Discrete series representation

From Wikipedia, the free encyclopedia

In mathematics, a discrete series representation is an irreducible unitary representation of a topological group G that is a subrepresentation of the left (or equivalently right) regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.

1 Properties
2 Semisimple groups
3 Further work
4 See also
5 External link

[edit] Properties

If G is unimodular, a unitary representation ρ of G is in the discrete series if and only if one matrix coefficient

$\langle \rho(g)\cdot v, w \rangle \,$

with v, w non-zero vectors is square-integrable on G, with respect to Haar measure.

[edit] Semisimple groups

A basic result of Harish-Chandra from 1966 classifies the discrete series representations of connected semisimple groups G. In particular, such a group has discrete series representations if and only if it has the same rank as a maximal compact subgroup K. In other words, a maximal torus in K must be a Cartan subgroup in G. (This result required that the center of G be finite, ruling out cases of the metaplectic group covers.) It applies in particular to special linear groups; of these only SL₂(R) has a discrete series.

Harish-Chandra went on to prove an analogue for these representations of the Weyl character formula. In the case where G is not compact, the representations have infinite dimension, and the notion of character is therefore more subtle to define since it is essentially a Schwartz distribution, with singularities.

[edit] Further work

Much subsequent work in the area of semisimple groups has revisited Harish-Chandra's foundational results. New methods have been found, in particular algebraic, with the use of homological algebra, and geometric.

An application of the index theorem, to construct all the discrete series representations in spaces of harmonic spinors has been given (Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, 1977). Before the Atiyah-Schmid work, Robert Langlands had conjectured, and several authors proved, a geometric analogue of the Borel-Bott-Weil theorem, for the discrete series. The difference in attacks was that the Atiyah-Schmid was a priori, not assuming Harish-Chandra's main results, while the earlier work of Narasimhan, Okamoto, Parasarathy and Schmid was a posteriori.