Joe Jenkins

On Gelfand pairs associated with solvable Lie groups

Let G be a locally compact group, and let K be a compact subgroup of Aut(G) , the group of automorphisms of G. There is a natural action of K on the convolution algebra L (G), and we denote by LK(G) the subalgebra of those elements in L (G) that are invariant under this action. The pair (K, G) is called a Gelfand pair if LI(G) is commutative. In this paper we consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group. We characterize such Gelfand pairs (K, G), and determine a moduli space for the associated K-spherical functions.

Bounded K-spherical functions on Heisenberg groups

Abstract Let Hn be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of Aut(Hn), the group of automorphisms of Hn. The pair (K, Hn) is called a Gelfand pair if LK1(Hn), the subalgebra of elements of L1(Hn) that are invariant under the action of K, is commutative. In this case, the continuous homomorphisms on LK1(Hn) are given by integrating against certain K-invariant functions on Hn. These functions are the K-spherical functions associated to the Gelfand pair (K, Hn). In this paper we show how to compute the bounded K-spherical functions on Hn.

The moment map for a multiplicity free action

Let K be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space V. One calls this a multiplicity-free action whenever the AT-isotypic components of C[V] are A-irreducible. We have shown that this is the case if and only if the moment map t : V —► t* for the action is finite-to-one on A-orbits. This is equivalent to a result concerning Gelfand pairs associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.

A fixed point theorem for exponentially bounded groups

Abstract A fixed point property for linear actions of locally compact groups is presented. It is shown, in particular, that if G is either a finitely generated, discrete solvable group or a connected group then G has this fixed point property if, and only if, G is exponentially bounded.

Primary projections on L2 of a nilmanifold

Let N denote a connected, simply connected nilpotent Lie group with discrete cocompact subgroup Γ. Let U denote the quasi-regular representation on N on L2(NΓ). L2(NΓ) can be written as a direct sum of primary subspaces with respect to U. A realization for the projections of L2(NΓ)) onto these primary summands is given in this paper.