On the geometric approach to the discrete series

TL;DR

This paper explores the geometric structure of discrete series representations of real semisimple Lie groups, establishing explicit formulas and correspondences that unify different parametrizations and provide new proofs of classical results.

Contribution

It introduces a geometric formula for the -homology of modules over complex semisimple Lie algebras and applies it to connect Harish-Chandra's and Beilinson-Bernstein's parametrizations of discrete series.

Findings

Established a geometric formula for -homology in terms of localization on the flag variety.

Provided a geometric proof of Schmid's -homology description for discrete series.

Deduced Blattner's conjecture on K-type multiplicities from the Borel-Weil-Bott theorem.

Abstract

Harish-Chandra classified discrete series representations of real semisimple Lie groups by describing their characters as tempered distributions with an explicit formula on the elliptic set. His approach was inspired by Weyl's proof of the character formula for irreducible representations of compact Lie groups. Hecht, Mili\v{c}i\'{c}, Schmid and Wolf gave an alternative construction using the localization theory of Beilinson and Bernstein: the discrete series are the global sections of standard Harish-Chandra sheaves on the flag variety attached to the closed orbits of the complexification of a maximal compact subgroup. Their approach was inspired by the Borel-Weil theorem. In this paper, we give an explicit correspondence between these parametrizations. First, for a nilpotent radical $\mathfrak{n}$ of any Borel subalgebra, we establish a geometric formula for the $\mathfrak{n}$-homology of a module over the universal enveloping algebra of a complex semisimple Lie algebra $\mathfrak{g}$, in terms of its localization on the flag variety of $\mathfrak{g}$. Then we give applications of the formula to the discrete series. First, we give a geometric proof of Schmid's result describing the $\mathfrak{n}$-homology of discrete series representations for the nilpotent radical $\mathfrak{n}$ attached to a Borel subalgebra containing the Lie algebra of a maximal torus. Using Osborne's formula, this gives the formula for the character on the elliptic set, and leads to the matching of Harish-Chandra's parameters for discrete series representations with the geometric parameters. This gives an alternative approach to the study of the discrete series. As an example, we deduce Blattner's conjecture on the multiplicities of K-types of the discrete series from the Borel-Weil-Bott theorem for the maximal compact subgroup.