math.RT — Representation Theory
We prove Lusztig’s conjectures P1–P15 for extended affine Weyl groups with positive integer-valued weight functions.
math.NT — Number Theory
Let be a split semisimple group whose simple factors are of classical type, over a global function field of arbitrary positive characteristic. We prove that every semisimple elliptic Langlands parameter is the parameter of a cusp form on , in the sense of V. Lafforgue's excursion operators; we call such a form an automorphic realization of the parameter. Neither nor its field of constants is extended. We start from L. Lafforgue's theorem for general linear groups and use automorphic descent and the theta correspondence. The work is to prove that the descent does not vanish over , to carry out the computations with root subgroups in small characteristic, in particular in characteristic two, and to identify the parameter itself and not only its unramified Hecke eigenvalues. This is the third of three papers. Together with the first, which contains the arithmetic foundations, and the second, which treats the exceptional types by a geometric method, it gives automorphic realizations of semisimple parameters on all split reductive groups, with a restriction on the characteristic for the exceptional types only.