Motivic Proof of a Character Formula for SL(2)

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.