Intersection complexes and unramified $L$-factors
HTML articles powered by AMS MathViewer
- by Yiannis Sakellaridis and Jonathan Wang HTML | PDF
- J. Amer. Math. Soc. 35 (2022), 799-910 Request permission
Abstract:
Let $X$ be an affine spherical variety, possibly singular, and $\mathsf L^+X$ its arc space. The intersection complex of $\mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in BravermanâFinkelbergâGaitsgoryâMirkoviÄ for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in BouthierâNgĂŽâSakellaridis for toric varieties and $L$-monoids. In this paper, we compute this intersection complex for the large class of those spherical $G$-varieties whose dual group is equal to $\check G$, and the stalks of its nearby cycles on the horospherical degeneration of $X$. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional $\check G$-representation determined by the set of $B$-invariant valuations on $X$. We prove the latter conjecture in many cases. Under the sheafâfunction dictionary, our calculations give a formula for the Plancherel density of the IC function of $\mathsf L^+X$ as a ratio of local $L$-values for a large class of spherical varieties.References
- Sergey Arkhipov and Roman Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, Israel J. Math. 170 (2009), 135â183. With an appendix by Bezrukavnikov and Ivan MirkoviÄ. MR 2506322, DOI 10.1007/s11856-009-0024-y
- S. Arkhipov, A. Braverman, R. Bezrukavnikov, D. Gaitsgory, and I. MirkoviÄ, Modules over the small quantum group and semi-infinite flag manifold, Transform. Groups 10 (2005), no. 3-4, 279â362. MR 2183116, DOI 10.1007/s00031-005-0401-5
- Ivan V. Arzhantsev and Dmitri A. Timashev, On the canonical embeddings of certain homogeneous spaces, Lie groups and invariant theory, Amer. Math. Soc. Transl. Ser. 2, vol. 213, Amer. Math. Soc., Providence, RI, 2005, pp. 63â83. MR 2140714, DOI 10.1090/trans2/213/04
- A. A. BeÄlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) AstĂ©risque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5â171 (French). MR 751966
- A. A. Beilinson and V. G. Drinfeld, Quantization of Hitchinâs fibration and Langlandsâ program, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 3â7. MR 1385674
- Alexander Beilinson and Vladimir Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MR 2058353, DOI 10.1090/coll/051
- A. A. BeÄlinson, How to glue perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984â1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 42â51. MR 923134, DOI 10.1007/BFb0078366
- A. A. BeÄlinson, On the derived category of perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984â1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27â41. MR 923133, DOI 10.1007/BFb0078365
- Alexander Braverman, Michael Finkelberg, and Dennis Gaitsgory, Uhlenbeck spaces via affine Lie algebras, The unity of mathematics, Progr. Math., vol. 244, BirkhĂ€user Boston, Boston, MA, 2006, pp. 17â135. MR 2181803, DOI 10.1007/0-8176-4467-9_{2}
- A. Braverman, M. Finkelberg, D. Gaitsgory, and I. MirkoviÄ, Intersection cohomology of Drinfeldâs compactifications, Selecta Math. (N.S.) 8 (2002), no. 3, 381â418. MR 1931170, DOI 10.1007/s00029-002-8111-5
- A. Braverman, M. Finkelberg, D. Gaitsgory, and I. MirkoviÄ, Erratum to: âIntersection cohomology of Drinfeldâs compactificationsâ [Selecta Math. (N.S.) 8 (2002), no. 3, 381â418; MR1931170], Selecta Math. (N.S.) 10 (2004), no. 3, 429â430. MR 2099075, DOI 10.1007/s00029-004-0383-5
- Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik, Character $D$-modules via Drinfeld center of Harish-Chandra bimodules, Invent. Math. 188 (2012), no. 3, 589â620. MR 2917178, DOI 10.1007/s00222-011-0354-3
- Alexander Braverman and Dennis Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), no. 3, 561â575. MR 1828302, DOI 10.1215/S0012-7094-01-10736-9
- A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), no. 2, 287â384. MR 1933587, DOI 10.1007/s00222-002-0237-8
- Alexander Braverman and Dennis Gaitsgory, Deformations of local systems and Eisenstein series, Geom. Funct. Anal. 17 (2008), no. 6, 1788â1850. MR 2399084, DOI 10.1007/s00039-007-0645-4
- Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris SĂ©r. I Math. 320 (1995), no. 3, 335â340 (French, with English and French summaries). MR 1320381
- M. Brion, D. Luna, and Th. Vust, Espaces homogĂšnes sphĂ©riques, Invent. Math. 84 (1986), no. 3, 617â632 (French). MR 837530, DOI 10.1007/BF01388749
- Victor Batyrev and Anne Moreau, Satellites of spherical subgroups, Algebr. Geom. 7 (2020), no. 1, 86â112. MR 4038405, DOI 10.14231/ag-2020-004
- A. Bouthier, B. C. NgĂŽ, and Y. Sakellaridis, On the formal arc space of a reductive monoid, Amer. J. Math. 138 (2016), no. 1, 81â108. MR 3462881, DOI 10.1353/ajm.2016.0004
- A. Bouthier, B. C. NgĂŽ, and Y. Sakellaridis, Erratum to: âOn the formal arc space of a reductive monoidâ [ MR3462881], Amer. J. Math. 139 (2017), no. 1, 293â295. MR 3619916, DOI 10.1353/ajm.2017.0006
- Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209â216. MR 1996415, DOI 10.1007/s00031-003-0606-4
- Michel Brion, The total coordinate ring of a wonderful variety, J. Algebra 313 (2007), no. 1, 61â99. MR 2326138, DOI 10.1016/j.jalgebra.2006.12.022
- Daniel Bump and Anne Schilling, Crystal bases, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Representations and combinatorics. MR 3642318, DOI 10.1142/9876
- David Ben-Zvi, Yiannis Sakellaridis, and Akshay Venkatesh, Duality in the relative Langlands program, In preparation.
- Justin Campbell, Nearby cycles of Whittaker sheaves on Drinfeldâs compactification, Compos. Math. 154 (2018), no. 8, 1775â1800. MR 3830552, DOI 10.1112/s0010437x18007285
- Justin Campbell, A resolution of singularities for Drinfeldâs compactification by stable maps, J. Algebraic Geom. 28 (2019), no. 1, 153â167. MR 3875364, DOI 10.1090/jag/727
- W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387â406. MR 571057
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, BirkhÀuser Boston, Inc., Boston, MA, 1997. MR 1433132
- W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207â231. MR 581582
- P. Deligne, ThĂ©orĂšmes de finitude en cohomologie $\ell$-adique, Cohomologie Ă©tale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977, pp. 233â261 (French). MR 3727439
- V. Drinfeld, Grinberg-Kazhdan theorem and Newton groupoids, arXiv:1801.01046v2, 2018.
- V. G. DrinfelâČd and Carlos Simpson, $B$-structures on $G$-bundles and local triviality, Math. Res. Lett. 2 (1995), no. 6, 823â829. MR 1362973, DOI 10.4310/MRL.1995.v2.n6.a13
- Boris Feigin, Michael Finkelberg, Alexander Kuznetsov, and Ivan MirkoviÄ, Semi-infinite flags. II. Local and global intersection cohomology of quasimapsâ spaces, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 113â148. MR 1729361, DOI 10.1090/trans2/194/06
- Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, 2005. Grothendieckâs FGA explained. MR 2222646, DOI 10.1090/surv/123
- Michael Finkelberg and Ivan MirkoviÄ, Semi-infinite flags. I. Case of global curve $\mathbf P^1$, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 81â112. MR 1729360, DOI 10.1090/trans2/194/05
- D. Gaitsgory, What acts on geometric Eisenstein series, Available at http://www.math.harvard.edu/~gaitsgde/GL/WhatActs.pdf, 2011.
- Dennis Gaitsgory, Outline of the proof of the geometric Langlands conjecture for $GL_2$, AstĂ©risque 370 (2015), 1â112 (English, with English and French summaries). MR 3364744
- I. Grojnowski and G. Lusztig, A comparison of bases of quantized enveloping algebras, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 11â19. MR 1247495, DOI 10.1090/conm/153/01304
- Dennis Gaitsgory and David Nadler, Spherical varieties and Langlands duality, Mosc. Math. J. 10 (2010), no. 1, 65â137, 271 (English, with English and Russian summaries). MR 2668830, DOI 10.17323/1609-4514-2010-10-1-65-137
- Frank D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math. 107 (1992), no. 1, 127â133. MR 1135467, DOI 10.1007/BF01231884
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- Birger Iversen, The geometry of algebraic groups, Advances in Math. 20 (1976), no. 1, 57â85. MR 399114, DOI 10.1016/0001-8708(76)90170-5
- V. Yu. Kaloshin, A geometric proof of the existence of Whitney stratifications, Mosc. Math. J. 5 (2005), no. 1, 125â133 (English, with English and Russian summaries). MR 2153470, DOI 10.17323/1609-4514-2005-5-1-125-133
- Masaki Kashiwara, The crystal base and Littelmannâs refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839â858. MR 1240605, DOI 10.1215/S0012-7094-93-07131-1
- Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383â413. MR 1262212, DOI 10.1215/S0012-7094-94-07317-1
- Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155â197. MR 1357199
- Friedrich Knop, Hanspeter Kraft, Domingo Luna, and Thierry Vust, Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, BirkhĂ€user, Basel, 1989, pp. 63â75. MR 1044585
- Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449â484. MR 1187560, DOI 10.1142/s0217751x92003896
- Friedrich Knop, The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) Manoj Prakashan, Madras, 1991, pp. 225â249. MR 1131314
- Friedrich Knop, The asymptotic behavior of invariant collective motion, Invent. Math. 116 (1994), no. 1-3, 309â328. MR 1253195, DOI 10.1007/BF01231563
- Friedrich Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285â309. MR 1324631, DOI 10.1007/BF02566009
- Friedrich Knop, Spherical roots of spherical varieties, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 6, 2503â2526 (English, with English and French summaries). MR 3331173, DOI 10.5802/aif.2919
- Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, BirkhĂ€user Boston, Boston, MA, 1995, pp. 335â368. MR 1363062, DOI 10.1007/978-1-4612-4264-2_{1}2
- F. Knop and B. Schalke, The dual group of a spherical variety, Trans. Moscow Math. Soc. 78 (2017), 187â216. MR 3738085, DOI 10.1090/mosc/270
- Mikhail Kapranov and Eric Vasserot, Vertex algebras and the formal loop space, Publ. Math. Inst. Hautes Ătudes Sci. 100 (2004), 209â269. MR 2102701, DOI 10.1007/s10240-004-0023-9
- Vincent Lafforgue, Quelques calculs reliés à la correspondance de Langlands géométrique pour ${\mathbb {P}}^1$, Available at http://vlafforg.perso.math.cnrs.fr/files/geom.pdf, 2009.
- Domingo Luna, Slices Ă©tales, Sur les groupes algĂ©briques, Bull. Soc. Math. France, Paris, MĂ©moire 33, Soc. Math. France, Paris, 1973, pp. 81â105 (French). MR 0342523, DOI 10.24033/msmf.110
- D. Luna, Grosses cellules pour les variĂ©tĂ©s sphĂ©riques, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 267â280 (French). MR 1635686
- G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175â201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, DOI 10.1143/PTPS.102.175
- D. Luna and Th. Vust, Plongements dâespaces homogĂšnes, Comment. Math. Helv. 58 (1983), no. 2, 186â245 (French). MR 705534, DOI 10.1007/BF02564633
- I. MirkoviÄ and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95â143. MR 2342692, DOI 10.4007/annals.2007.166.95
- V. L. Popov, Contractions of actions of reductive algebraic groups, Mat. Sb. (N.S.) 130(172) (1986), no. 3, 310â334, 431 (Russian); English transl., Math. USSR-Sb. 58 (1987), no. 2, 311â335. MR 865764, DOI 10.1070/SM1987v058n02ABEH003106
- Sam Raskin, Chiral principal series categories II: the factorizable Whittaker category, Available at https://web.ma.utexas.edu/users/sraskin/cpsii.pdf.
- R. W. Richardson, Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc. 9 (1977), no. 1, 38â41. MR 437549, DOI 10.1112/blms/9.1.38
- Yiannis Sakellaridis, On the unramified spectrum of spherical varieties over $p$-adic fields, Compos. Math. 144 (2008), no. 4, 978â1016. MR 2441254, DOI 10.1112/S0010437X08003485
- Yiannis Sakellaridis, Spherical varieties and integral representations of $L$-functions, Algebra Number Theory 6 (2012), no. 4, 611â667. MR 2966713, DOI 10.2140/ant.2012.6.611
- Yiannis Sakellaridis, Spherical functions on spherical varieties, Amer. J. Math. 135 (2013), no. 5, 1291â1381. MR 3117308, DOI 10.1353/ajm.2013.0046
- Yiannis Sakellaridis, Inverse Satake transforms, Geometric aspects of the trace formula, Simons Symp., Springer, Cham, 2018, pp. 321â349. MR 3969880
- Simon Schieder, The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeldâs compactifications, Selecta Math. (N.S.) 21 (2015), no. 3, 763â831. MR 3366920, DOI 10.1007/s00029-014-0161-y
- Simon Schieder, Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for $\rm SL_2$, Duke Math. J. 167 (2018), no. 5, 835â921. MR 3782063, DOI 10.1215/00127094-2017-0044
- Simon Schieder, Geometric Bernstein asymptotics and the DrinfeldâLafforgueâVinberg degeneration for arbitrary reductive groups, preprint, available at: https://arxiv.org/abs/1607.00586v2, 2021.
- Groupes de monodromie en gĂ©omĂ©trie algĂ©brique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois-Marie 1967â1969 (SGA 7 II); DirigĂ© par P. Deligne et N. Katz. MR 0354657
- Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Ătudes Sci. Publ. Math. 25 (1965), 49â80. MR 180554, DOI 10.1007/BF02684397
- Yiannis Sakellaridis and Akshay Venkatesh, Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), viii+360 (English, with English and French summaries). MR 3764130
- Dmitry A. Timashev, Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, vol. 138, Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR 2797018, DOI 10.1007/978-3-642-18399-7
- Ă. B. Vinberg and B. N. KimelâČfelâČd, Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 12â19, 96 (Russian). MR 509380
- J. Wang, Universal local acyclicity, Available at https://www.jonathanpwang.com/notes/ULA.pdf, 2021.
- B. Wasserman, Wonderful varieties of rank two, Transform. Groups 1 (1996), no. 4, 375â403. MR 1424449, DOI 10.1007/BF02549213
- Xinwen Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2017, pp. 59â154. MR 3752460
Additional Information
- Yiannis Sakellaridis
- Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- MR Author ID: 796283
- ORCID: 0000-0003-3924-286X
- Email: sakellar@jhu.edu
- Jonathan Wang
- Affiliation: Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
- MR Author ID: 912751
- Email: jwang4@perimeterinstitute.ca
- Received by editor(s): February 8, 2021
- Received by editor(s) in revised form: July 14, 2021
- Published electronically: October 5, 2021
- Additional Notes: The first author was supported by NSF grants DMS-1801429 and DMS-1939672, and by a stipend to the IAS from the Charles Simonyi Endowment. The second author was supported by NSF grant DMS-1803173
- © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc. 35 (2022), 799-910
- MSC (2020): Primary 22E57, 11F67; Secondary 14D24, 14M27, 43A85
- DOI: https://doi.org/10.1090/jams/990
- MathSciNet review: 4433079