Motivic Proof of a Character Formula for SL(2)

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 sl(2). We then find identities in the ring of virtual Chow motives over 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.

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