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

Abstract

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. In the first part of the article we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G. We describe the weight-monodromy filtration in terms of the combinatorics of the Langlands dual group of G and generalizations of the Picard-Lefschetz oscillators found in [Sch1]. Our proofs use certain local models for the principal degeneration whose geometry is studied in the second part. Our local models simultaneously provide two types of degenerations of the Zastava spaces, which 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 we call Vinberg fusion: It is obtained not by degenerating divisors on the curve, but by degenerating the group G via the Vinberg semigroup. On the level of cohomology the Vinberg fusion gives rise to an algebra structure, while the Beilinson-Drinfeld fusion gives rise to a coalgebra structure; the Hopf algebra axiom is a consequence of the underlying geometry. It is natural to conjecture that this Hopf algebra agrees with the universal enveloping algebra of the positive part of the Langlands dual Lie algebra. The above procedure would then yield a novel and highly geometric way to pass to the Langlands dual side: Elements of the Langlands dual Lie algebra are represented as cycles on the above moduli spaces, and the Lie bracket of two elements is obtained by deforming the cartesian product cycle along the Vinberg degeneration.