Discrete Series Representation - Semisimple Groups

Semisimple Groups

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 SL₂(R).) It applies in particular to special linear groups; of these only SL₂(R) has a discrete series (for this, see the representation theory of SL₂(R)).

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

L + ρ,

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 group W_K 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 (mod the Weyl group W_G of G) under the Harish-Chandra correspondence identifying infinitesimal characters of G with points of

t⊗C/W_G.

So for each discrete series representation, there are exactly

|W_G|/|W_K|

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 (represented by a locally integrable function), 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 (where the product is over roots α having positive inner product with the vector v).

Harish-Chandra's regularity theorem implies that the character of a discrete series representation is a locally integrable function on the group.