Monodromy and Vinberg fusion for the principal degeneration of the space of G-bundles

We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of G-bundles Bun_G for an arbitrary reductive group G, and their relationship to the Langlands dual group of G. The article consists of two parts. In the first and main part, we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G and relate it to the Langlands dual group of G. We determine the weight-monodromy filtration on the nearby cycles, generalizing the results of [Sch1] from the case G=SL_2 to the case of an arbitrary reductive group G: Our description is given in terms of the combinatorics of the Langlands dual group of G and generalizations of the Picard-Lefschetz oscillators found in [Sch1]. Our proofs in the first part use certain local models for the principal degeneration of Bun_G whose geometry may be of independent interest and is studied in the second part. Our local models simultaneously provide two types of degenerations of the Zastava spaces; these degenerations are of very different nature, and together equip the Zastava spaces with the geometric analog of a Hopf algebra structure. The first degeneration corresponds to the usual Beilinson-Drinfeld fusion of divisors on the curve. The second degeneration is new and corresponds to what one might call Vinberg fusion: It is obtained not by degenerating divisors on the curve, but by degenerating the group G via the Vinberg semigroup. Furthermore, on the level of cohomology the degeneration corresponding to the Vinberg fusion gives rise to an algebra structure, while the degeneration corresponding to the Beilinson-Drinfeld fusion gives rise to a coalgebra structure; the compatibility between the two degenerations yields the Hopf algebra axiom.