Journal of the Mathematical Society of Japan

Local theta correspondence of depth zero representations and theta dichotomy

Shu-Yen PAN

Full-text: Open access

Abstract

In this paper, we prove that depth zero representations are preserved by local theta correspondence for any type I reductive dual pairs over a $p$-adic field. Moreover, the minimal $K$-types of the paired depth zero irreducible admissible representations are paired by the theta correspondence for finite reductive dual pairs. As a consequence, we prove that the Iwahori-spherical representations are preserved by the local theta correspondence. Then we obtain some partial result of theta dichotomy for finite reductive dual pairs and $p$-adic reductive dual pairs of symplectic and orthogonal group, which is analogous to S. Kudla and S. Rallis' result for $p$-adic unitary groups.

Article information

Source
J. Math. Soc. Japan Volume 54, Number 4 (2002), 793-845.

Dates
First available in Project Euclid: 5 October 2007

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1191591993

Mathematical Reviews number (MathSciNet)
MR1921088

Zentralblatt MATH identifier
1039.22012

Digital Object Identifier
doi:10.2969/jmsj/1191591993

Subjects
Primary: 11F27: Theta series; Weil representation; theta correspondences
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 20C33: Representations of finite groups of Lie type

Keywords
local theta correspondence depth of a representation theta dichotomy minimal K-types

Citation

PAN, Shu-Yen. Local theta correspondence of depth zero representations and theta dichotomy. J. Math. Soc. Japan 54 (2002), no. 4, 793--845. doi:10.2969/jmsj/1191591993. http://projecteuclid.org/euclid.jmsj/1191591993.


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