Geometrization of principal series representations of reductive groups  [ Géométrisation des représentations de la série principale des groupes reductifs ]
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, p. 2273-2330
En théorie des représentations, on cherche souvent à écrire des représentations réalisées dans des espaces de fonctions invariantes comme les fonctions trace de faisceaux pervers équivariants. Dans le cas des représentations de la série principale d’un groupe réductif G connexe scindé sur un corps local, il existe une description des familles de telles représentations realisées dans des espaces de fonctions sur G invariantes sous l’action de translation du sous-groupe d’Iwahori ou d’un sous-groupe compact ouvert plus petit approprié, comme l’ont etudié Howe, Bushnell et Kutzko, Roche, et d’autres. Dans cet article, nous construisons des catégories de faisceaux pervers dont les traces redonnent les families associées aux caractères réguliers de T(𝔽 q [[t]]), et démontrons des conjectures de Drinfeld pour leur structure. Nous proposons également des conjectures sur la géométrisation des familles associées à des caractères plus généraux.
In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected split reductive group G over a local field, there is a description of families of these representations realized on spaces of functions on G invariant under the translation action of the Iwahori subgroup, or a suitable smaller compact open subgroup, studied by Howe, Bushnell and Kutzko, Roche, and others. In this paper, we construct categories of perverse sheaves whose traces recover the families associated to regular characters of T(𝔽 q [[t]]), and prove conjectures of Drinfeld on their structure. We also propose conjectures on the geometrization of families associated to more general characters.
DOI : https://doi.org/10.5802/aif.2988
Classification:  22E50,  20G25
Mots clés: Séries principales, isomorphisme géométrique de Satake, sous-groupes compacts ouverts, algèbre de Hecke, géométrisation, faisceaux pervers propres
@article{AIF_2015__65_5_2273_0,
     author = {Kamgarpour, Masoud and Schedler, Travis},
     title = {Geometrization of principal series representations of reductive groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     pages = {2273-2330},
     doi = {10.5802/aif.2988},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_5_2273_0}
}
Kamgarpour, Masoud; Schedler, Travis. Geometrization of principal series representations of reductive groups. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2273-2330. doi : 10.5802/aif.2988. http://www.numdam.org/item/AIF_2015__65_5_2273_0/

[1] Adler, J. D. Refined anisotropic K-types and supercuspidal representations, Pacific J. Math., Tome 185 (1998) no. 1, pp. 1-32 | Zbl 0924.22015

[2] Adler, J. D.; Roche, A. An intertwining result for p-adic groups, Canad. J. Math., Tome 52 (2000) no. 3, pp. 449-467 | Article | Zbl 1160.22304

[3] Arkhipov, S.; Bezrukavnikov, R. Perverse sheaves on affine flags and Langlands dual group, Israel J. Math., Tome 170 (2009), pp. 135-183 (With an appendix by Bezrukavrikov and I. Mirković) | Article | Zbl 1214.14011

[4] Arkhipov, S.; Braverman, A.; Bezrukavnikov, R.; Gaitsgory, D.; Mirković, I. Modules over the small quantum group and semi-infinite flag manifold, Transform. Groups, Tome 10 (2005) no. 3-4, pp. 279-362 | Article | Zbl 1122.17007

[5] Beilinson, A.; Drinfeld, V. Quantization of Hitchin’s integrable system and Hecke eigensheaves (http://www.math.uchicago.edu/~mitya/langlands.html)

[6] Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 100 (1982), pp. 5-171

[7] Bernstein, J.; Lunts, V. Equivariant sheaves and functors, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1578 (1994), pp. iv+139 | Zbl 0808.14038

[8] Bernstein, J. N. Le “centre” de Bernstein, Representations of reductive groups over a local field, Hermann, Paris (Travaux en Cours) (1984), pp. 1-32 (Edited by P. Deligne) | Zbl 0599.22016

[9] Bezrukavnikov, R. On tensor categories attached to cells in affine Weyl groups, Representation theory of algebraic groups and quantum groups, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 40 (2004), pp. 69-90 | Zbl 1078.20044

[10] Bezrukavnikov, R. Noncommutative counterparts of the Springer resolution, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich (2006), pp. 1119-1144 (arXiv:math/0604445) | Zbl 1135.17011

[11] Bezrukavnikov, R. Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, Israel J. Math., Tome 170 (2009), pp. 185-206 | Article | Zbl 1214.14012

[12] Bezrukavnikov, R.; Finkelberg, M.; Ostrik, V. On tensor categories attached to cells in affine Weyl groups. III, Israel J. Math., Tome 170 (2009), pp. 207-234 | Article | Zbl 1210.20004

[13] Bezrukavnikov, R.; Ostrik, V. On tensor categories attached to cells in affine Weyl groups. II, Representation theory of algebraic groups and quantum groups, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 40 (2004), pp. 101-119 | Zbl 1078.20045

[14] Boyarchenko, Mitya Characters of unipotent groups over finite fields, Selecta Math. (N.S.), Tome 16 (2010) no. 4, pp. 857-933 | Article | Zbl 1223.20038

[15] Boyarchenko, Mitya; Drinfeld, Vladimir A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic (2006) (http://arxiv.org/abs/math/0609769)

[16] Boyarchenko, Mitya; Drinfeld, Vladimir A duality formalism in the spirit of Grothendieck and Verdier, Quantum Topol., Tome 4 (2013) no. 4, pp. 447-489 | Article

[17] Boyarchenko, Mitya; Drinfeld, Vladimir Character sheaves on unipotent groups in positive characteristic: foundations, Selecta Math. (N.S.), Tome 20 (2014) no. 1, pp. 125-235 | Article

[18] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972) no. 41, pp. 5-251 | Numdam | Zbl 0254.14017

[19] Bushnell, C. J.; Henniart, G. The local Langlands conjecture for GL ( 2 ) , Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 335 (2006), pp. xii+347 | Article | Zbl 1100.11041

[20] Bushnell, C. J.; Kutzko, P. C. Smooth representations of reductive p-adic groups: structure theory via types, Proc. London Math. Soc. (3), Tome 77 (1998) no. 3, pp. 582-634 | Article | Zbl 0911.22014

[21] Bushnell, C. J.; Kutzko, P. C. Semisimple types in GL n , Compositio Math., Tome 119 (1999) no. 1, pp. 53-97 | Article | Zbl 0933.22027

[22] Deligne, P. Cohomologie étale, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 569 (1977), pp. iv+312pp (Séminaire de Géométrie Algébrique du Bois-Marie SGA 41øer 2, Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier) | Zbl 0345.00010

[23] Deligne, P. La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980) no. 52, pp. 137-252 | Numdam | Zbl 0456.14014

[24] Dimca, A. Sheaves in topology, Springer-Verlag, Berlin, Universitext (2004), pp. xvi+236 | Zbl 1043.14003

[25] Feigin, B.; Finkelberg, M.; Kuznetsov, A.; Mirković, I. Semi-infinite flags. II. Local and global intersection cohomology of quasimaps’ spaces, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 194 (1999), pp. 113-148 | Zbl 1076.14511

[26] Finkelberg, M.; Mirković, I. Semi-infinite flags. I. Case of global curve 1 , Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 194 (1999), pp. 81-112 | Zbl 1076.14512

[27] Frenkel, E. Lectures on the Langlands program and conformal field theory, Frontiers in number theory, physics, and geometry. II, Springer, Berlin (2007), pp. 387-533 | Article | Zbl 1196.11091

[28] Frenkel, E.; Gaitsgory, D. Local geometric Langlands correspondence and affine Kac-Moody algebras, Algebraic geometry and number theory, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 253 (2006), pp. 69-260 | Article | Zbl 1184.17011

[29] Frenkel, E.; Gaitsgory, D.; Vilonen, K. Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math. (2), Tome 153 (2001) no. 3, pp. 699-748 | Article | Zbl 1070.11050

[30] Gaitsgory, D. Notes on 2D conformal field theory and string theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI (1999), pp. 1017-1089 | Zbl 1170.81429

[31] Gaitsgory, D. Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math., Tome 144 (2001) no. 2, pp. 253-280 | Article | Zbl 1072.14055

[32] Ginzburg, V. Perverse sheaves on a loop group and Langlands’ duality (2000) (http://arxiv.org/abs/alg-geom/9511007)

[33] Giraud, J. Cohomologie non abélienne, Springer-Verlag, Berlin (1971), pp. ix+467 (Die Grundlehren der mathematischen Wissenschaften, Band 179) | Zbl 0226.14011

[34] Goresky, M.; Macpherson, R. Intersection homology. II, Invent. Math., Tome 72 (1983) no. 1, pp. 77-129 | Article | Zbl 0529.55007

[35] Greenberg, Marvin J. Schemata over local rings, Ann. of Math. (2), Tome 73 (1961), pp. 624-648 | Zbl 0115.39004

[36] Haines, T. J.; Rostami, S. The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory, Tome 14 (2010), pp. 264-284 | Article | Zbl 1251.22013

[37] Howe, Roger E. On the principal series of Gl n over p-adic fields, Trans. Amer. Math. Soc., Tome 177 (1973), pp. 275-286 | Zbl 0257.22018

[38] Kamgarpour, Masoud Stacky abelianization of algebraic groups, Transform. Groups, Tome 14 (2009) no. 4, pp. 825-846 | Article | Zbl 1225.14038

[39] Kamgarpour, Masoud Compatibility of the Feigin–Frenkel Isomorphism and the Harish–Chandra Isomorphism for jet algebras, Trans. Amer. Math. Soc. (2014) (published electronically, www.ams.org/journals/tran/0000-000-00/S0002-9947-2014-06419-2/)

[40] Kamgarpour, Masoud; Schedler, Travis Ramified Satake isomorphisms for strongly parabolic characters., Doc. Math., J. DMV, Tome 18 (2013), pp. 1275-1300

[41] Kiehl, R.; Weissauer, R. Weil conjectures, perverse sheaves and l ’adic Fourier transform, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Tome 42 (2001), pp. xii+375 | Zbl 0988.14009

[42] Kreidl, Martin On p-adic lattices and Grassmannians, Math. Z., Tome 276 (2014) no. 3-4, pp. 859-888 | Article | Zbl 1304.13010

[43] Laszlo, Y.; Olsson, M. The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. Inst. Hautes Études Sci. (2008) no. 107, pp. 169-210 | Article | Numdam | Zbl 1191.14003

[44] Laumon, G. Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. (1987) no. 65, pp. 131-210 | Numdam | Zbl 0641.14009

[45] Lusztig, G. Singularities, character formulas, and a q-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 101 (1983), pp. 208-229 | Zbl 0561.22013

[46] Lusztig, G. Intersection cohomology complexes on a reductive group, Invent. Math., Tome 75 (1984) no. 2, pp. 205-272 | Article | Zbl 0547.20032

[47] Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2), Tome 166 (2007) no. 1, pp. 95-143 | Article | Zbl 1138.22013

[48] Nadler, David Perverse sheaves on real loop Grassmannians, Invent. Math., Tome 159 (2005) no. 1, pp. 1-73 | Article | Zbl 1089.14008

[49] Reich, Ryan Cohen Twisted geometric Satake equivalence via gerbes on the factorizable Grassmannian, Represent. Theory, Tome 16 (2012), pp. 345-449 | Zbl 1302.22014

[50] Roche, A. Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. École Norm. Sup. (4), Tome 31 (1998) no. 3, pp. 361-413 | Article | Numdam | Zbl 0903.22009

[51] Roche, A. The Bernstein decomposition and the Bernstein centre, Ottawa lectures on admissible representations of reductive p -adic groups, Amer. Math. Soc., Providence, RI (Fields Inst. Monogr.) Tome 26 (2009), pp. 3-52 | Zbl 1176.22015

[52] Saavedra Rivano, N. Catégories Tannakiennes, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 265 (1972), pp. ii+418 | Zbl 0241.14008

[53] Yu, Jiu-Kang Construction of tame supercuspidal representations, J. Amer. Math. Soc., Tome 14 (2001) no. 3, p. 579-622 (electronic) | Article | Zbl 0971.22012

[54] Yu, Jiu-Kang Smooth models associated to concave functions in Bruhat-Tits Theory (2002) (http://www.math.purdue.edu/~jyu/prep/model.pdf)

[55] Zhu, Xinwen The geometric Satake correspondence for ramified groups (2011) (http://arxiv.org/abs/1107.5762v1)

[56] Zhu, Xinwen On the coherence conjecture of Pappas and Rapoport, Ann. of Math. (2), Tome 180 (2014) no. 1, pp. 1-85 | Article | Zbl 1300.14042