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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The nonarchimedean theta correspondence
for $\operatorname{GS\lowercase{p}}(2)$ and $\operatorname{GO}(4)$


Author: Brooks Roberts
Journal: Trans. Amer. Math. Soc. 351 (1999), 781-811
MSC (1991): Primary 11F27; Secondary 22E50
MathSciNet review: 1458334
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Abstract: In this paper we consider the theta correspondence between the sets $\operatorname{Irr} (\operatorname{GSp} (2,k))$ and $\operatorname{Irr} (\operatorname{GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname{Irr} (\operatorname{GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname{Irr} (\operatorname{O} (X))$ that occur in the theta correspondence between $\operatorname{Irr} (\operatorname{Sp} (2,k))$ and $\operatorname{Irr} (\operatorname{O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.


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Additional Information

Brooks Roberts
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada
Address at time of publication: Department of Mathematics, Brink Hall, University of Idaho, Moscow, Idaho 83844-1103
Email: brooks@member.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02126-1
PII: S 0002-9947(99)02126-1
Keywords: Nonarchimedean theta correspondence, \protect{$\GSp (2)$, $\GO (4)$}
Received by editor(s): June 3, 1996
Received by editor(s) in revised form: February 6, 1997
Additional Notes: During the period of this work the author was a Research Associate with the NSF 1992–1994 special project Theta Functions, Dual Pairs, and Automorphic Forms at the University of Maryland, College Park, and was supported by a Stipendium at the Max-Planck-Institut für Mathematik.
Article copyright: © Copyright 1999 American Mathematical Society