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, pp. 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 : 10.5802/aif.2988
Classification : 22E50, 20G25
Keywords: Principal series representations, geometric Satake isomorphism, compact open subgroups, Hecke algebras, geometrization, clean perverse sheaves
Mot clés : Séries principales, isomorphisme géométrique de Satake, sous-groupes compacts ouverts, algèbre de Hecke, géométrisation, faisceaux pervers propres
Kamgarpour, Masoud 1 ; Schedler, Travis 2

1 School of Mathematics and Physics The University of Queensland St. Lucia, Brisbane 4072 (Australia)
2 Department of Mathematics, 1 University Station C1200 Austin, TX 78712-0257 (USA)
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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/articles/10.5802/aif.2988/

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

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

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

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

[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) (Astérisque), Volume 100, Soc. Math. France, Paris, 1982, pp. 5-171

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

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

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

[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

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

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

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

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

[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., Volume 4 (2013) no. 4, pp. 447-489 | DOI

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

[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

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

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

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

[22] Deligne, P. Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin, 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

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

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

[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. Transl. Ser. 2), Volume 194, Amer. Math. Soc., Providence, RI, 1999, pp. 113-148 | Zbl

[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. Transl. Ser. 2), Volume 194, Amer. Math. Soc., Providence, RI, 1999, pp. 81-112 | Zbl

[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 | DOI | Zbl

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

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

[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

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

[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

[34] Goresky, M.; MacPherson, R. Intersection homology. II, Invent. Math., Volume 72 (1983) no. 1, pp. 77-129 | DOI | Zbl

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

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

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

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

[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, Volume 18 (2013), pp. 1275-1300

[41] Kiehl, R.; Weissauer, R. Weil conjectures, perverse sheaves and l ’adic Fourier transform, 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], 42, Springer-Verlag, Berlin, 2001, pp. xii+375 | Zbl

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

[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 | DOI | Numdam | Zbl

[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

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

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

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

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

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

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

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

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

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

[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), Volume 180 (2014) no. 1, pp. 1-85 | DOI | Zbl

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