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WHITTAKER FUNCTIONS FROM MOTIVIC CHERN CLASSES

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Abstract

We prove a motivic analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized ag manifold G/B multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure–Lusztig (D–L) operators, identifies an Euler characteristic above to a formula of Brubaker, Bump and Licata for the Iwahori–Whittaker functions of the principal series representation of the p-adic Langlands dual group. As a corollary, we recover the classical Casselman–Shalika formula for the spherical Whittaker function. The proofs are based on localization in equivariant K-theory, and require a geometric interpretation of how the Hecke inverse of a D–L operator acts on the class of a point. We prove that the Hecke inverse operators give Grothendieck–Serre dual classes of the motivic classes, a result which might be of independent interest. In an Appendix jointly authored with Dave Anderson, we show that if the line bundle is trivial, we recover a generalization of a classical formula by Kostant, Macdonald, Shapiro and Steinberg for the Poincaré polynomial of G/B; the generalization we consider is due to Akyıldız and Carrell and replaces G/B by any smooth Schubert variety.

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References

  1. Akyıldız, E., Carrell, J.: Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant, Macdonald, Shapiro, and Steinberg. Michigan Math. J. 61(3), 543–553 (2012)

    Article  MathSciNet  Google Scholar 

  2. P. Aluffi, L. Mihalcea, J. Schürmann, C. Su, Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells, arXiv:1709.08697 (2017).

  3. P. Aluffi, L. Mihalcea, J. Schürmann, C. Su, Motivic Chern classes of Schu-bert cells, Hecke algebras, and applications to Casselman's problem, to appear in Annales Sci. de L’Ecole Normale Superieure, arXiv:1902.10101 (2019).

  4. Andersen, H.H.: Schubert varieties and Demazure’s character formula. Invent. Math. 79(3), 611–618 (1985)

    Article  MathSciNet  Google Scholar 

  5. Biaƚynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)

    Article  MathSciNet  Google Scholar 

  6. Brasselet, J., Schürmann, J., Yokura, S.: Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal. 2(1), 1–55 (2010)

    Article  MathSciNet  Google Scholar 

  7. Brion, M.: Equivariant Chow groups for torus actions. Transform. Groups. 2, 225–267 (1997)

    Article  MathSciNet  Google Scholar 

  8. B. Brubaker, V. Buciumas, D. Bump, H. Gustafsson, Colored vertex models and Iwahori Whittaker functions, arXiv:1906.04140 (2019).

  9. Brubaker, B., Bump, D., Licata, A.: Whittaker functions and Demazure operators. J. Number Theory. 146, 41–68 (2015)

    Article  MathSciNet  Google Scholar 

  10. Casselman, W., Shalika, J.: The unramified principal series of p-adic groups. II. The Whittaker function. Compos. Math. 41(2), 207–231 (1980)

    MathSciNet  MATH  Google Scholar 

  11. N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Modern Birkhäuser Classics, Birkhäuser Boston, Boston, MA, 2010.

  12. Demazure, M.: Sur la formule des caractères de H. Weyl. Invent. Math. 9, 249–252 (1969)

    Article  MathSciNet  Google Scholar 

  13. M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88.

  14. Feher, L., Rimanyi, R., Weber, A.: Motivic Chern classes and K-theoretic stable envelopes. Proc. London Math. Soc. 122(1), 153–189 (2021)

    Article  MathSciNet  Google Scholar 

  15. F. Hirzebruch, Topological Methods in Algebraic Geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

  16. Ikeda, T., Mihalcea, L.C., Naruse, H.: Double Schubert polynomials for the classical groups. Adv. Math. 226(1), 840–886 (2011)

    Article  MathSciNet  Google Scholar 

  17. B. Ion, Nonsymmetric Macdonald polynomials and matrix coefficients for un-ramified principal series, Adv. Math. 201 (2006), no. 1, 36–62, 2006.

  18. A. Knutson, A Schubert calculus recurrence from the noncomplex W-action on G/B, arXiv:0306304 (2003).

  19. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81(4), 973–1032 (1959)

    Article  MathSciNet  Google Scholar 

  20. Kostant, B., Kumar, S.: T-equivariant K-theory of generalized flag varieties. J. Diff. Geom. 32(2), 549–603 (1990)

    MathSciNet  MATH  Google Scholar 

  21. S. Kumar, Kac–Moody Groups, Their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser, Boston, 2002.

  22. Lee, K.H., Lenart, C., Liu, D.: Whittaker functions and Demazure characters. J. Inst. Math. Jussieu. 18(4), 759–781 (2019)

    Article  MathSciNet  Google Scholar 

  23. Lusztig, G.: Equivariant K-theory and representations of Hecke algebras. Proc. Amer. Math. Soc. 94(2), 337–342 (1985)

    MathSciNet  MATH  Google Scholar 

  24. Macdonald, I.G.: The Poincaré series of a Coxeter group. Math. Ann. 199, 161–174 (1972)

    Article  MathSciNet  Google Scholar 

  25. Maxim, L.G., Schürmann, J.: Characteristic classes of singular toric varieties. Comm. Pure Appl. Math. 68(12), 2177–2236 (2015)

    Article  MathSciNet  Google Scholar 

  26. V. B. Mehta, A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. 122 (1985), no. 1, 27–40.

  27. L. C. Mihalcea, H. Naruse, C. Su, Left Demazure–Lusztig operators on equivariant (quantum) cohomology and K-theory, Internat. Math. Res. Notices (2021), rnab049, https://doi.org/https://doi.org/10.1093/imrn/rnab049.

  28. Nielsen, H.A.: Diagonalizably linearized coherent sheaves. Bull. Soc. Math. France. 102, 85–97 (1974)

    Article  MathSciNet  Google Scholar 

  29. A. Okounkov, A. Smirnov, Quantum difference equation for Nakajima varieties, arXiv:1602.09007 (2016).

  30. Reeder, M.: p-adic Whittaker functions and vector bundles on ag manifolds. Compos. Math. 85(1), 9–36 (1993)

    MathSciNet  MATH  Google Scholar 

  31. F. Rodier, Modèle de Whittaker des représentations admissibles des groupes réductifs p-adiques déployés, C. R. Acad. Sci. Paris Sér. A–B 275 (1972), 1045–1048.

  32. Steinberg, R.: Finite refection groups. Transact. Amer. Math. Soc. 91(3), 493–504 (1959)

    Article  Google Scholar 

  33. C. Su, G. Zhao, C. Zhong, On the K-theory stable bases of the Springer resolution, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 3, 663–711.

  34. Tymoczko, J.S.: Permutation actions on equivariant cohomology of flag varieties. Contemp. Math. 460, 365–384 (2008)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech University, Blacksburg, VA, 24061, USA

    LEONARDO C. MIHALCEA

  2. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    CHANGJIAN SU

  3. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    DAVE ANDERSON

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  1. LEONARDO C. MIHALCEA
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  2. CHANGJIAN SU
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  3. DAVE ANDERSON
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Correspondence to LEONARDO C. MIHALCEA.

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Leonardo C. Mihalcea is supported by a Simons Collaboration Grant.

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MIHALCEA, L.C., SU, C. & ANDERSON, D. WHITTAKER FUNCTIONS FROM MOTIVIC CHERN CLASSES. Transformation Groups 27, 1045–1067 (2022). https://doi.org/10.1007/s00031-022-09731-x

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