Strongly Harmonic Forms for Representations in the Discrete Series

✍ Scribed by L. Barchini

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 172 KB
• Volume: 161
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3327

✦ Synopsis

Let G be a semisimple connected Lie group and let K be a maximal compact subgroup. Assume that rank G=rank K, and let T/K be a Cartan subgroup of G. The quotient GÂT carries an indefinite G-invariant hermitian form. The standard Dolbeault operator has a formal adjoint differential operator * inv with respect to the invariant hermitian form. Let s denote the complex dimension of KÂT. We form the indefinite harmonic space H s (GÂT, L /+2\ )=[(0, s)+L /+2\ _valued forms in Ker & Ker * inv ]. In this paper we show that under some positivity conditions on / the cohomology space H s (GÂT, L / ) contains a copy of the representation in the discrete series of G with parameter /.

1999 Academic Press

0. Introduction

Let G be a semisimple connected Lie group with finite center and K/G a maximal compact subgroup. We assume that G contains a compact Cartan subgroup T. In this situation G has a nonempty discrete series. Representations in the discrete series are associated to regular elliptic coadjoint orbits O. These orbits can be turned into homogeneous complex manifolds. With no loss of generality we can set O#GÂT. Once an invariant complex structure on GÂT has been fixed, every homogeneous line bundle L / Ä GÂT can be turned uniquely into a homogeneous holomorphic line bundle. The orbit GÂT carries two hermitian forms, one indefinite G-invariant, the other positive definite. Since T is compact, the positive definite form happens to be G-invariant. This is the metric used by Schmid in [S1] and [S2] to realize discrete series representations in the space of L 2 harmonic forms. In particular, if is the standard Dolbeault operator and if * pos denotes its formal adjoint differential operator with respect to the positive metric, then Schmid proved that, under certain positivity conditions on L / , every K-finite Dolbeault cohomology class admits a square integrable representative in Ker & Ker * pos .

In this paper we consider * inv , the formal adjoint operator to with respect to the invariant hermitian form on GÂT, and prove the following theorem.