Nilpotent Gelfand pairs and the Heisenberg fan

Gelfand pairs play an important role across representation theory, from the classification of the irreducible representations of S_n to the computation of Euler products of L-functions in number theory. In this talk, I focus on the representation theory of the Gelfand pair (U(n)H_n, H_n), where H_n is the n+1-dimensional real Heisenberg group. I review the representation theory of the Heisenberg group H_n as well as the semi-direct product U(n)H_n. This involves a (brief) discussion of Kirillov's orbit method classifying unitary representations of nilpotent, simply connected, real Lie groups and a classification theorem coming from Mackey theory. The talk culminates with a discussion of the "Heisenberg Fan," a beautiful visualization of the structure of L^2(H_n) as a U(n)H_n-module. Time permitting, I will discuss an extension of the orbit method to the group U(n)H_n, as well as generalizations of the theory to other nilpotent Gelfand pairs.

This talk is geared towards the usual attendees of this seminar. While some Lie theory will necessarily come up, the talk should be coherent to those without significant prior knowledge.