Compatible Discrete Series

Several very interesting results connecting the theory of abelian ideals of Borel subalgebras, some ideas of D. Peterson relating the previous theory to the combinatorics of affine Weyl groups, and the theory of discrete series are stated in a recent paper (\cite{Ko2}) by B. Kostant. In this paper we provide proofs for most of Kostant's results extending them to $ad$-nilpotent ideals and develop one direction of Kostant's investigation, the compatible discrete series.Comment: AmsTex file, 27 Pages; minor corrections; to appear in Pacific Journal of Mathematic

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

On the K-structure of the discrete series

Preprin

The annihilator of invariant vectors for a quantized parabolic subalgebra

Preprin

A construction of singular unitary representations of reductive Lie groups

Tesi di Dottorat

Decomposizione del prodotto esterno di una rappresentazione del gruppo Sp(n) × Sp(1)

Tesi di Laure

Abelian subalgebras in Z2-graded Lie algebras and affine Weyl groups

Abstract Let $G=G_0 \oplus G_1$ be a simple ℤ2-graded Lie algebra and let $b_0$ be a fixed Borel subalgebra of $G_0$. We describe and enumerate the abelian $b_0$-stable subalgebras of $G_1$. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved

Ad-nilpotent ideals containing a fixed number of simple root spaces

We give formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a Lie algebra of type B or D containing a fixed number of root spaces attached to simple roots. This result solves positively a conjecture of Panyushev (cf. D. Panyushev, ad-nilpotent ideals: generators and duality, J. of Alg., to appear) and affords a complete knowledge of the above statistics for any simple Lie algebra. We also study the restriction of the above statistics to the abelian ideals of a Borel subalgebra, obtaining uniform results for any simple Lie algebra.Comment: Latex, 9 page, revised and expanded version: a uniform formula for the statistics in the abelian case is provide