On the geometric approach to the discrete series
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, Miličić, 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.