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 $\PS_\Delta {\rm L} (2,\mathbb{H})$.

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie 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 $\bigoplus_{i\in I} L(G_i)$.

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 ``Jacquet-Langlands'' 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).

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.

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 self-contained and full proofs are given of all the fundamental results.

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.

We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi$ of a unipotent subgroup, for the non-tempered 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.

Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\ggo$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\ggo(F)$ sont des distributions $J_M(X,f)$---$f$ est une fonction test---indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d'Arthur. Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont bornées sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un résultat bien connu de Harish-Chandra. Si $M \subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_M^b(X,f)$, égales à $J_M(X,f)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, {\it etc.}) restent bornées quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

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.

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 non-Archimedean local field, and $\psi$ a non-trivial 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 modular---Shimura or Drinfeld---varieties.

The representation parabolically induced from an irreducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations.

We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split 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 Rankin-Selberg 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.

In this paper it is shown that inclusions inside the Segal-Wilson 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.

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let $\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ 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 $\chi := (\Tr_{\La_1},\dots, \Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega \colon \C^{n} \times (\C^\times)^{\,n-l} \to C$. 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~$C$.

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 Harish-Chandra 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.''

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 quasi-split classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a quadratic extension $E/F$. We prove the self-duality of unitary supercuspidal data of standard Levi subgroups of $G_n(F)$ which give discrete series representations of $G_n(F)$.

J.~Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D.~Kazhdan in the connected case.

For a reductive $p$-adic group $G$, we compute the supports of the Hecke algebras for the $K$-types for $G$ lying in a certain frequently-occurring class. When $G$ is classical, we compute the intertwining between any two such $K$-types.

In this paper, we use Jacquet module methods to study the problem of classifying discrete series for the classical $p$-adic groups $\Sp(2n,F)$ and $\SO(2n+1,F)$.

The main new results in this paper are contained in the geometric Theorems 1 and~2 of Section~0.1 below and they are related to previous results of M.~Gromov and of myself (\cf\ \cite{1},~\cite{2}). These results are used to prove some general potential theoretic estimates on Lie groups (\cf\ Section~0.3) that are related to my previous work in the area (\cf\ \cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos (\cf\ \cite{5},~\cite{21}).

In the first part of this paper generalizations of Hesselink's $q$-analog of Kostant's multiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$-analog of the Kostant-Rallis theorem is given for the real group $\SL(4,\mathbb{R})$ (that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals---an 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 Moy-Prasad filtrations of $p$-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

The trace formula contains terms on the spectral side that are constructed from unramified automorphic $L$-functions. We shall establish an identify that relates these terms with corresponding terms attached to endoscopic groups of $G$. In the process, we shall show that the $L$-functions of $G$ that come from automorphic representations of endoscopic groups have meromorphic continuation.

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.

Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.