Dirac cohomology and unipotent representations of complex

groups

Dan Barbasch and Pavle Pandˇ zi´c

This paper is dedicated to Henri Moscovici.

Abstract. This paper studies unitary representations with Dirac cohomology

for complex groups, in particular relations to unipotent representations.

1. Introduction

In this paper we will study the problem of classifying unitary representations

with Dirac cohomology. We will focus on the case when the group G is a complex

reductive group viewed as a real group. It will easily follow that a necessary condi-

tion for having nonzero Dirac cohomology is that twice the inﬁnitesimal character

is regular and integral. The main conjecture is the following.

Conjecture 1.1. Let G be a complex reductive Lie group viewed as a real

group, and π be an irreducible unitary representation such that twice the inﬁnites-

imal character of π is regular and integral. Then π has nonzero Dirac cohomology

if and only if π is cohomologically induced from an essentially unipotent represen-

tation with nonzero Dirac cohomology. Here by an essentially unipotent represen-

tation we mean a unipotent representation tensored with a unitary character.

We start with some background and motivation.

Let G be the real points of a linear connected reductive group. Its Lie algebra

will be denoted by g0. Fix a Cartan involution θ and write g0 = k0 + s0 for the

Cartan decomposition. Denote by K the maximal compact subgroup of G with Lie

algebra k0. The complexiﬁcation g := (g0)C decomposes as g = k + s.

A representation (π, H) on a Hilbert space is called unitary if H admits a

G-invariant positive deﬁnite inner product. One of the major problems of represen-

tation theory is to classify the irreducible unitarizable modules of G. As motivation

for why this problem is important, we present an example from automorphic forms.

2010 Mathematics Subject Classiﬁcation. Primary 22E47 ; Secondary 22E46.

Key words and phrases. Dirac cohomology, unipotent representations.

The ﬁrst author was supported in part by NSF Grants #0967386 and #0901104.

The second author was supported in part by a grant from the Ministry of Science, Education

and Sports of the Republic of Croatia.

1

Contemporary Mathematics

Volume 546, 2011

c 2011 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/546/10782