## Journal of the Mathematical Society of Japan

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

### Local theta correspondence of depth zero representations and theta dichotomy

#### 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**

https://projecteuclid.org/euclid.jmsj/1191591993

**Digital Object Identifier**

doi:10.2969/jmsj/1191591993

**Mathematical Reviews number (MathSciNet)**

MR1921088

**Zentralblatt MATH identifier**

1039.22012

**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. https://projecteuclid.org/euclid.jmsj/1191591993