A Normal Form for Admissible Characters in the Sense of Lynch

Parabolic subalgebras p of semisimple Lie algebras define a Z-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of g on which the adjoint group of p acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of g1 is exactly one that involves all simple root spaces. It is however difficult to write down such nilpotent elements for general parabolic subalgebras. In this paper we give an explicit construction of admissible elements in g1 that uses as few root spaces as possible. Introduction Let g be a semisimple Lie algebra over C, p ⊂ g a parabolic subalgebra. There is a Z-grade of g, g = ∑ j gj such that p = ∑ j≥0 gj and n := ∑ j>0 gj is the nilradical of p. By a theorem of Richardson ([R]) there is always a Richardson element in n, i.e. an element X ∈ n satisfying [p, X ] = n. We say that p is nice if there is a Richardson element in the first graded part g1. Nice parabolic subalgebras have been classified in [BW]. If p is a Borel subalgebra, then a Richardson element of g1 involves all simple root spaces. For arbitrary nice parabolic subalgebras, the support of a Richardson element in the first graded part may consist of all roots of g1. In this sense it is far from being a simple representative of a Richardson element. The goal of this paper is to give a normal form of Richardson elements for nice parabolic subalgebras in the classical case. The construction uses as few root spaces of the nilradical as possible. It turns out that in many cases, the support of this normal form spans a simple system of roots. Since Richardson elements correspond to admissible characters ν : n → C, the normal form describes how admissible characters of the (opposite) nilradical actually look like. In his thesis [L], Lynch studied Whittaker modules for which there is an admissible homomorphism ν : n → C: Let U be the universal enveloping algebra of g and let V be a U-module. A vector v ∈ V is called a Whittaker vector if there exists a nonsingular character ν : n → C such that xv = ν(x)v for all x ∈ n. The module V is a Whittaker module if V is cyclically generated by a Whittaker vector. An element x ∈ g is called admissible if x ∈ g−1 and n x = {0}. In particular, x is a nilpotent element of g. There is a natural bijection between g−1 and the characters Date: October 13, 2004.