## Journal of the Mathematical Society of Japan

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

Shu-Yen PAN

#### 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

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

Mathematical Reviews number (MathSciNet)
MR1921088

Zentralblatt MATH identifier
1039.22012

Digital Object Identifier
doi:10.2969/jmsj/1191591993

#### 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.