Title: 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.