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

Schieder, Simon

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 821-866.

Nous étudions la géométrie et les singularités de la direction principale de la dégénérescence de Drinfeld-Lafforgue-Vinberg de l'espace moduli de G-torseurs Bun G pour un groupe réductif arbitraire G, et leur relation avec le groupe dual de Langlands Gˇ.

L'article est constitué de deux parties. Dans la première partie, nous étudions l'action de monodromie sur les cycles proches de la dégénérescence principale de Bun G et la relions au groupe dual de Langlands Gˇ. Nous décrivons la filtration par monodromie sur les cycles proches et généralisons les résultats de [37] du cas G= SL 2 au cas d'un groupe réductif arbitraire G. Notre description est donnée en termes de combinatoire du groupe dual de Langlands Gˇ et de généralisations des oscillateurs de Picard-Lefschetz trouvés dans [37]. Nos preuves dans la première partie utilisent certains modèles locaux pour la dégénérescence principale de Bun G dont la géométrie est étudiée dans la seconde partie.

Nos modèles locaux fournissent deux types de dégénérescence des espaces Zastava; ces dégénérations sont de nature très différente, et équipent les espaces de Zastava avec l'analogue géométrique d'une structure d'algèbre de Hopf. La première dégénérescence correspond à la fusion Beilinson-Drinfeld des diviseurs. La deuxième dégénérescence est nouvelle et correspond à ce que nous appelons Vinberg fusion: Elle est obtenue non pas par des diviseurs dégénérés sur la courbe, mais en dégénérant le groupe G via le semigroupe de Vinberg. De plus, au niveau de la cohomologie, la dégénérescence correspondant à la fusion de Vinberg donne lieu à une structure de algebra, tandis que la dégénérescence correspondant à la fusion de Beilinson-Drinfeld donne lieu à une structure de coalgebra; la compatibilité entre les deux dégénérations donne l'axiome de l'algèbre de Hopf.

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 Gˇ 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 Gˇ. We describe the weight-monodromy filtration on the nearby cycles and generalize the results of [37] 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 Gˇ and generalizations of the Picard-Lefschetz oscillators found in [37]. Our proofs in the first part use certain local models for the principal degeneration of Bun G whose geometry 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 we 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.

Classification : 14D24; 14D20, 14D23, 11R39, 14H60, 16T05, 14D05, 14D06
Keywords: Geometric representation theory, geometric Langlands program, moduli spaces of $G$-bundles, nearby cycles, Picard-Lefschetz theory, weight-monodromy theory, Vinberg semigroup, Langlands duality.
Mot clés : Théorie géométrique des représentations, programme géométrique de Langlands, espaces de moduli de $G$-torseurs, cycles proches, théorie de Picard-Lefschetz, théorie de la monodromie, semigroupe de Vinberg, dualité de Langlands.

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