Representation theory on the open Bruhat cell

摘要

The action of a connected reductive algebraic group G on G/P_, where P_ is a parabolic subgroup, differentiates to a representation of the Lie algebra g of G by vector fields on U_+, the unipotent radical of a parabolic opposite to P_. The classical instances of this setting that we study in detail are the actions of GL_n on the Grassmannian of k-planes (1 ≤ k ≤ n), of SO_n on the quadric of isotropic lines, and of SO_(2n) or SP_(2n) on their respective Grassmannians of maximal isotropic spaces; in each instance, U_+ is one of the usual affine charts. We show that both the polynomials on U_+ and the polynomial vector fields on U_+ form g-modules dual to parabolically induced modules, construct an explicit composition chain of the former module in the case where G is classical simple and U_+ is Abelian—these are exactly the cases above—and indicate how this chain can be used to analyse the module of vector fields, as well. We present two proofs of our main theorems: one uses the results of Enright and Shelton on classical Hermitian pairs, and the other is independent of their work. The latter proof mixes classical (and briefly reviewed) facts of representation theory with combinatorial and computational arguments, and is accessible to readers unfamiliar with the vast modern literature on category O.