Equal rank local theta correspondence as a strong Morita equivalence

Let (G, H) be one of the equal rank reductive dual pairs (Mp2n, O2n+1) or (Un, Un) over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say G θ and H θ , of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C∗-algebras whose spectra are Gθ and Hθ . This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees

Arithmetic Aspects of Bianchi Groups

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data