Unitary representations of Lie groups and operators of finite rank. (English) Zbl 0828.22013

Poguntke, Detlev

Ann. Math. (2) 140, No. 3, 503-556 (1994).

Let \(H\) be a connected Lie group, let \(C^*(H)\) be its \(C^*\)-algebra and let \(\pi \in \widehat {H}\) be an irreducible representation of \(H\) such that \(\pi (C^*(H))\) contains the compact operators. In the semi- simple case, there exist smooth functions \(f\) with compact support on \(H\) such that \(\pi(f)\) is a rank one projection. This fact is due to the bounded “finiteness” of the \(K\)-types. In the nilpotent case, Dixmier’s functional calculus gives us Schwartz-functions \(f\) such that \(\pi (f)\) is of finite rank. In this paper the author treats the general case and he shows that there exists for any \(H\) and \(\pi\) as above a smooth function \(f\) on \(H\), such that \(r \cdot f\) is integrable for any representative function \(r\) of \(H\) and such that \(\pi(f)\) is an operator of rank 1. Representative functions are matrix coefficients of finite dimensional representations of \(H\). These functions play an essential role in the proof. In the first part of the paper the author discusses representative functions on groups, he introduces a special class of Lie groups, the class \([MB]\), which are semidirect products of abelian groups \(B\) and a group \(M\), whose connected component \(M_0\) allows a locally faithful finite dimensional representation.
In sections 3 to 7 the (long and difficult) proof of the theorem for groups of type \([MB]\) is given. It proceeds by induction on the dimension of \(M_0\) and considers the different possibilities for the restriction of \(\pi\) to the nilradical of \(H\). The techniques used in these proofs are too involved to be explained here. Finally in section 8 it is shown that any connected simply connected Lie group \(H\) can be imbedded into a larger \([BM]\) group \(G\) and \(\pi\) can be extended to a representation \(\rho\) of \(G\) in such a way that the existence of a rapidly decreasing \(f \in L^1(G)\) with \(\text{rank}(\rho (f)) < \infty\) implies the existence of a \(g \in L^1(H)\) such that \(\pi(g)\) is a projection of rank 1.

Reviewer: J.Ludwig (Metz)

MSC:

22E27

Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)

Keywords:

connected Lie groups; operators of finite rank in \(\pi(C^* (G))\); \(L^ 1(G)\); irreducible representation