On a conjecture of Braverman-Kazhdan

Chen, Tsao-Hsien

doi:10.1090/jams/992

On a conjecture of Braverman-Kazhdan
HTML articles powered by AMS MathViewer

by Tsao-Hsien Chen

J. Amer. Math. Soc. 35 (2022), 1171-1214

DOI: https://doi.org/10.1090/jams/992

Published electronically: December 2, 2021

HTML | PDF | Request permission

Abstract:

In this article we prove a conjecture of Braverman-Kazhdan in [Geom. Funct. Anal. Special Volume (2000), pp. 237–278] on acyclicity of $\rho$-Bessel sheaves on reductive groups. We do so by proving a vanishing conjecture proposed in our previous work [A vanishing conjecture: the GLn case, arXiv:1902.11190]. As a corollary, we obtain a geometric construction of the non-linear Fourier kernel for a finite reductive group as conjectured by Braverman and Kazhdan. The proof uses the theory of Mellin transforms, Drinfeld center of Harish-Chandra bimodules, and a construction of a class of character sheaves in mixed-characteristic.

References

Sivaramakrishna Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Sur les groupes algébriques, Bull. Soc. Math. France, Mém. 33, Soc. Math. France, Paris, 1973, pp. 5–79 (French). MR 0335524, DOI 10.24033/msmf.109
A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1–50. MR 1237825
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik, Character $D$-modules via Drinfeld center of Harish-Chandra bimodules, Invent. Math. 188 (2012), no. 3, 589–620. MR 2917178, DOI 10.1007/s00222-011-0354-3
Alexander Beilinson and Victor Ginzburg, Wall-crossing functors and $\scr D$-modules, Represent. Theory 3 (1999), 1–31. MR 1659527, DOI 10.1090/S1088-4165-99-00063-1
A. Braverman and D. Kazhdan, $\gamma$-functions of representations and lifting, Geom. Funct. Anal. Special Volume (2000), 237–278. With an appendix by V. Vologodsky; GAFA 2000 (Tel Aviv, 1999). MR 1826255, DOI 10.1007/978-3-0346-0422-2_{9}
Alexander Braverman and David Kazhdan, $\gamma$-sheaves on reductive groups, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 27–47. MR 1985192
Roman Bezrukavnikov and Alexander Yom Din, On parabolic restriction of perverse sheaves, Publ. Res. Inst. Math. Sci. 57 (2021), no. 3-4, 1089–1107. MR 4322008, DOI 10.4171/prims/57-3-12
D. Ben-Zvi and S. Gunningham, Symmetries of categorical representations and the quantum Ngô action, arXiv:1712.01963, 2017.
T.-H. Chen, A vanishing conjecture: the $GL_n$ case, arXiv:1902.11190, 2019.
Shuyang Cheng and B\hook{a}o Châu Ngô, On a conjecture of Braverman and Kazhdan, Int. Math. Res. Not. IMRN 20 (2018), 6177–6200. MR 3872321, DOI 10.1093/imrn/rnx052
P. Deligne, Applications de la formule des traces aux sommes trigonométriques, Cohomologie étale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977, pp. 168–232 (French). MR 3727438
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
P. Deligne, Le formalisme des cycles évanescents, Groupes de Monodromie en Géométrie Algébrique, SGA 7, Volume II, Lecture Notes in Mathematics, vol 340, Springer, 1973.
Vladimir Drinfeld, On a conjecture of Kashiwara, Math. Res. Lett. 8 (2001), no. 5-6, 713–728. MR 1879815, DOI 10.4310/MRL.2001.v8.n6.a3
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
L. Fargues and P. Scholze, Geometrization of the local Langlands correspondence, arXiv:2102.13459, 2021.
Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5, Institut des Hautes Études Scientifiques, Paris, 1963. Troisième édition, corrigée; Séminaire de Géométrie Algébrique, 1960/61. MR 0217087
Victor Ginsburg, Admissible modules on a symmetric space, Astérisque 173-174 (1989), 9–10, 199–255. Orbites unipotentes et représentations, III. MR 1021512
Victor Ginzburg, Nil-Hecke algebras and Whittaker $\mathcal D$-modules, Lie groups, geometry, and representation theory, Progr. Math., vol. 326, Birkhäuser/Springer, Cham, 2018, pp. 137–184. MR 3890208, DOI 10.1007/978-3-030-02191-7_{6}
Sam Gunningham, Generalized Springer theory for $D$-modules on a reductive Lie algebra, Selecta Math. (N.S.) 24 (2018), no. 5, 4223–4277. MR 3874694, DOI 10.1007/s00029-018-0443-x
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
Ofer Gabber and François Loeser, Faisceaux pervers $l$-adiques sur un tore, Duke Math. J. 83 (1996), no. 3, 501–606 (French). MR 1390656, DOI 10.1215/S0012-7094-96-08317-9
Gus Lonergan, A Fourier transform for the quantum Toda lattice, Selecta Math. (N.S.) 24 (2018), no. 5, 4577–4615. MR 3874699, DOI 10.1007/s00029-018-0419-x
Ting-Yu Lee, Adjoint quotients of reductive groups, Autour des schémas en groupes. Vol. III, Panor. Synthèses, vol. 47, Soc. Math. France, Paris, 2015, pp. 131–145 (English, with English and French summaries). MR 3525843
G. Lusztig, Character sheaves, I, Adv. Math. 56 (1985), 193–237, https://doi.org/10.1016/0001-8708(85)90034-9.
G. Laumon and E. Letellier, Note on a conjecture of Braverman-Kazhdan, arXiv:1906.07476, 2019.
Dragan Miličić and Wolfgang Soergel, The composition series of modules induced from Whittaker modules, Comment. Math. Helv. 72 (1997), no. 4, 503–520. MR 1600134, DOI 10.1007/s000140050031
I. Mirković and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), no. 2, 405–418. MR 948107, DOI 10.1007/BF01394339
Thomas Nevins, Descent of coherent sheaves and complexes to geometric invariant theory quotients, J. Algebra 320 (2008), no. 6, 2481–2495. MR 2437511, DOI 10.1016/j.jalgebra.2008.04.011
B\hook{a}o Châu Ngô, Hankel transform, Langlands functoriality and functional equation of automorphic $L$-functions, Jpn. J. Math. 15 (2020), no. 1, 121–167. MR 4068833, DOI 10.1007/s11537-019-1650-8
T. A. Springer, Reductive groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR 546587
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. 59–154. MR 3752460
X. Zhu, Coherent sheaves on the stack of Langlands parameters, arXiv:2008.02998, 2020.