Limits of weight spaces, Lusztig’s 𝑞-analogs, and fiberings of adjoint orbits

Abstract

Let G be a connected complex semisimple algebraic group, and T a maximal torus inside a Borel subgroup B, with g, t, and b their Lie algebras. Let V be a representation in the category a for g. The t-decomposition V = met* VA of V into a direct sum of finite-dimensional weight spaces is central in the representation theory of g. In this paper, we introduce on weight spaces a new structure, the principal filtration Je (V) , where e is a principal nilpotent in g chosen to be compatible with t; for example, e can be the sum of the simple root vectors relative to (t, b) . This filtration is constructed in a very simple way by taking J' (V') to be the space of vectors annihilated by the (p + 1)th power of e, for p > 0. Our approach is motivated by Kostant's fundamental work [Ki, K2] on actions of the principal TDS (three-dimensional subalgebra) and coordinate rings of regular adjoint orbits. In Theorem 3.4, we give a new description, in terms of the dimension jumps of the principal filtration of the weight space, of Lusztig's [L] q-analog mm (q) of dominant ,-weight multiplicity in a finite-dimensional irreducible g-representation V>. The polynomial m (q) was defined algebraically as an alternating sum over the Weyl group, through a q-analog of Kostant's weight multiplicity formula. We prove that m (q) is equal to the jump polynomial