Geometric Langlands for hypergeometric sheaves

Kamgarpour, Masoud; Yi, Lingfei

doi:10.1090/tran/8509

Geometric Langlands for hypergeometric sheaves
HTML articles powered by AMS MathViewer

by Masoud Kamgarpour and Lingfei Yi PDF

Trans. Amer. Math. Soc. 374 (2021), 8435-8481 Request permission

Abstract:

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340 (French, with English and French summaries). MR 1320381
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves (1997). https://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf.
David Ben-Zvi and David Nadler, Betti geometric Langlands, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 3–41. MR 3821166
Frits Beukers, Henri Cohen, and Anton Mellit, Finite hypergeometric functions, Pure Appl. Math. Q. 11 (2015), no. 4, 559–589. MR 3613122, DOI 10.4310/PAMQ.2015.v11.n4.a2
Tsao-Hsien Chen, Vinberg’s $\theta$-groups and rigid connections, Int. Math. Res. Not. IMRN 23 (2017), 7321–7343. MR 3801422, DOI 10.1093/imrn/rnw235
Alessio Corti and Vasily Golyshev, Hypergeometric equations and weighted projective spaces, Sci. China Math. 54 (2011), no. 8, 1577–1590. MR 2824960, DOI 10.1007/s11425-011-4218-5
V. Chernousov, P. Gille, and A. Pianzola, Torsors over the punctured affine line, Amer. J. Math. 134 (2012), no. 6, 1541–1583. MR 2999288, DOI 10.1353/ajm.2012.0051
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
L. Dembélé, A. Panchishkin, J. Voight, and W. Zudilin, Special hypergeometric motives and their $L$-functions: Asai recognition (2019), available at https://arxiv.org/abs/1906.07384.
Edward Frenkel, Recent advances in the Langlands program, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 151–184. MR 2043750, DOI 10.1090/S0273-0979-04-01001-8
Edward Frenkel and Benedict Gross, A rigid irregular connection on the projective line, Ann. of Math. (2) 170 (2009), no. 3, 1469–1512. MR 2600880, DOI 10.4007/annals.2009.170.1469
D. Gaitsgory, On de Jong’s conjecture, Israel J. Math. 157 (2007), 155–191. MR 2342444, DOI 10.1007/s11856-006-0006-2
Dennis Gaitsgory, Progrès récents dans la théorie de Langlands géométrique, Astérisque 390 (2017), Exp. No. 1109, 139–168 (French). Séminaire Bourbaki. Vol. 2015/2016. Exposés 1104–1119. MR 3666025
V. Ginzburg, Perverse sheaves on a Loop group and Langlands’ duality (1995), available at https://arxiv.org/pdf/alg-geom/9511007.pdf.
Vassily Gorbounov and Maxim Smirnov, Some remarks on Landau-Ginzburg potentials for odd-dimensional quadrics, Glasg. Math. J. 57 (2015), no. 3, 481–507. MR 3395329, DOI 10.1017/S0017089514000433
John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, DOI 10.1090/S0002-9947-1987-0879564-8
Benedict H. Gross and Mark Reeder, Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154 (2010), no. 3, 431–508. MR 2730575, DOI 10.1215/00127094-2010-043
T. Haines and M. Rapoport, On parahoric subgroups, Adv. Math. 219 (2008), no. 1, 188–198. Appendix to Pappas and Rapoport’s Twisted loop groups and their affine flag varieties.
Jochen Heinloth, Uniformization of $\scr G$-bundles, Math. Ann. 347 (2010), no. 3, 499–528. MR 2640041, DOI 10.1007/s00208-009-0443-4
Jochen Heinloth, Bao-Châu Ngô, and Zhiwei Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), no. 1, 241–310. MR 2999041, DOI 10.4007/annals.2013.177.1.5
Richard Paul Horja, Hypergeometric functions and mirror symmetry in toric varieties, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–Duke University. MR 2700280
K. Jakob and Z. Yun, Euphotic representations and rigid automorphic data (2020), available at https://arxiv.org/abs/2008.04029.
Masoud Kamgarpour, Stacky abelianization of algebraic groups, Transform. Groups 14 (2009), no. 4, 825–846. MR 2577200, DOI 10.1007/s00031-009-9067-8
Masoud Kamgarpour and Travis Schedler, Ramified Satake isomorphisms for strongly parabolic characters, Doc. Math. 18 (2013), 1275–1300. MR 3138847
Masoud Kamgarpour and Daniel S. Sage, A geometric analogue of a conjecture of Gross and Reeder, Amer. J. Math. 141 (2019), no. 5, 1457–1476. MR 4011806, DOI 10.1353/ajm.2019.0038
Masoud Kamgarpour and Daniel S. Sage, Rigid connections on $\Bbb P^1$ via the Bruhat-Tits building, Proc. Lond. Math. Soc. (3) 122 (2021), no. 3, 359–376. MR 4230058, DOI 10.1112/plms.12346
Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536, DOI 10.1515/9781400882434
Nicholas M. Katz, Rigid local systems, Annals of Mathematics Studies, vol. 139, Princeton University Press, Princeton, NJ, 1996. MR 1366651, DOI 10.1515/9781400882595
Tom H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp. 1–85. MR 774055
Gérard Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987), no. 2, 309–359 (French). MR 899400, DOI 10.1215/S0012-7094-87-05418-4
G. Laumon, M. Rapoport, and U. Stuhler, ${\scr D}$-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), no. 2, 217–338. MR 1228127, DOI 10.1007/BF01244308
T. Lam and N. Templier, The mirror conjecture for minuscule flag varieties (2017), available at https://arxiv.org/abs/1705.00758.
I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), no. 1, 95–143. MR 2342692, DOI 10.4007/annals.2007.166.95
Ken Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205–1223. MR 1407498, DOI 10.1090/S0002-9947-98-01887-X
Eric M. Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, vol. 8, Mathematical Society of Japan, Tokyo, 2000. With a preface by Toshio Oshima. MR 1805058
C. Pech, K. Rietsch, and L. Williams, On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections, Adv. Math. 300 (2016), 275–319. MR 3534834, DOI 10.1016/j.aim.2016.03.020
M. Reeder, P. Levy, J. Yu, and B. Gross, Gradings of positive rank on simple Lie algebras, Transform. Groups 17 (2012), no. 4, 1123–1190.
Mark Reeder and Jiu-Kang Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27 (2014), no. 2, 437–477. MR 3164986, DOI 10.1090/S0894-0347-2013-00780-8
W. Sawin and N. Templier, On the Ramanujan conjecture for automorphic forms over function fields I. Geometry (2018), available at https://arxiv.org/abs/1805.12231.
Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
Alexander Varchenko, Special functions, KZ type equations, and representation theory, CBMS Regional Conference Series in Mathematics, vol. 98, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. MR 2016311, DOI 10.1090/cbms/098
È. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709 (Russian). MR 0430168
D. Xu and X. Zhu, Bessel $F$-isocrystals for reductive groups (2019), available at https://arxiv.org/abs/1910.13391.
Jiu-Kang Yu, Smooth models associated to concave functions in Bruhat-Tits theory, Autour des schémas en groupes. Vol. III, Panor. Synthèses, vol. 47, Soc. Math. France, Paris, 2015, pp. 227–258 (English, with English and French summaries). MR 3525846
Zhiwei Yun, Motives with exceptional Galois groups and the inverse Galois problem, Invent. Math. 196 (2014), no. 2, 267–337. MR 3193750, DOI 10.1007/s00222-013-0469-9
Zhiwei Yun, Rigidity in automorphic representations and local systems, Current developments in mathematics 2013, Int. Press, Somerville, MA, 2014, pp. 73–168. MR 3307715
Zhiwei Yun, Epipelagic representations and rigid local systems, Selecta Math. (N.S.) 22 (2016), no. 3, 1195–1243. MR 3518549, DOI 10.1007/s00029-015-0204-z
Don Zagier, The arithmetic and topology of differential equations, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2018, pp. 717–776. MR 3890449
Don Zagier and Aleksey Zinger, Some properties of hypergeometric series associated with mirror symmetry, Modular forms and string duality, Fields Inst. Commun., vol. 54, Amer. Math. Soc., Providence, RI, 2008, pp. 163–177. MR 2454324
Wadim Zudilin, Hypergeometric heritage of W. N. Bailey, ICCM Not. 7 (2019), no. 2, 32–46. With an appendix containing letters from Bailey to Freeman Dyson. MR 3995129, DOI 10.4310/ICCM.2019.v7.n2.a4