76. CJM 2004 (vol 56 pp. 945)
 Helminck, Aloysius G.; Schwarz, Gerald W.

Smoothness of Quotients Associated \\With a Pair of Commuting Involutions
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.
Categories:20G15, 20G20, 22E15, 22E46 

77. CJM 2004 (vol 56 pp. 963)
78. CJM 2004 (vol 56 pp. 883)
 Tandra, Haryono; Moran, William

Kirillov Theory for a Class of Discrete Nilpotent Groups
This paper is concerned with the Kirillov map for a class of
torsionfree nilpotent groups $G$. $G$ is assumed to be discrete,
countable and $\pi$radicable, with $\pi$ containing the primes
less than or equal to the nilpotence class of $G$. In addition,
it is assumed that all of the characters of $G$ have idempotent
absolute value. Such groups are shown to be plentiful.
Category:22D10 

79. CJM 2004 (vol 56 pp. 293)
 Khomenko, Oleksandr; Mazorchuk, Volodymyr

Structure of modules induced from simple modules with minimal annihilator
We study the structure of generalized Verma modules over a
semisimple complex finitedimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
BernsteinGelfandGelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of HarishChandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.
Keywords:parabolic induction, generalized Verma module, simple module, Ha\rish\Chand\ra bimodule, equivalent categories Categories:17B10, 22E47 

80. CJM 2004 (vol 56 pp. 168)
 Pogge, James Todd

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 nonmaximal, nonBorel parabolic subgroups,
other than $\GL_n$.
Categories:11F70, 22E55 

81. CJM 2003 (vol 55 pp. 1155)
 Đoković, Dragomir Ž.; Litvinov, Michael

The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$
The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$orbit (or $K^0$orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$orbit (resp.\ $K^0$orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$orbits (resp.\ $K^0$orbits) of smaller dimensions. The
description of the closure of nilpotent $K$orbits has been known for
some time, but not so for the nilpotent $K^0$orbits. A conjecture
describing the closure of nilpotent $K^0$orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 

82. CJM 2003 (vol 55 pp. 1121)
 Bettaïeb, Karem

Classification des reprÃ©sentations tempÃ©rÃ©es d'un groupe $p$adique
Soit $G$ le groupe des points d\'efinis sur un corps $p$adique d'un
groupe r\'eductif connexe. A l'aide des caract\`eres virtuels
supertemp\'er\'es de $G$, on prouve (conjectures de Clozel) que toute
repr\'esentation irr\'eductible temp\'er\'ee de $G$ est
irr\'eductiblement induite d'une essentielle d'un sousgroupe de
L\'evi de~$G$.
Category:22E 

83. CJM 2003 (vol 55 pp. 1080)
 Kellerhals, Ruth

Quaternions and Some Global Properties of Hyperbolic $5$Manifolds
We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 

84. CJM 2003 (vol 55 pp. 969)
 Glöckner, Helge

Lie Groups of Measurable Mappings
We describe new construction principles for infinitedimensional Lie
groups. In particular, given any measure space $(X,\Sigma,\mu)$ and
(possibly infinitedimensional) Lie group $G$, we construct a Lie
group $L^\infty (X,G)$, which is a Fr\'echetLie group if $G$ is so.
We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an
arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie
group, modelled on the locally convex direct sum $\bigoplus_{i\in I}
L(G_i)$.
Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 

85. 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 

86. CJM 2002 (vol 54 pp. 1100)
 Wood, Peter J.

The Operator Biprojectivity of the Fourier Algebra
In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
Keywords:locally compact group, Fourier algebra, operator space, projective Categories:13D03, 18G25, 43A95, 46L07, 22D99 

87. CJM 2002 (vol 54 pp. 795)
 Möller, Rögnvaldur G.

Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
Willis's structure theory of totally disconnected locally compact groups
is investigated in the context of permutation actions. This leads to new
interpretations of the basic concepts in the theory and also to new proofs
of the fundamental theorems and to several new results. The treatment of
Willis's theory is selfcontained and full proofs are given of all the
fundamental results.
Keywords:totally disconnected locally compact groups, scale function, permutation groups, groups acting on graphs Categories:22D05, 20B07, 20B27, 05C25 

88. CJM 2002 (vol 54 pp. 828)
 Moriyama, Tomonori

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 MellinBarnes type for the radial part of our
spherical function.
Categories:22E45, 11F70 

89. CJM 2002 (vol 54 pp. 769)
 Miyazaki, Takuya

Nilpotent Orbits and Whittaker Functions for Derived Functor Modules of $\Sp(2,\mathbb{R})$
We study the moderate growth generalized Whittaker functions,
associated to a unitary character $\psi$ of a unipotent subgroup,
for the nontempered cohomological representation of $G = \Sp
(2,\mathbb{R})$. Through an explicit calculation of a holonomic
system which characterizes these functions we observe that their
existence is determined by the including relation between the real
nilpotent coadjoint $G$orbit of $\psi$ in
$\mathfrak{g}_{\mathbb{R}}^\ast$ and the asymptotic support of the
cohomological representation.
Categories:22E46, 22E30 

90. CJM 2002 (vol 54 pp. 263)
 Chaudouard, PierreHenri

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 testindexÃ©es par les
sousgroupes de LÃ©vi $M$ de $G$ et les Ã©lÃ©ments semisimples 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 HarishChandra. 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 semisimples rÃ©guliers, se dÃ©composent en produits de ces nouvelles
intÃ©grales locales.
Categories:22E35, 11F70 

91. 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 

92. 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 

93. CJM 2001 (vol 53 pp. 675)
94. CJM 2001 (vol 53 pp. 278)
 Helminck, G. F.; van de Leur, J. W.

Darboux Transformations for the KP Hierarchy in the SegalWilson Setting
In this paper it is shown that inclusions inside the SegalWilson
Grassmannian give rise to Darboux transformations between the
solutions of the $\KP$ hierarchy corresponding to these planes. We
present a closed form of the operators that procure the transformation
and express them in the related geometric data. Further the
associated transformation on the level of $\tau$functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 

95. 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 

96. CJM 2001 (vol 53 pp. 195)
 Mokler, Claus

On the Steinberg Map and Steinberg CrossSection for a Symmetrizable Indefinite KacMoody Group
Let $G$ be a symmetrizable indefinite KacMoody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2nl}}$ be the characters of the
fundamental irreducible representations of $G$, defined as convergent
series on a certain part $G^{\tralg} \subseteq G$. Following
Steinberg in the classical case and Br\"uchert in the affine case, we
define the Steinberg map $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2nl}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,nl} \to C$. We investigate the local behaviour of
$\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1)
\bigr)$, and we show that there exists a neighborhood of $(0,\dots,0)
\times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular
analytical map, satisfying a certain functional identity. This
identity has its origin in an action of the center of $G$ on~$C$.
Categories:22E65, 17B65 

97. CJM 2000 (vol 52 pp. 1192)
 Herb, Rebecca A.

Orbital Integrals on $p$Adic Lie Algebras
Let $G$ be a connected reductive $p$adic group and let $\frakg$ be its
Lie algebra. Let $\calO$ be any $G$orbit in $\frakg$. Then the orbital
integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution
on $\frakg $, and HarishChandra proved that its Fourier transform $\hat
\mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular
semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of
$\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give
a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$
sufficiently large. In the case that $\calO$ is a regular semisimple orbit,
the formula is already known by work of Waldspurger. In the case that
$\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at
infinity is already known because of its homogeneity properties. The
general case combines aspects of these two extreme cases. The formula
for $\hat\mu _{\calO}$ at infinity can be used to formulate a ``theory
of the constant term'' for the space of distributions spanned by the
Fourier transforms of orbital integrals. It can also be used to show
that the Fourier transforms of orbital integrals are ``linearly
independent at infinity.''
Categories:22E30, 22E45 

98. CJM 2000 (vol 52 pp. 1101)
 Zhang, Yuanli

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 quasisplit
classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a
quadratic extension $E/F$. We prove the selfduality of unitary
supercuspidal data of standard Levi subgroups of $G_n(F)$ which
give discrete series representations of $G_n(F)$.
Category:22E35 

99. CJM 2000 (vol 52 pp. 804)
100. CJM 2000 (vol 52 pp. 449)