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.
Publié le : 2009-05-15
Classification:
Motivic integration,
supercuspidal representations,
characters,
orbital integrals,
22E50,
03C10
@article{1243430527,
author = {Cunningham, Clifton and Gordon, Julia},
title = {Motivic Proof of a Character Formula for SL(2)},
journal = {Experiment. Math.},
volume = {18},
number = {1},
year = {2009},
pages = { 11-44},
language = {en},
url = {http://dml.mathdoc.fr/item/1243430527}
}
Cunningham, Clifton; Gordon, Julia. Motivic Proof of a Character Formula for SL(2). Experiment. Math., Tome 18 (2009) no. 1, pp. 11-44. http://gdmltest.u-ga.fr/item/1243430527/