Motivic proof of a character formula for SL(2)

Details Motivic proof of a character formula for SL(2) Motivic proof of a character formula for SL(2)

2006-09-09

arxiv.org/abs/math/0609260v3

Mathematics

Representation Theory

51 pages; to appear in Experimental Mathematics

Scientific paper

We give a motivic proof of a character formula for depth zero supercuspidal representations of $p$-adic SL(2). We begin by finding the virtual Chow motives for the character values of all depth zero supercuspidal representations of $p$-adic SL(2), at topologically unipotent elements. Then we find the virtual Chow motives for the values of the Fourier transform of all regular elliptic orbital integrals with depth zero in their Cartan subalgebra, at topologically nilpotent elements. Finally, we prove a character formula for depth zero supercuspidal representations by showing that the formula corresponds to three identities in the ring of virtual Chow motives over $\mathbb{Q}$.

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