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Representation Theory

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Coincidence of algebraic and smooth theta correspondences

Authors: YiXin Bao and BinYong Sun
Journal: Represent. Theory 21 (2017), 458-466
MSC (2010): Primary 22E46, 22E50
Published electronically: November 1, 2017
MathSciNet review: 3718456
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Abstract: An “automatic continuity” question has naturally occurred since Roger Howe established the local theta correspondence over $\mathbb R$: Does the algebraic version of local theta correspondence over $\mathbb R$ agree with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor.

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Additional Information

YiXin Bao
Affiliation: School of Sciences, Harbin Institute of Technology, Shenzhen, 518055, China
MR Author ID: 947459

BinYong Sun
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences –and– School of Mathematics, University of Chinese Academy of Sciences, Beijing, 100190, China
MR Author ID: 805605

Keywords: Reductive dual pair, theta correspondence, Casselman-Wallach representation, Harish-Chandra module.
Received by editor(s): January 13, 2017
Received by editor(s) in revised form: May 26, 2017, and August 10, 2017
Published electronically: November 1, 2017
Additional Notes: The second author was supported in part by the National Natural Science Foundation of China (No. 11525105, 11688101, 11621061 and 11531008).
Article copyright: © Copyright 2017 American Mathematical Society