Stable Automorphic Forms on Semisimple Groups

Advances in Mathematics, 2006

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semisimple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on "counting" (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups,

Groups, Geometry, and Dynamics, 2012

Notices of the American Mathematical Society, 2021

Journal of the European Mathematical Society, 2013

We provide partial results towards a conjectural generalization of a theorem of Lubotzky -Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for the most part, independent of whether the underlying global field is a number field or a function field. Our main result is Theorem 4 below. Before stating it, we provide some background. Let K be a global field (number field or function field), and let S be a nonempty set of finitely many inequivalent valuations of K including one from each class of archimedean valuations. The ring O S ⊆ K will denote the corresponding ring of S-integers. For any v ∈ S, we let K v be the completion of K with respect to v so that K v is a locally compact field. Let G be a noncommutative, absolutely almost simple, K-isotropic K-group. Let G be the semisimple Lie group endowed with a left-invariant metric. Notice that |S| is the number of simple factors of G. Under the diagonal embedding, the arithmetic group G(O S ) is a lattice in G. The lattice being noncocompact is equivalent to the assumption that G is K-isotropic. The metric on G restricts to a metric on G(O S ). Denote the Euclidean, or geometric, rank of G by k(G, S), so that k(G, S) = v∈S rank K v G.

Transactions of the American Mathematical Society, 2010

We construct automorphic representations of non-linear twofold covers of simply connected Chevalley groups via residues of Eisenstein series. In the process, we establish some basic results in representation theory of local groups.

Comptes Rendus Mathematique, 2016

We present several finiteness results for absolutely almost simple algebraic groups over finitely generated fields that are more general than global fields. We also discuss the relations between the various finiteness properties involved in these results, such as the properness of the global-to-local map in the Galois cohomology of a given K-group G relative to a certain natural set V of discrete valuations of K, and the finiteness of the number of isomorphism classes of K-forms of G having, on the one hand, smooth reduction at V and, on the hand, the same isomorphism classes of maximal K-tori as G. Résumé. Nous présentons plusieurs résultats de finitude pour les groupes algébriques absolument presque simples définis sur des corps de type fini plus généraux que les corps globaux. Nous discutons aussi des liens entre les propriétés de finitude divers qui entrent dans le cadre de notre analyse, tels que la propreté de l'application globale-locale dans la cohomologie galoisienne d'un K-groupe G par rapportà un ensemble convenable V de valuations discrètes de K, et la finitude du nombre de K-formes de G ayant, d'une part, bonne réduction en V , et, d'autre part, possédant les même classes d'isomorphisme de K-tores maximaux que G. même ensemble marcheégalement pour les groupes de type G 2. Dans les sections 2 et 3, il s'agit de l'analyse des groupes algébriques absolument presque simples possédant les mêmes classes d'isomorphisme de tores maximaux sur le corps de définition. Plus précisément, si G est un groupe algébrique simplement connexe absolument presque simple défini sur 1

Acta Mathematica Sinica, English Series, 2011

Let G = Spec A be an affine K-group scheme andà = {w ∈ A * : dim K A * · w · A * < ∞}. Let −, − : A * ×à → K, w,w := tr(ww), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A * .

2012

2011

Transactions of the American Mathematical Society, 1975

Let F F be a local field of characteristic zero and G {\text {G }} a connected algebraic group defined over F F . Let G G be the locally compact group of F F -rational points. One characterizes the group B ( G ) B(G) of g ∈ G g \in G whose conjugacy class is relatively compact. For instance, if G {\text {G}} is F F -split or reductive without anisotropic factors then B ( G ) B(G) is the center of G G . If H H is a closed subgroup of G G such that G / H G/H has finite volume, then the centralizer of H H in G G is contained in B ( G ) B(G) . If, moreover, H H is the centralizer of some x ∈ G x \in G then G / H G/H is compact.