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Representation Theory

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On the theta correspondence for $(\mathrm {GSp}(4), \mathrm {GSO}(4,2))$ and Shalika periods

Author: Kazuki Morimoto
Journal: Represent. Theory 18 (2014), 28-87
MSC (2010): Primary 11F27; Secondary 22E50
Published electronically: April 16, 2014
MathSciNet review: 3193382
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Abstract: We consider both local and global theta correspondences for $\mathrm {GSp}_4$ and $\mathrm {GSO}_{4,2}$. Because of the accidental isomorphism $\mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2}$, these correspondences give rise to those between $\mathrm {GSp}_4$ and $\mathrm {GU}_{2,2}$ for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on $\mathrm {GU}(2,2)$ by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.

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Additional Information

Kazuki Morimoto
Affiliation: Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
Address at time of publication: Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Received by editor(s): March 11, 2013
Received by editor(s) in revised form: November 1, 2013
Published electronically: April 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society