Graded characters of induced representations, II, classification of principal series modules for complex groups

Abstract

Let G be a connected complex reductive algebraic group, which we view here as a real Lie group by taking its complex points and then ignoring its complex structure. The corresponding real Lie group obtained in this way will be denoted G,. The purpose of this paper is the classification of principal series modules for G, with integral regular infinitesimal character, up to isomorphism of Harish-Chandra modules. Using translation functors, this parametrizes all Lie algebra modules that occur, as the space of K,-finite sections, of an integral regular line bundle on the flag manifold of G,. (See 2.14.) Here the term “integral regular” means that we consider only line bundles that are GR-equivariant and whose space of C” sections have an integral regular infinitesimal character (the same as that of a finite dimensional G,-module). Our main result is the proof, in this complex case, of a conjecture that was stated in [4]. (See (OS).) We also give some estimates on Ext Groups (2.16) at the end of Section 2.