Abstract

The talk will concern an aspect of the representation theory of Lie groups motivated by applications to bounds for period integrals.

I will introduce a notion of a vector in an irreducible unitary representation being “localized”. I will then plead the case that this definition is interesting.

For example, recall that Schur’s lemma says that the equivariant operators on an irreducible unitary representation are constant multiples of the identity. I will describe an asymptotic classification of “approximately equivariant” operators when restricted to localized vectors.

I will indicate some applications of this simple-sounding result, particularly to establishing the invariance of certain limiting measures that arise in semiclassical analysis and number theory.