Geometric Bernstein Asymptotics and the Drinfeld-Lafforgue-Vinberg degeneration for arbitrary reductive groups

We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] which was confined with the case G = SL_2. The multi-parameter degeneration VinBun_G gives rise to, for each parabolic of G, a corresponding nearby cycles functor. Our main theorem expresses the stalks of these nearby cycles in terms of the cohomology of the parabolic Zastava spaces. From this description we deduce that the nearby cycles of VinBun_G correspond, under the sheaf-function correspondence, to Bernstein's asymptotics map on the level of functions. This had been speculated by Bezrukavnikov-Kazhdan [BK] and Chen-Yom Din [CY] and conjectured in a precise form by Sakellaridis [Sak2].