Limit of discrete series representation

Let $G$ be a linear connected semisimple real Lie group with compact Cartan subalgebra $\mathfrak t$ , and choose a Weyl chamber $C$ in the root system of $(\mathfrak g_{\mathbb C},\mathfrak t_{\mathbb C})$ . Let $\rho_C$ be half the sum of the $C$ -positive roots, and let $\lambda$ be dominant on the closure of $C$ , with $\lambda-\rho_C$ exponentiating to a character of the corresponding compact Cartan subgroup. The Harish-Chandra construction extends from regular $\lambda$ , which gives discrete series, to singular $\lambda$ . A limit of discrete series representation is a nonzero irreducible unitary representation $\pi(\lambda,C)$ obtained from such a singular parameter.

Nonvanishing and temperedness

The continued character $\pi(\lambda,C)$ is zero exactly when $\lambda$ is orthogonal to a compact $C$ -simple root. Every nonzero limit is irreducible and tempered. These criteria, and equivalence under the Weyl group of the compact roots, are part of the Knapp–Zuckerman classification.

Example

For $SU(1,1)$ , the compact Cartan has no compact roots. Moving a regular Harish-Chandra parameter to the singular value $0$ from either of the two Weyl chambers produces two distinct nonzero limits of discrete series. They are tempered but not square-integrable, and provide the archimedean representations associated with weight-one phenomena.

Conventions and scope

References

Anthony W. Knapp and Gregg J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Annals of Mathematics 116 (1982), 389–455, with appendix 493–501. Journal record. Relevant: Theorem 1.1 and the limit-of-discrete-series parameters.
Henri Carayol and Anthony W. Knapp, Limits of discrete series with infinitesimal character zero, Transactions of the American Mathematical Society 359 (2007), 5611–5651. DOI record. Relevant: introduction and §2 for the chamber construction, nonvanishing criterion, and examples.