Richardson elements for classical Lie algebras

Abstract

Parabolic subalgebras of semi-simple Lie algebras decompose as p = m ⊕ n where m is a Levi factor and n the corresponding nilradical. By Richardson's theorem [R.W. Richardson, Bull. London Math. Soc. 6 (1974) 21–24], there exists an open orbit under the action of the adjoint group P on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for Richardson elements in the classical case. This generalizes a construction for gl N of Brüstle et al. [Algebr. Represent. Theory 2 (1999) 295–312] to the other classical Lie algebra and it extends the authors normal forms of Richardson elements for nice parabolic subalgebras of simple Lie algebras to arbitrary parabolic subalgebras of the classical Lie algebras [K. Baur, Represent. Theory 9 (2005) 30–45]. As applications we obtain a description of the support of Richardson elements and we recover the Bala–Carter label of the orbit of Richardson elements.