Discrete series representation - Imported Knowlpedia Content

Definition

Let $G$ be a second-countable locally compact group with left Haar measure. A discrete series representation of $G$ is an irreducible unitary representation whose equivalence class occurs as a closed $G$ -invariant subspace of the left regular representation on $L^2(G)$ . Equivalently, it is an irreducible summand in the discrete part of that regular representation. The word “discrete” describes its occurrence as a Hilbert direct summand, not discreteness of the group, the representation space, or its parameter set.

Coefficient criterion

When $G$ is unimodular, $\pi$ belongs to the discrete series exactly when it has a nonzero matrix coefficient in $L^2(G)$ . Irreducibility then yields square integrability and Schur orthogonality for all coefficients, with a positive formal degree depending on the normalization of Haar measure Duflo–Moore, introduction and §§2–3. This characterization explains the common alternative name “square-integrable representation.”

Examples and existence

Every irreducible unitary representation of a compact group is discrete series after Haar measure is normalized. For connected semisimple real Lie groups with finite center, Harish-Chandra's criterion says that discrete series exists exactly when the group has a compact Cartan subgroup, equivalently when its rank equals the rank of a maximal compact subgroup Knapp, Chapter XII.

Relationship to relative discrete series

If the center is noncompact, central characters make a nonzero coefficient constant in modulus along central cosets, obstructing membership in $L^2(G)$ . The corresponding notion is then a square-integrable representation modulo the center. Neither notion should be confused with the discrete spectrum of an automorphic quotient, where occurrence is measured in a different $L^2$ -representation.

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: introduction and §§2–3 on the discrete part, 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 Harish-Chandra's discrete series.