
\bigr)$ et $\bigl( \Sp(p,q),\Og^*(2n) \bigr)$ avec $p+q = r+s = n$.
Categories:22E30, 22E46 

40. CJM 2009 (vol 61 pp. 779)
 Grbac, Neven

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 quasisplit. 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 

41. CJM 2009 (vol 61 pp. 708)
 Zelenyuk, Yevhen

Regular Homeomorphisms of Finite Order on Countable Spaces
We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.
\item If $G$ is a countably infinite Abelian group with finitely many
elements of order $2$ and $\beta G$ is the Stone\v Cech
compactification of $G$ as a discrete semigroup, then for every
idempotent $p\in\beta G\setminus\{0\}$, the subset
$\{p,p\}\subset\beta G$ generates algebraically the free product of
oneelement semigroups $\{p\}$ and~$\{p\}$.
\end{compactenum}
Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone\v Cech compactification Categories:22A30, 54H11, 20M15, 54A05 

42. CJM 2009 (vol 61 pp. 691)
 Yu, Xiaoxiang

Prehomogeneity on QuasiSplit Classical Groups and Poles of Intertwining Operators
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a nonArchimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is nonabelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.
Categories:22E50, 20G05 

43. CJM 2009 (vol 61 pp. 373)
 McKee, Mark

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 

44. CJM 2009 (vol 61 pp. 427)
 Tadi\'c, Marko

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 

45. CJM 2009 (vol 61 pp. 351)
 Graham, William; Hunziker, Markus

Multiplication of Polynomials on Hermitian Symmetric spaces and LittlewoodRichardson Coefficients
Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicityfree. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual LittlewoodRichardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, LittlewoodRichardson coefficients, Jack polynomials Categories:14L30, 22E46 

46. CJM 2009 (vol 61 pp. 222)
47. CJM 2008 (vol 60 pp. 1306)
 Mui\'c, Goran

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 nonarchimedean 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 

48. CJM 2008 (vol 60 pp. 1001)
49. CJM 2008 (vol 60 pp. 1067)
 Kariyama, Kazutoshi

On Types for Unramified $p$Adic Unitary Groups
Let $F$ be a nonarchimedean local field of residue characteristic
neither 2 nor 3 equipped with a galois involution with fixed field
$F_0$, and let $G$ be a symplectic group over $F$ or an unramified
unitary group over $F_0$. Following the methods of BushnellKutzko for
$\GL(N,F)$, we define an analogue of a simple type attached to a
certain skew simple stratum, and realize a type in $G$. In
particular, we obtain an irreducible supercuspidal representation of
$G$ like $\GL(N,F)$.
Keywords:$p$adic unitary group, type, supercuspidal representation, Hecke algebra Categories:22E50, 22D99 

50. CJM 2008 (vol 60 pp. 790)
 Blasco, Laure

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 sousgroupe 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 
