An algebraic construction of a class of representations of a semi-simple Lie algebra

Abstract

It is weltknown that the discrete class representations of a semisimple Lie group form the building blocks for the representation theory of semisimple Lie groups. Several attempts have been made to realize these representations by proving analogoues of the Borel-Weil-Bott theorem for noncompact symmetric spaces. We note that in any such analogue there is no "ab initio" proof that the space of "LZ-harmonic forms" concerned is nonzero. In this paper we give a straightforward algebraic construction of a class of irreducible, infinitesimally unitarizable representations of a semisimple Lie algebra. This class contains a special subseries of the discrete series. Our method is by explicitly constructing (through algebraic results about existence and uniqueness) certain operators on the direct sum of some cohomology spaces (of bundles on a compact flag manifold); the operators so defined will represent the given Lie algebra. Going into the details of this paper one can see that our construction has some applications (among them, for example, is a proof of Blattner's conjecture for the special subseries of the discrete series). In [1] Enright and Varadarajan obtained some modules which include all discrete classes. Recently (i.e. at the time of writing up this paper) Schmid has also obtained some modules which include all discrete series. We now begin to describe our results in detail. Let G be a connected noncompact semisimple Lie group with finite center. Let K be a maximal compact subgroup of G. Assume rank of of K = rank of G. Let T C K C G be a Cartan subgroup of G. Let t c k c g denote the Lie algebras of T, K, and G respectively, Let tCc kCc gc denote the complexifications of t C k C g. Let S be the set of roots of( t c, gC) and P C S a positive system of roots. Let Pk and P, denote the set of compact and noncompact roots in P respectively so that P = P k u P , . Let b c k c be the Borel subalgebra of k c, defined by