Square-Integrable Factor Representations of Locally Compact Groups

Abstract

The well-known theory of square-integrable representations is generalized to the case of primary representations (not necessarily type I) quasi-contained in either the regular representation or the representation induced from a character of the center of a (not necessarily unimodular) locally compact group, and relations with the topology of the primitive ideal space of the group ${C^\ast }$-algebra are obtained. The cases of discrete and almost connected groups are examined in more detail, and it is shown that for such groups, square-integrable factor representations must be traceable. For connected Lie groups, these representations can (in principle) be determined up to quasi-equivalence using a complicated construction of L. Pukanszky-for type I simply connected solvable Lie groups, the characterization reduces to that conjectured by C. C. Moore and J. Wolf. In the case of unimodular exponential groups, essentially everything is as in the nilpotent case (including a result on multiplicities in the decomposition of ${L^2}(G/\Gamma )$, $\Gamma$ a discrete uniform subgroup of G). Finally, it is shown that the same criterion as for type I solvable Lie groups characterizes the squareintegrable representations of certain solvable $\mathfrak {p}$-adic groups studied by R. Howe.