If G is unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one matrix coefficient with v, w non-zero vectors is square-integrable on G, with respect toHaar 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.
Semisimple groups
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 subgroupK. In other words, a maximal torusT in K must be a Cartan subgroup in G. It applies in particular to special linear groups; of these only SL has a discrete series. Harish-Chandra's classification of the discrete series representations of a semisimple connected Lie group is given as follows. If L is the weight lattice of the maximal torus T, a sublattice of it where t is the Lie algebra of T, then there is a discrete series representation for every vector v of where ρ is the Weyl vector of G, that is not orthogonal to any root of G. Every discrete series representation occurs in this way. Two such vectors v correspond to the same discrete series representation if and only if they are conjugate under the Weyl groupWK of the maximal compact subgroup K. If we fix a fundamental chamber for the Weyl group of K, then the discrete series representation are in 1:1 correspondence with the vectors of L + ρ in this Weyl chamber that are not orthogonal to any root of G. The infinitesimal character of the highest weight representation is given by v under the Harish-Chandra correspondence identifying infinitesimal characters of G with points of So for each discrete series representation, there are exactly discrete series representations with the same infinitesimal character. 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 a Schwartz distribution, with singularities. The character is given on the maximal torus T by When G is compact this reduces to the Weyl character formula, with v = λ + ρ for λ the highest weight of the irreducible representation. Harish-Chandra's regularity theorem implies that the character of a discrete series representation is a locally integrable function on the group.
Limit of discrete series representations
Points v in the coset L + ρ orthogonal to roots of G do not correspond to discrete series representations, but those not orthogonal to roots of K are related to certain irreducible representations called limit of discrete series representations. There is such a representation for every pair where v is a vector of L + ρ orthogonal to some root of G but no orthogonal to any root of K corresponding to a wall of C, and C is a Weyl chamber of G containing v. Two pairs give the same limit of discrete series representation if and only if they are conjugate under the Weyl group of K. Just as for discrete series representations v gives the infinitesimal character. There are at most |WG|/|WK| limit of discrete series representations with any given infinitesimal character. Limit of discrete series representations are tempered representations, which means roughly that they only just fail to be discrete series representations.
Constructions of the discrete series
Harish-Chandra's original construction of the discrete series was not very explicit. Several authors later found more explicit realizations of the discrete series.
constructed most of the discrete series representations in the case when the symmetric space of G is hermitian.
constructed many of the discrete series representations for arbitrary G.
conjectured, and proved, a geometric analogue of the Borel-Bott-Weil theorem, for the discrete series, using L2 cohomology instead of the coherent sheaf cohomology used in the compact case.
An application of the index theorem, constructed all the discrete series representations in spaces of harmonic spinors. Unlike most of the previous constructions of representations, the work of Atiyah and Schmid did not use Harish-Chandra's existence results in their proofs.
