Unitary highest weight modules

Abstract

The unitary highest weight modules of a semisimple Lie group G have been classified [S, 93. Unfortunately this classification does not fit well with a more general picture of the unitary dual of G. We give such a classification for G a classical group, using derived functors and the philosophy of unipotent representations. In particular these representations are associated to coadjoint orbits of G. Let 8, = Lie(G), 0 a Cartan involution of 6 = 8, @C. A derived functor module is defined by the data q = I + u a O-stable parabolic subalgebra of 0, and 17 an irreducible unitary representation of L, the stabilizer of L in G. Considering Z7 as a representation of I, write Z7= (A, rc) where A E c* (c = center of I, * denotes dual), and rt = Z7l r,,,, . We define A(& rc) = &(,I, rc) (cf. Definition 6.1). Now fix n: and vary 1. Under certain restrictions on I, which amount to a positivity condition on the corresponding bundle over G/L, A(I, rr) is known to be irreducible and unitary. As II varies within this range the Langlands parameters of A(A, rr) vary smoothly. However, one cannot obtain all unitary highest weight modules under these restrictions. The orbit method suggests how to obtain the remaining unitary highest weight modules. The modules A(& rr) are associated to coadjoint orbits as follows. If rc is trivial we say A(]“, rr) is associated to the elliptic orbit 0 = G. II. Not all unitary highest weight modules may be obtained with rc trivial, an example of which is the oscillator representation of $(2n, R) (the metapletic group). We define 2n irreducible unitary highest weight modules rc,+ (1 (2(n-k),R)) is 113 OOOl-8708/87 $7.50