Theta lifting of holomorphic discrete series: The case of

Authors:
Kyo Nishiyama and Chen-bo Zhu

Journal:
Trans. Amer. Math. Soc. **353** (2001), 3327-3345

MSC (2000):
Primary 22E46, 11F27

DOI:
https://doi.org/10.1090/S0002-9947-01-02830-6

Published electronically:
April 9, 2001

MathSciNet review:
1828608

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be a reductive dual pair in the stable range. We investigate theta lifts to of unitary characters and holomorphic discrete series representations of , in relation to the geometry of nilpotent orbits. We give explicit formulas for their -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme -type, we show that the structure of the resulting representations of is almost identical to the -module structure of the regular function rings on the closure of the associated nilpotent -orbits in , where is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

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Additional Information

**Kyo Nishiyama**

Affiliation:
Faculty of Integrated Human Studies, Kyoto University, Sakyo, Kyoto 606-8501, Japan

Email:
kyo@math.h.kyoto-u.ac.jp

**Chen-bo Zhu**

Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

Email:
matzhucb@nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9947-01-02830-6

Keywords:
Reductive dual pair,
theta lifting,
holomorphic discrete series,
nilpotent orbits,
associated cycles

Received by editor(s):
August 11, 2000

Received by editor(s) in revised form:
November 8, 2000

Published electronically:
April 9, 2001

Article copyright:
© Copyright 2001
American Mathematical Society