分野別セミナー

Let G be a Lie group with a bi-invariant positive measure and let H be a
closed subgroup of G satisfying the same conditions. Set S=G/H. Let D(S)
be the space of test functions on S and let D'(S) be its continuous anti
-dual. The group G acts on both spaces by left translations. Let R be a
continuous unitary irreducible representation of G on a Hilbert space V.
We shall discuss the following two problems:
1. Can R be realized on D'(S), i.e. does there exist a continuous linear
injection from V to D'(S), commuting with the action of G?
2. If such a realization exists, is it unique (up to scaling)?
We will focus on Problem 2 (which is related to the classical theory of
Gelfand pairs). We shall show recent answers and indicate an interesting
link with number theory.