Advertisement

Selecta Mathematica

, 25:32 | Cite as

Affine quiver Schur algebras and p-adic \({\textit{GL}}_n\)

  • Vanessa Miemietz
  • Catharina StroppelEmail author
Article

Abstract

In this paper we consider the (affine) Schur algebra which arises as the endomorphism algebra of certain permutation modules for the Iwahori–Matsumoto Hecke algebra. This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of \({\text {GL}}_2({\mathbb {Q}}_5)\) in characteristic 3.

Mathematics Subject Classification

20C08 33D80 20G43 14M15 22E57 

Notes

Acknowledgements

We thank Günter Harder, David Helm, Peter Scholze, Shaun Stevens and Torsten Wedhorn for useful discussions on the background material of this paper, Ruslan Maksimau and Andrew Mathas for sharing their insight into Hecke algebras, and the referees for their advice. This work was partly supported by the DFG Grant SFB/TR 45 and EPSRC Grant EP/K011782/1.

References

  1. 1.
    Bernstein, I.N.: Le centre de Bernstein. In: Deligne, P. (ed.) Representations of Reductive Groups Over a Local Field, pp. 1–32. Travaux en Cours, Hermann, Paris (1984)Google Scholar
  2. 2.
    Bernstein, I.N., Gelfand, I.M.: Schubert cells and the cohomology of a flag space. Funkcional. Anal. i Priloen. 7(1), 64–65 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernstein, I.N., Zelevinsky, A.: Representations of the group \({{\rm GL}}(n,F)\), where \(F\) is a local non-Archimedean field. Uspehi Mat. Nauk 31, 5–70 (1976)MathSciNetGoogle Scholar
  4. 4.
    Blondel, C.: Basic representation theory of reductive \(p\)-adic groups. In: Lecture Series Morningside Center of Mathematics. Beijing (2011)Google Scholar
  5. 5.
    Brundan, J., Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras. Invent. Math. 178(3), 451–484 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (2010)CrossRefGoogle Scholar
  7. 7.
    Demazure, M.: Désingularisation des variétés de Schubert généralisées. Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday. I. Ann. Sci. École Norm. Sup. (4) 7, 53–88 (1974)CrossRefGoogle Scholar
  8. 8.
    Doty, S.R., Green, R.M.: Presenting affine \(q\)-Schur algebras. Math. Z. 256(2), 311–345 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dipper, R., James, G.: The \(q\)-Schur algebra. Proc. Lond. Math. Soc. (3). 59(1), 23–50 (1989)CrossRefGoogle Scholar
  10. 10.
    Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press (1997)Google Scholar
  11. 11.
    Geck, M., Pfeiffer, G.: Characters of finite Coxeter groups and Iwahori–Hecke algebras. In: LMS Monographs. New Series, vol. 21. Oxford University Press (2000)Google Scholar
  12. 12.
    Ginzburg, V., Vasserot, E.: Langlands reciprocity for affine quantum groups of type \(A_ n\). Int. Math. Res. Not. 3, 67–85 (1993)CrossRefGoogle Scholar
  13. 13.
    Green, R.M.: On 321-avoiding permutations in affine Weyl groups. J. Algebr. Comb. 15(3), 241–252 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Green, R.M.: The affine \(q\)-Schur algebra. J. Algebra 215(2), 379–411 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Green, J.A.: Polynomial representations of \({{\rm GL}}_n\). In: Springer Lecture Notes, vol. 830. Springer (1980)Google Scholar
  16. 16.
    Harris, M.: The local Langlands conjecture for \({{\rm GL}}(n)\) over a \(n< p\)-adic field. Invent. Math. 134(1), 177–210 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Harris, M., Taylor, R.: The geometry and cohomology of some simple shimura varieties. In: Annals of Mathematics Studies, vol. 151. Princeton University Press (2001)Google Scholar
  18. 18.
    Henniart, G.: Une preuve simple des conjectures de Langlands pour \(p\) sur un corps \(p\)-adique. Invent. Math. 139(2), 439–455 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Iwahori, N., Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups. Inst. Hautes tudes Sci. Publ. Math. 25, 5–48 (1965)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kang, S.-J., Kashiwara, M., Park, E.: Geometric realization of Khovanov–Lauda–Rouquier algebras associated with Borcherds–Cartan data. Proc. Lond. Math. Soc. (3) 107(4), 907–931 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Khovanov, M., Lauda, A., Mackaay, M., Stošić, M.: Extended graphical calculus for categorified quantum \({{\mathfrak{s}}}{{\mathfrak{l}}}(2)\). Mem. AMS 2019(126) (2012)Google Scholar
  22. 22.
    Khovanov, M., Lauda, A.: A categorification of quantum \({{\mathfrak{s}}}{{\mathfrak{l}}}(n)\). Quantum Topol. 1(1), 1–92 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory Am. Math. Soc. 13, 309–347 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group. In: University Lecture Series, vol. 15. American Mathematical Society (1999)Google Scholar
  25. 25.
    Lusztig, G.: Quivers, perverse sheaves and quantized enveloping algebras. JAMS 4(2), 365–421 (1991)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lusztig, G.: Affine Hecke algebras and their graded version. JAMS 2(3), 599–685 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lusztig, G.: Some examples of square integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277, 623–653 (1983)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group. In: University Lecture Series, vol. 15. AMS (1999)Google Scholar
  29. 29.
    Ménguez, A., Sécherre, V.: Représentations lisses modulo \(\ell \) de \({{\rm GL}}_m(D)\). Duke Math. J. 163(4), 795–887 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Przezdziecki, T.: Cohomological Hall algebras and Quiver Schur algebras, in PhD Thesis, Department of Mathematics University of Bonn (2019) (in preparation)Google Scholar
  31. 31.
    Rouquier, R.: Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19(2), 359–410 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rouquier, R.: 2-Kac–Moody algebras. arXiv:0812.5023
  33. 33.
    Schiffmann, O.: Lectures on Hall algebras. In: Geometric Methods in Representation Theory. II, pp. 1–141. Seminars and Conferences, 24-II, Society Mathematics, Paris (2012)Google Scholar
  34. 34.
    Scholze, P.: The local Langlands correspondence for \({{\rm GL}}_n\) over \(p\)-adic fields. Invent. Math. 192(3), 663–715 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sécherre, V., Stevens, S.: Block decomposition of the category of \(\ell \)-modular smooth representations of \({{\rm GL}}_{n}(F)\) and its inner forms. Preprint arXiv:1402.5349, to appear in Annales scientifiques de l’Ecole Normale Supérieure
  36. 36.
    Stroppel, C., Webster, B.: Quiver Schur algebras and \(q\)-Fock space. arXiv:1110.1115
  37. 37.
    Stroppel, C.: Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126(3), 547–596 (2005)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Takeuchi, M.: The group ring of \({{\rm GL}}_n({\mathbb{F}}_q)\) and the \(q\)-Schur algebra. J. Math. Soc. Jpn. 48, 259–274 (1996)CrossRefGoogle Scholar
  39. 39.
    Vignéras, M.-F.: Schur algebras of reductive \(p\)-adic groups. I. Duke Math. J. 116(1), 35–75 (2003)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Vignéras, M.-F.: Induced \(R\)-representations of \(p\)-adic reductive groups. Selecta Math. (N.S.) 4(4), 549–623 (1998)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Varagnolo, M., Vasserot, E.: Canonical bases and KLR-algebras. J. Reine Angew. Math. 659, 67–100 (2011)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Varagnolo, M., Vasserot, E.: From double affine Hecke algebras to quantized affine Schur algebras. Int. Math. Res. Not. 26, 1299–1333 (2004)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Webster, B.: A note on isomorphisms between Hecke algebras. arXiv:1305.0599v1
  44. 44.
    Wedhorn, T.: The local Langlands correspondence for \({{\rm GL}}(n)\) over \(p\)-adic fields. In: School on Automorphic Forms on \({{\rm GL}}(n)\), ICTP Lecture Notes, vol. 21, pp. 237–320. Trieste (2008)Google Scholar
  45. 45.
    Zelevinsky, A.V.: Induced representations of reductive \(p\)-adic groups II. On irreducible representations of O. Ann. Sci. Ec. Norm. Super. 13(2), 165–210 (1980)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NorwichUK
  2. 2.BonnGermany

Personalised recommendations