Theta correspondence is a major theme in the theory of automorphic forms and in representation theory. The local version of the correspondence sets up a bijection between certain subsets of admissible duals of suitable pairs of reductive groups. There are two special cases in which the correspondence is known to enjoy extra features, the ‘equal rank’ case where temperedness is preserved and the ‘stable range’ case where unitarity is preserved.
In joint work with Bram Mesland (Leiden), we show that in these special cases, the local theta correspondence is actually given by a Morita equivalence of certain C*-algebras. There are interesting applications and some global questions that follow this result. Time permitting, I will discuss some of these.