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.
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.
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
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