Motivic Proof of a Character Formula for SL(2)
Cunningham, Clifton ; Gordon, Julia
Experiment. Math., Tome 18 (2009) no. 1, p. 11-44 / Harvested from Project Euclid
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/