Square Integrable Representations of Nilpotent Groups

Abstract

We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if $\Gamma$ is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in ${L_2}(N/\Gamma )$, and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.