On automorphic sheaves on BunG

Let X be a smooth projective connected curve over an algebraically closed field k of positive characteristic. Let G be a reductive group over k, γ be a dominant coweight for G, and E be an l-adic Ǧ-local system on X , where Ǧ denotes the Langlands dual group (over Q̄l). Let BunG be the moduli stack of G-bundles on X . Under some conditions on the triple (G, γ,E) we propose a conjectural construction of a distinguished E-Hecke automorphic sheaf on BunG. We are motivated by a construction of automorphic forms suggested by Ginzburg, Rallis and Soudry in [6, 7]. We also generalize Laumon’s theorem ([10], Theorem 4.1) for our setting. Finally, we formulate an analog of the Vanishing Conjecture of Frenkel, Gaitsgory and Vilonen for Levi subgroups of G.