Character formulas for discrete series on semisimple Lie groups

Rebecca A. Herb

doi:10.1017/s0027763000017542

Abstract

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

Highlights

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G
G always has discrete series representations. The characters of these representations are distributions on the group G, realizable as locally integrable functions. Formulas for these characters are known up to certain integer constants which have only been evaluated for a few special cases
Denote by ϊ the subalgebra of © corresponding to K, and let t be a Cartan subalgebra of © such that t c ϊ

Summary

Introduction

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. For any connected component ψ of Ij'OR) = {if e §: α(ff) ^fc 0 for all real roots aeΦ(®c,§c)}, there are integers cy(w: λ: ζ+) such that for H e ψ Π $', ψ = ίj Π©', (1.2) Γ2(fl) = ^(fl)"[1] Σ det w φ : J: $+) exp

Objectives

Results

Conclusion