Further Generalization of Generalized Verma Modules

0.1. Let G be a complex semisimple Lie group, B a Borel subgroup, P a parabolic subgroup containing B, g=Lie(G), 6=Lie(£), |)=Lie(P), and E a finite dimensional irreducible £/(}))-module, where U(—) denotes the enveloping algebra. A £7(g)-module of the form U($)®vwE is called a generalized Verma module [24] and, in the special case where £=&, it is called a Verma module (cf. [9] and its references). In the course of proving the Kazhdan-Lusztig conjecture [21], it was shown [1], [8] that the Verma modules correspond to the local cohomologies at the 5-orbits on G/B via the localization functor. Thus it is natural to ask what are the £/(g)-modules corresponding to the local cohomologies at the 5-orbits on G/P. In this paper, we shall give an answer to this problem. It turns out that here appears a further generalization of the generalized Verma modules. We shall construct these £/(g)-modules in a purely algebraic way as follows. Let ;JT be the set of linear characters of the Lie algebra p, A the ring of polynomial functions on Jf, and c: p—>.4 the canonical homomorphism, which we shall consider as an ,4-valued character of a Cartan subalgebra, say t, contained in b=LiQ(B). Let 1 be the lowest weight of a finite dimensional irreducible pmodule, W the Weyl group, Wi the Weyl subgroup of W corresponding to P, and w an element of W which is longest in the coset wWi. Let UA(—)—U(—} ®c,4 and define the 'universal' Verma module MA(w(c+X—p)—p) by MA(w(c+ A—p)—p) = UA(Q)®UAa,).w<.c+i-P)-pA, where p is the half of the sum of the positive roots. Note that w(c+X—p)—p is not fully universal as a character of i but it is universal among the characters lying on a certain facet with respect to the reflection group W (translated by w(X—p)—p). Hence MA(w(c-t X—p)—p) resembles to reducible Verma modules and we can construct its quotient VA(w, c+A, p) in the same way as the construction of the simple quotient