Skip to main content
SpringerLink
Log in

Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori–Hecke algebras. This is commonly referred to as Kottwitz’s conjecture. The reductive groups associated with the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of Haines and Ngô (Compos Math 133:117–150, 2002). Upon completion of the first version of this paper, Pappas and Zhu released a preprint (now published as Pappas and Zhu in Invent Math 194(1):147–254, 2013) which contained within its scope the main theorem of this paper. However, the methods of Pappas and Zhu (2013) are very different, and some of the proofs from this paper have been useful in forthcoming work of Haines–Stroh.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Theorem 5.8 in [10] assumes constant parameter systems, but this is not necessary to the conclusion; see Lemma 5.3.3 in [25] for some explanation of why this is so.

  2. The notion of duality here is similar to the one occurring in ELM5: It is required that \( \mathcal {F}_0 = \{ x \in \mathcal {V}(R) \;\vert \;\phi _R ( \mathcal {F}_0, x ) \subset (t+p)^{\nu -\mu } \mathcal {R}[t] \} \) and similarly for \( \mathcal {F}_{d/2}\).

  3. It is a slight abuse of notation to suppress the indices \( \mu \) and \( \nu \) from the name of the functor.

  4. The sense in which \( \widetilde{g} \) induces g is that multiplication by \( (t+p)^{\mu +1} \) within \( \mathcal {V} \) yields an identification of \( t^{-m} \widetilde{\mathcal {V}}_0 \) with \( \overline{\mathcal {W}}_{\text {sup}}\). In the rest of the paper, this principle will be referred to as a “shift.”

  5. The ordinary product \( \overline{\phi }_R \) is well-defined on \( \widetilde{\mathcal {L}}_0 \) due to ELM5.

  6. It is a slight abuse of notation to suppress the indices m and n from the name of the functor.

  7. The notion of duality here is similar to the one occurring in ELM5: It is required that \( \mathcal {K}_0 = \{ x \in \mathcal {V}(R) \;\vert \;\phi _R ( \mathcal {K}_0, x ) \subset t^{n-m} (t+p)^{\nu -\mu } \mathcal {R}[t] \} \) and similarly for \( \mathcal {K}_{d/2}\).

  8. The notion of duality here is identical to that for \( \mathbf {Conv}^{(m, n \, ; \, \mu , \nu )} \).

  9. The necessary content of the Appendix of [24], especially Proposition A.43, applies in the equi-characteristic case \( \mathbf {F}((t)) / \mathbf {F}_p((t)) \) here despite the use of the mixed-characteristic case in [24]. The dualizing shift “ a ” of [24] depends on the multiplier from AFV5.

  10. The meaning of “Zariski-locally” here is as in the definition of \( \widetilde{\mathbf {M}}^{(m,n)} \).

  11. The equality in hand, \( h_1 ( \gamma ( g_2 ( t^{-m} (t+p)^{-\mu } \mathcal {R}[t]^d ) ) ) = h_1 ( h_2 ( t^{-m} (t+p)^{-\mu } \mathcal {R}[t]^d ) ) \), is logically weaker than the element-wise equality that was used to conclude the proof of Lemma 10.5.1 but is nonetheless sufficient for that conclusion.

  12. Note that the reversed convolution product occurs on the right-hand side here.

References

  1. Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  2. Bourbaki, N.: Commutative Algebra. Chapters 1–7. Elements of Mathematics (Berlin). Springer, Berlin (1998). Translated from the French; Reprint of the 1989 English translation

  3. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100. Soc. Math. France, Paris, pp. 5–171 (1982) (French)

  4. Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups. North-Holland Mathematics Studies, vol. 39. North-Holland Publishing Co, Amsterdam, New York (1980). Translated from the French by J. Bell

  5. Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

    MATH  Google Scholar 

  6. Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144(2), 253–280 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Görtz, U.: Flatness of local models of certain Shimura varieties of PEL type. Math. Ann. 321, 689–727 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Görtz, U.: Affine Springer fibers and affine Deligne-Lusztig varieties. Affine flag manifolds and principal bundles. Trends Math., Birkhäuser/Springer Basel AG, Basel, pp. 1–50 (2010)

  9. Görtz, U., Wedhorn, T.: Algebraic geometry I. Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden (2010) Schemes with examples and exercises

  10. Haines, T.J.: The combinatorics of Bernstein functions. Trans. Am. Math. Soc. 353(3), 1251–1278 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. Haines, T.J.: Introduction to Shimura varieties with bad reduction of parahoric type. Harmonic analysis, the trace formula, and Shimura varieties. Clay Math. Proc., vol. 4, Am. Math. Soc., Providence, RI, pp. 583–642 (2005)

  12. Haines, T., Ngô, B.C.: Nearby cycles for local models of some Shimura varieties. Compos. Math. 133, 117–150 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kazhdan, D., Lusztig, G.: Schubert varieties and Poincaré duality. Proc. Symp. Pure Math. 36, 185–203 (1980)

    Article  MATH  Google Scholar 

  15. Kottwitz, R.: On the \( \lambda \)-adic representations associated to some simple Shimura varieties. Invent. Math. 108, 653–665 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kottwitz, R., Rapoport, M.: Minuscule alcoves for \({{\rm GL}}_n\) and \(G{{\rm Sp}}_{2n}\). Manuscr. Math. 102(4), 403–428 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lusztig, G.: Cells in affine Weyl groups and tensor categories. Adv. Math. 129, 85–98 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Algebraic Geom. 9, 577–605 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Pappas, G., Rapoport, M.: Local models in the ramified case 1: the EL case. J. Algebraic Geom. 12, 107–145 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pappas, G., Rapoport, M.: Local models in the ramified case 2: splitting models. Duke Math. J. 127, 193–250 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pappas, G., Rapoport, M.: Local models in the ramified case 3: unitary groups. J. Inst. Math. Jussieu 8, 507–564 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pappas, G., Zhu, X.: Local models of Shimura varieties and a conjecture of Kottwitz. Invent. Math. 194(1), 147–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rapoport, M.: On the bad reduction of Shimura varieties. In: Automorphic forms, Shimura varieties, and \(L\)-functions. Perspect. Math., vol. 11, pp. 253–321 (1988)

  24. Rapoport, M., Zink, T.: Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ (1996)

  25. Rostami, S.: The Bernstein presentation for general connected reductive groups. J. Lond. Math. Soc. 9(2), 514–536 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Scharlau, W.: Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270. Springer, Berlin (1985)

  27. Smithling, B.D.: Topological flatness of orthogonal local models in the split, even case. I. Math. Ann. 350(2), 381–416 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Smithling, B.D.: Topological flatness of local models for ramified unitary groups. I. The odd dimensional case. Adv. Math. 226(4), 3160–3190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Smithling, B.D.: Topological flatness of local models for ramified unitary groups. II. The even dimensional case. J. Inst. Math. Jussieu 13(2), 303–393 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I thank my PhD adviser, Thomas Haines, for suggesting this question in the first place, for all his support and advice during and after graduate school, for many helpful discussions during the original resolution of this question, and for numerous improvements to the final version (including the recent identification of an error in one of the proofs and the suggestion of a way to fix it). I thank Niranjan Ramachandran for urging me to finally submit this paper for publication. Most of the additions and improvements to this version were accomplished during the Fall 2014 semester at MSRI (DMS-1440140), in consultation with Matthias Strauch, and I also thank the organizers and staff for inviting me to the program and for all their help throughout the semester. Finally, I thank Selecta Kojak for all the good times and great riddims.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, WI, 53706, USA

    Sean Rostami

Authors
  1. Sean Rostami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sean Rostami.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rostami, S. Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties. Sel. Math. New Ser. 23, 643–719 (2017). https://doi.org/10.1007/s00029-016-0252-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-016-0252-z

Mathematics Subject Classification

Search

Navigation