Beilinson–Drinfeld Schubert varieties of parahoric group schemes and twisted global Demazure modules

• Jiuzu Hong ^[1] ; Huanhuan Yu ^[2]
  1. [1] Department of Mathematics, University of North Carolina, USA
  2. [2] Beijing International Center for Mathematical Research, Peking University, People’s Republic of China
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 1, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-01011-8
• Enlaces
  • Texto completo
• Resumen

  • Let G be a parahoric Bruhat–Tits group scheme arising from a -curve C and a certain -action on a simple algebraic group G for some finite cyclic group . We prove the flatness of Beilinson–Drinfeld Schubert varieties of G, we determine the rigidified Picard group of the Beilinson–Drinfeld Grassmannian GrG,Cn of G, and we establish the factorizable and equivariant structures on rigidified line bundles over GrG,Cn . We develop an algebraic theory of global Demazure modules of twisted current algebras, and using our geometric results we prove that when C = A1, the spaces of global sections of line bundles on BD Schubert varieties of G are dual to the twisted global Demazure modules. This generalizes the work of Dumanski–Feigin–Finkelberg in the untwisted setting.

• Referencias bibliográficas
  • . Atiyah, M., Macdonald, I.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)
  • 2. Brookner, A., Corwin, D., Etingof, P., Sam, S.: On Cohen–Macaulayness of Sn-invariant subspace arrangements. Int. Math. Res. Not. IMRN...
  • 3. Braverman, A., Finkelberg, M.: Twisted zastava and q-Whittaker functions. J. Lond. Math. Soc. 96(2), 309–325 (2017)
  • 4. Besson, M., Hong, J.: Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules. arXiv:2010.11357
  • 5. Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions. Commun. Math. Phys. 164(2), 385–419 (1994)
  • 6. Broshi, M.: G-torsors over a Dedekind scheme. J. Pure Appl. Algebra 217(1), 11–19 (2013)
  • 7. Chari, V., Fourier, G., Khandai, T.: A categorical approach to Weyl modules. Transform. Groups 15(3), 517–549 (2010)
  • 8. Chari, V., Ion, B., Kus, D.: Weyl modules for the hyperspecial current algebra. Int. Math. Res. Not. IMRN 15, 6470–6515 (2015)
  • 9. Dumanski, I., Feigin, E.: Reduced arc schemes for Veromese embeddings and global Demazure modules. Commun. Contemp. Math. (2022)
  • 10. Dumanski, I., Feigin, E., Finkelberg, M.: Beilinson–Drinfeld Schubert varieties and global Demazure modules. Forum Math. Sigma 9(e42),...
  • 11. Damiolini, C., Hong, J.: Local types of(, G)-bundles and parahoric group scheme. Proc. Lond. Math. Soc. 127(2), 261–294 (2023)
  • 12. Deligne, P., Milne, J.S.: Tannakian Categories. https://www.jmilne.org/math/xnotes/tc2018.pdf
  • 13. Dracula, C.:When is a morphism of varieties which induces isomorphism on each fiber an isomorphism. https://math.stackexchange.com/q/769892
  • 14. Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. (JEMS) 5(1), 41–68 (2003)
  • 15. Fourier, G., Kus, D.: Demazure modules and Weyl modules: the twisted current case. Trans. Am. Math. Soc. 365(11), 6037–6064 (2013)
  • 16. Fourier, G., Khandai, T., Kus, D., Savage, A.: Local Weyl modules for equivariant map algebras with free abelian group actions. J. Algebra...
  • 17. Feigin, B., Loktev, S.: On generalized Kostka polynomials and the quantum Verlinde rule. Amer. Math. Soc. Transl. Ser. 2, 194, Adv. Math....
  • 18. Feigin, B., Loktev, S.: Multi-dimensional Weyl modules and symmetric functions. Commun. Math. Phys. 251(3), 427–445 (2004)
  • 19. Feigin, E., Makedonskyi, I.: Vertex algebras and coordinate rings of semi-infinite flags. Commun. Math. Phys. 369(1), 221–244 (2019)
  • 20. Fourier, G., Littelmann, P.: Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211(2), 566–593...
  • 21. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg (1977)
  • 22. Heinloth, J.: Uniformization of G -bundles. Math. Ann. 347(3), 499–528 (2010)
  • 23. Hong, J., Kumar, S.: Conformal blocks for Galois covers of algebraic curves. Compositio Math. 159, 2191–2259 (2023)
  • 24. Hong, J., Kumar, S.: Twisted conformal blocks and their dimension. Math. Z. 306, 76 (2024)
  • 25. Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
  • 26. Kumar, S.: Kac–Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics Vol. 214, Birkhauser (2002)
  • 27. Kus, D., Venkatesh, R.: Twisted Demazure modules, fusion product decomposition and twisted Qsystems. Represent. Theory 20, 94–127 (2016)
  • 28. Mumford, D.: Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, 5 Published for the Tata Institute of Fundamental...
  • 29. Nori, M.V.: On the representations of the fundamental group. Compositio Math. 33(1), 29–41 (1976)
  • 30. Ogus, A., Bergman, G.: Nakayama’s lemma for half-exact functors. Proc. Am. Math. Soc. 31, 67–74 (1972)
  • 31. Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118–198 (2008)
  • 32. Reeves, W.: On the ind-flatness of the Beilinson–Drinfeld Grassmannian. In preparation
  • 33. Sorger, C.: Lectures on moduli of principal G-bundles over algebraic curves. School on Algebraic Geometry (Trieste, 1999), 1–57, ICTP...
  • 34. Springer, T.A.: Linear algebraic groups. Second edition. Progress in Mathematics, 9. Birkhäuser Boston, Inc., Boston, MA (1998)
  • 35. Springer, T.A.: Twisted conjugacy in simply connected groups. Transform. Groups 11(3), 539–545 (2006)
  • 36. The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu (2022)
  • 37. Zhu, X.: Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian. Adv. Math. 221(2), 570–600 (2009)
  • 38. Zhu, X.: On the coherence conjecture of Pappas and Rapoport. Ann. Math. 180(1), 1–85 (2014)
  • 39. Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. Geometry of moduli spaces and representation theory,...