Contemporary I\fathematics

Volume 467, 2008

The Omega-Regular Unitary Dual

of the Metaplectic Group of Rank 2

Alessandra Pantano, Annegret Paul, and Susana A. Salamanca-Riba

This paper is dedicated to the dear memory of Professor Fokko du Cloux.

ABSTRACT.

In this paper we formulate a conjecture about the unitary dual

of the metaplectic group. We prove this conjecture for the case of Mp(4,R).

The result is a strengthening, for this case, of the following result by the third

author: any unitary representation of a real reductive Lie group with strongly

regular infinitesimal character can be obtained by cohomological induction

from a one dimensional representation. Strongly regular representations are

those whose infinitesimal character is at least as regular as that of the trivial

representation. We are extending the result to representations with omega-

regular infinitesimal character: those whose infinitesimal character is at least

as regular as that of the oscillator representation. The proof relies heavily

on Parthasarathy's Dirac operator inequality. In one exception we explicitly

calculate the signature of an intertwining operator to establish nonunitarity.

Some of the results on intertwining operators presented in section 5.2 are joint

work of Dan M. Barbasch and the first author.

1.

Introduction

This paper is based on a presentation by the third author at the 13th Confer-

ence of African American Researchers in the Mathematical Sciences (CAARl\fS13).

The presentation was intentionally expository, aimed at non-experts in the field of

representation theory. \Vith this in mind, an introductory survey of the funda-

mental concepts underlying this work was provided. A brief extract of the original

presentation appears in the appendix. \Ve have limited the introductory remarks to

a discussion about

S£(2, JR),

as some results relative to this group are paramount

for understanding the main ideas of the paper.

2000 Mathematics Subject Classification. Primary 22E46.

Key words and phrases. Unitary dual, Dirac operator inequality, intertwining operators,

derived functor modules.

This material is based upon work supported by the National Science Foundation under Grants

No. DMS 0554278 and DMS 0201944.

@2008 American r...fathematical Society

http://dx.doi.org/10.1090/conm/467/09130