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 ${\mathrm{Bun}}_G$ pour un groupe réductif arbitraire $G$, et leur relation avec le groupe dual de Langlands $\check{G}$.

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 ${\mathrm{Bun}}_G$ et la relions au groupe dual de Langlands $\check{G}$. Nous décrivons la filtration par monodromie sur les cycles proches et généralisons les résultats de [37] du cas $G={\mathrm{SL}}_2$ au cas d'un groupe réductif arbitraire $G$. Notre description est donnée en termes de combinatoire du groupe dual de Langlands $\check{G}$ 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 ${\mathrm{Bun}}_G$ 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 ${\mathrm{Bun}}_G$ for an arbitrary reductive group $G$, and their relationship to the Langlands dual group $\check{G}$ 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 ${\mathrm{Bun}}_G$ and relate it to the Langlands dual group $\check{G}$. We describe the weight-monodromy filtration on the nearby cycles and generalize the results of [37] from the case $G={\mathrm{SL}}_2$ to the case of an arbitrary reductive group $G$. Our description is given in terms of the combinatorics of the Langlands dual group $\check{G}$ 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 ${\mathrm{Bun}}_G$ 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.

Beilinson, A.; Bernstein, J., I. M. Gel'fand Seminar (Adv. Soviet Math.), Volume 16, Amer. Math. Soc., 1993, pp. 1-50 | MR | Zbl

Beilinson, A.; Bernstein, J.; Deligne, P., Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Soc. Math. France, 1982, pp. 5-171 | Numdam | MR | Zbl

Beilinson, A.; Drinfel'd, V. Quantization of Hitchin's integrable system and Hecke eigensheaves (preprint http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf )

Beilinson, A.; Drinfel'd, V., American Mathematical Society Colloquium Publications, 51, Amer. Math. Soc., 2004, 375 pages (ISBN: 0-8218-3528-9) | DOI | MR | Zbl

Beilinson, A., $K$-theory, arithmetic and geometry (Moscow, 1984–1986) (Lecture Notes in Math.), Volume 1289, Springer, 1987, pp. 42-51 | DOI | MR | Zbl

Braverman, A.; Finkelberg, M.; Gaitsgory, D.; Mirković, I. Intersection cohomology of Drinfeld's compactifications, Selecta Math., Volume 8 (2002), pp. 381-418 (ISSN: 1022-1824) | DOI | MR | Zbl

Braverman, A.; Gaitsgory, D. Geometric Eisenstein series, Invent. math., Volume 150 (2002), pp. 287-384 (ISSN: 0020-9910) | DOI | MR | Zbl

Braverman, A.; Gaitsgory, D. Deformations of local systems and Eisenstein series, Geom. Funct. Anal., Volume 17 (2008), pp. 1788-1850 (ISSN: 1016-443X) | DOI | MR | Zbl

Brion, M.; Kumar, S., Progress in Math., 231, Birkhäuser, 2005, 250 pages (ISBN: 0-8176-4191-2) | MR | Zbl

Bezrukavnikov, R.; Kazhdan, D. Geometry of second adjointness for $p$-adic groups, Represent. Theory, Volume 19 (2015), pp. 299-332 (ISSN: 1088-4165) | DOI | MR | Zbl

Campbell, J. Nearby cycles of Whittaker sheaves on Drinfeld's compactification (preprint arXiv:1702.04375 ) | MR

Chen, T.-H.; Yom Din, A. A formula for the geometric Jacquet functor and its character sheaf analogue, Geom. Funct. Anal., Volume 27 (2017), pp. 772-797 (ISSN: 1016-443X) | DOI | MR | Zbl

De Concini, C.; Procesi, C., Invariant theory (Montecatini, 1982) (Lecture Notes in Math.), Volume 996, Springer, 1983, pp. 1-44 | DOI | MR | Zbl

Deligne, P. La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math., Volume 52 (1980), pp. 137-252 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Drinfel'd, V.; Gaitsgory, D. Geometric constant term functor(s), Selecta Math., Volume 22 (2016), pp. 1881-1951 (ISSN: 1022-1824) | DOI | MR | Zbl

Drinfel'd, V. Cohomology of compactified moduli varieties of $F$-sheaves of rank 2, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 162 (1987), p. 107-158, 189 (ISSN: 0373-2703) | DOI | MR | Zbl

Drinfel'd, V. Moduli varieties of $F$-sheaves, Funktsional. Anal. i Prilozhen., Volume 21 (1987), pp. 23-41 (ISSN: 0374-1990) | DOI | MR | Zbl

Drinfel'd, V.; Wang, J. On a strange invariant bilinear form on the space of automorphic forms, Selecta Math., Volume 22 (2016), pp. 1825-1880 (ISSN: 1022-1824) | DOI | MR | Zbl

Feigin, B.; Finkelberg, M.; Kuznetsov, A.; Mirković, I. Semiinfinite Flags II. Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, Volume 194 (1999), pp. 113-148 | MR | Zbl

Finkelberg, M.; Mirković, I. Semiinfinite Flags I. Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, Volume 194 (1999), pp. 81-112 | MR | Zbl

Gaitsgory, D. Notes on the Geometric Langlands Program (preprint http://www.math.harvard.edu/~gaitsgde/GL/ )

Gaitsgory, D. On de Jong's conjecture, Israel J. Math., Volume 157 (2007), pp. 155-191 (ISSN: 0021-2172) | DOI | MR | Zbl

Gaitsgory, D. Outline of the proof of the geometric Langlands conjecture for $G{L}_2$ , Astérisque, Volume 370 (2015), pp. 1-112 (ISBN: 978-2-85629-806-0, ISSN: 0303-1179) | MR | Zbl

Gaitsgory, D. A “strange” functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017), pp. 1123-1162 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Gaitsgory, D.; Nadler, D. Spherical varieties and Langlands duality, Mosc. Math. J., Volume 10 (2010), p. 65-137, 271 (ISSN: 1609-3321) | DOI | MR | Zbl

Lafforgue, L. Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc., Volume 11 (1998), pp. 1001-1036 (ISSN: 0894-0347) | DOI | MR | Zbl

Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math., Volume 166 (2007), pp. 95-143 (ISSN: 0003-486X) | DOI | MR | Zbl

Putcha, M. S., London Mathematical Society Lecture Note Series, 133, Cambridge Univ. Press, 1988, 171 pages (ISBN: 0-521-35809-4) | DOI | MR | Zbl

Renner, L. E., Encyclopaedia of Math. Sciences, 134, Springer, 2005, 246 pages (ISBN: 3-540-24241-4) | MR | Zbl

Rittatore, A. Very flat reductive monoids, Publ. Mat. Urug., Volume 9 (2001), pp. 93-121 (ISSN: 0797-1443) | MR | Zbl

Rittatore, A. Algebraic monoids with affine unit group are affine, Transform. Groups, Volume 12 (2007), pp. 601-605 (ISSN: 1083-4362) | DOI | MR | Zbl

Rittatore, A. Monoïdes algébriques et plongements de groupes (1997)

Rittatore, A. Algebraic monoids and group embeddings, Transform. Groups, Volume 3 (1998), pp. 375-396 (ISSN: 1083-4362) | DOI | MR | Zbl

Sakellaridis, Y. Inverse Satake transforms (preprint arXiv:1410.2312 ) | MR

Sakellaridis, Y. Non-categorical structures in harmonic analysis (2014) (MSRI talk, video https://www.msri.org/workshops/708/schedules/19168 )

Schieder, S. Geometric Bernstein Asymptotics and the Drinfel'd-Lafforgue-Vinberg degeneration for arbitrary reductive groups (preprint arXiv:1607.00586 )

Schieder, S. The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld's compactifications, Selecta Math., Volume 21 (2015), pp. 763-831 (ISSN: 1022-1824) | DOI | MR | Zbl

Schieder, S. Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for ${\mathrm{SL}}_2$ , Duke Math. J., Volume 167 (2018), pp. 835-921 (ISSN: 0012-7094) | DOI | MR | Zbl

Vinberg, E. B., Lie groups and Lie algebras: E. B. Dynkin's Seminar (Amer. Math. Soc. Transl. Ser. 2), Volume 169, Amer. Math. Soc., 1995, pp. 145-182 | DOI | MR | Zbl

Wang, J. P. On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics, ISBN: 978-0355-07666-0, ProQuest LLC, Ann Arbor, MI (2017) http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:10266436 | MR

Wang, J. On the reductive monoid associated to a parabolic subgroup, J. Lie Theory, Volume 27 (2017), pp. 637-655 (ISSN: 0949-5932) | MR | Zbl