Proof of Theorems 16.22 and 16.24

Abstract

When the infinitesimal character is integral, Theorem 16.22 is the main result (Theorem 7.3) of [55]; it is a more or less straightforward consequence of [8] and [40]. The proofs in [55] can be modified easily to cover the case when the simple root system Δ (O) of (16.9) is contained in a set of simple roots for R(G, T(O). Unfortunately this is not always the case. That the general case can be treated has been known to various experts for many years, but there does not seem to be an account of it in print. The geometric part of the argument for the case of Verma modules may be found in the first chapter of [37]. The outline below is gleaned from conversations with Bernstein, Brylinski, Kashiwara, and Lusztig; it is due to them and to Beilinson. To simplify the notation, we take the central element to (z in Z(zG)0z (be trivial; this changes nothing.