Monday, Feb. 9, 4 pm in MS 6221


Martin Weissman, UCSC


Title: Backwards functoriality from classical groups to G_2


Abstract: I will discuss ongoing work with collaborators, on "backwards lifting" from PGSp_6 to G_2 over p-adic fields. If one believes the local Langlands conjectures for G_2, as refined by Arthur and Vogan, every generic supercuspidal representation of G_2 should arise as an endoscopic lift from PGL_3, or else should lift to a generic supercuspidal representation of PGSp_6. Therefore, to construct every generic supercuspidal reprsentation of G_2, one should understand the endoscopic lifts from PGL_3 (previous work of Savin and Gan), and one should understand how to construct a generic supercuspidal representation of G_2 from certain representations of PGSp_6 (current work of the speaker, Savin, and others on "backwards lifting"). Such constructions should eventually reduce the local Langlands conjectures for G_2 to the local Langlands conjectures for classical groups. I will describe some exceptional groups and dual reductive pairs, in order to describe the exceptional theta correspondences we use to construct representations of G_2 from some representations of PGSp_6. I will state some general strategies for studying these exceptional theta correspondences, the philosophy of Rallis on towers, the specific results we have obtained, and some potential applications to arithmetic.