Infinite characters on GL(n,Q), on SL(n,Z), and on groups acting on trees

Caractères infinis de GL(n,Q), de SL(n,Z) et de groupes agissant sur un arbre

Bachir Bekka (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Answering a question of J. Rosenberg from [Ros–89], we construct the first examples of infinite characters on GL n (K) for a global field K and n ≥ 2. The case n = 2 is deduced from the following more general result. Let G a non amenable countable subgroup acting on locally finite tree X. Assume either that the stabilizer in G of every vertex of X is finite or that the closure of the image of G in Aut(X) is not amenable. We show that G has uncountably many infinite dimensional irreducible unitary representations (π, H) of G which are traceable, that is, such that the C *-subalgebra of B(H) generated by π(G) contains the algebra of the compact operators on H. In the case n ≥ 3, we prove the existence of infinitely many characters for G = GL n (R), where n ≥ 3 and R is an integral domain such that G is not amenable. In particular, the group SL n (Z) has infinitely many such characters for n ≥ 2.

Domaines: Mathématiques [math]; Analyse fonctionnelle [math.FA]

Bachir Bekka. Infinite characters on GL(n,Q), on SL(n,Z), and on groups acting on trees. Journal of Functional Analysis, 2019, 277 (7), pp.2160-2178. ⟨10.1016/j.jfa.2018.10.003⟩. ⟨hal-01824201⟩