Journal of the Mathematical Society of Japan

Local theta correspondence of depth zero representations and theta dichotomy

Shu-Yen PAN

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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

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

First available in Project Euclid: 5 October 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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