## Experimental Mathematics

- Experiment. Math.
- Volume 18, Issue 1 (2009), 11-44.

### Motivic Proof of a Character Formula for SL(2)

Clifton Cunningham and Julia Gordon

#### Abstract

This paper provides a proof of a $p$-adic character formula by means of motivic integration. We use motivic integration to produce virtual Chow motives that control the values of the characters of all depth-zero supercuspidal representations on all topologically unipotent elements of $p$}-adic $\SL(2)$; likewise, we find motives for the values of the Fourier transform of all regular elliptic orbital integrals having minimal nonnegative depth in their own Cartan subalgebra, on all topologically nilpotent elements of $p$-adic $\mathfrak{sl}(2)$. We then find identities in the ring of virtual Chow motives over $\mathbb{Q}$ that relate these two classes of motives. These identities provide explicit expressions for the values of characters of all depth-zero supercuspidal representations of $p$}-adic $\SL(2)$ as linear combinations of Fourier transforms of semisimple orbital integrals, thus providing a proof of a $p$-adic character formula.

#### Article information

**Source**

Experiment. Math., Volume 18, Issue 1 (2009), 11-44.

**Dates**

First available in Project Euclid: 27 May 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1243430527

**Mathematical Reviews number (MathSciNet)**

MR2548984

**Zentralblatt MATH identifier**

1179.22018

**Subjects**

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

Secondary: 03C10: Quantifier elimination, model completeness and related topics

**Keywords**

Motivic integration supercuspidal representations characters orbital integrals

#### Citation

Cunningham, Clifton; Gordon, Julia. Motivic Proof of a Character Formula for SL(2). Experiment. Math. 18 (2009), no. 1, 11--44. https://projecteuclid.org/euclid.em/1243430527