Generalized Gelfand pairs attached to groups of Heisenberg type

FERNANDO LEVSTEIN; LINDA SAAL

San Pedro, Brasil

Congreso; Latin-American Colloquium in Algebra; 2009

USP

Groups of Heisenberg type were introduced by A. Kaplan in the 80´s. They were used to construct examples of isospectral varieties that are non- isometric. The group Aut(N) of automorphisms of a group of Heisenberg type was computed by Riehm (orthogonal ones) and Saal (the full goup). They contain a subgroup of the form K= Spin(m) x U where U are the automorphisms that fix every element in the center Z(N) and m=dim Z(N). The problem of classifying the pairs (K x N, N) that are Gelfand pairs was solved by F. Ricci when U is compact. In the non-compact case one can ask whether or not the pair is a generalized Gelfand pair. We were able to answer this question when N is irreducible and it turns out that this happens only for m less than 10. Finally we will show an example where it is possible to determine the spherical distributions that correspond to a generalized Gelfand pair.