Discrete series

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left 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.

If G is unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one (and hence all) matrix coefficient

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

When G is unimodular, the discrete series representation has a formal dimension d, with the property that

for v, w, x, y in the representation. When G is compact this coincides with the dimension when the Haar measure on G is normalized so that G has measure 1.

Harish-Chandra (1965, 1966) classified 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 T in K must be a Cartan subgroup in G. (This result required that the center of G be finite, ruling out groups such as the simply connected cover of SL(2,R).) It applies in particular to special linear groups; of these only SL(2,R) has a discrete series (for this, see the representation theory of SL(2,R)).

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