Square-integrable representation modulo the center

Definition

Let $G$ be a second-countable unimodular locally compact group with center $Z$ , and fix an invariant measure on $G/Z$ . An irreducible unitary representation $\pi$ of $G$ is square-integrable modulo the center if some nonzero matrix coefficient satisfies

\int_{G/Z}\left|\langle \pi(g)u,v\rangle\right|^2\,d(gZ)<\infty.

Schur's lemma gives a unitary central character $\chi_\pi$ with $\pi(z)=\chi_\pi(z)I$ , so the absolute value of the coefficient is constant on central cosets and the integrand is well defined. Such representations are also called the relative discrete series.

Equivalent formulations and orthogonality

For an irreducible unitary representation in this setting, existence of one nonzero square-integrable coefficient modulo $Z$ implies the corresponding orthogonality relations for all coefficients. After a normalization of measure, these relations involve a positive formal degree. This is the central-quotient analogue of the coefficient criterion for the discrete series; the general operator-valued orthogonality theory is developed by Duflo–Moore, §§2–3.

Why the quotient is necessary

If $Z$ is noncompact, every coefficient of an irreducible representation has constant modulus along $Z$ . A nonzero coefficient therefore cannot usually belong to $L^2(G)$ , even when it is square-integrable on $G/Z$ . For a real reductive group with noncompact center, relative discrete series is consequently the useful replacement for ordinary discrete series.

Conventions and scope

References

Michel Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, Journal of Functional Analysis 21 (1976), 209–243. DOI record. Relevant: §§2–3 on square-integrable representations and orthogonality operators.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained book record. Relevant: Chapter XII on discrete series and square integrability modulo the center.