##### Department of Mathematics,

University of California San Diego

****************************

### University of California Lie Theory Workshop

## Joseph Wolf

#### University of California, Berkeley

## Infinite Dimensional Gelfand Pairs

##### Abstract:

The simplest infinite dimensional Gelfand pairs are the ones of the form $(G,K) = \varinjlim (G_n,K_n)$ where the $(G_n,K_n)$ are finite dimensional Gelfand pairs. Here we take "Gelfand pair" to mean that the action of $G$ on a suitable Hilbert space $L^2(G/K)$ is multiplicity free, and we study several cases where that multiplicity free property holds. The strongest results are for cases where the $G_n$ are semidirect products $N_n\rtimes K_n$ with $N_n$ nilpotent. Then the $N_n$ are commutative or $2$--step nilpotent. In many cases where the derived algebras $[\mathfrak{n}_n,\mathfrak{n}_n]$ are of bounded dimension we construct $G_n$--equivariant isometric maps $\zeta_n : L^2(G_n/K_n) \to L^2(G_{n+1}/K_{n+1})$ and prove that the left regular representation of $G$ on the Hilbert space $L^2(G/K) := \varinjlim L^2(G_n/K_n),\zeta_n$ is a multiplicity free direct integral of irreducible unitary representations.

Host: Efim Zelmanov

### February 16, 2008

### 1:00 PM

### NSB 1205

****************************