An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type

Abstract

Let G be a connected non-compact semi-simple Lie group admitting a finite dimensional faithful representation. Let K be a maximal compact subgroup of G. We suppose that GIK is hermitian symmetric. When G -SL(2, R), V. Bargmann [31 constructed the discrete series for G on spaces of square-integrable holomorphic functions on GIK (which is isomorphic to the unit disc) and in the general case Harish-Chandra (cf. [5]) constructed part of the discrete series for G in a similar way. (See also [6].) However, in the general case, this method does not yield all the discrete series. In analogy with the Borel-Weil-Bott theorem [4, 8(a)], it was suggested in [91 that all the discrete classes (which were obtained by Harish-Chandra in [7(f)]) might be realized on spaces of square-integrable harmonic forms of type (0, q) (square-integrable D-cohomology spaces) with coefficients in holomorphic vector bundles on GIK arising from finite dimensional irreducible unitary representations of K. (When q = 0, we have the case which was considered in [5].) We prove in this paper that most of the discrete classes are obtained in this way. We proceed to describe the results of this paper in more detail. Let g and f be the Lie algebras of G and K respectively. Let t be a Cartan subalgebra of t. Then t is also a Cartan subalgebra of g. Choose an ordering on the roots of (gC, tc) compatible with the complex structure on GIK. For an irreducible unitary representation z-A of K with the highest weight A, we denote by the holomorphic vector bundle on GIK associated to the contragredient representation. Let H,,q(EA) denote the Hilbert space of square-integrable harmonic forms of type (0, q) with coefficients in (these are the squareintegrable a-cohomology spaces attached to EA defined in [11]). The unitary representation wr of G on H,,q(EA) decomposes into a finite number of irreducible representations each of which belongs to the discrete series for G (Proposition 3.1). Let p denote the half sum of positive roots of (gC, tc) and let qA