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Spherical Hecke algebras for Kac-Moody groups over local fields. (English) Zbl 1315.20046

In this paper the authors use the hovel associated to an almost split Kac-Moody group \(G\) over a local non-archimedean field to define an analogue \(\mathcal H\) of the classical spherical Hecke algebra in this situation. The authors show that the structure constants of \(\mathcal H\) are polynomials in the cardinality of the residue field and have integer coefficients depending on the geometry of the standard apartments of the hovel. The paper establishes an analogue of Satake isomorphism from \(\mathcal H\) to the algebra of certain invariant elements in a certain completion of a Laurent polynomial ring. In particular, this shows that \(\mathcal H\) is commutative. The paper also contains a direct proof of commutativity of \(\mathcal H\) in the case of a split group \(G\). The proof of the Satake isomorphism involves parabolic retraction.

MSC:

20G44 Kac-Moody groups
20C08 Hecke algebras and their representations
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
20E42 Groups with a \(BN\)-pair; buildings
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[1] I. Satake, ”Theory of spherical functions on reductive algebraic groups over \({\mathfrak p}\)-adic fields,” Inst. Hautes Études Sci. Publ. Math., vol. 18, pp. 5-69, 1963. · Zbl 0122.28501
[2] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local,” Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972. · Zbl 0254.14017
[3] J. Parkinson, ”Buildings and Hecke algebras,” J. Algebra, vol. 297, iss. 1, pp. 1-49, 2006. · Zbl 1095.20003 · doi:10.1016/j.jalgebra.2005.08.036
[4] A. Braverman and D. Kazhdan, ”The spherical Hecke algebra for affine Kac-Moody groups I,” Ann. of Math., vol. 174, iss. 3, pp. 1603-1642, 2011. · Zbl 1235.22027 · doi:10.4007/annals.2011.174.3.5
[5] A. Braverman and D. Kazhdan, ”Representation of affine Kac-Moody groups over local and global fields: a survey of some recent results,” in European Congress of Mathematics, Zürich: Eur. Math. Soc., 2014, pp. 91-117. · Zbl 1364.22008 · doi:10.4171/120-1/6
[6] S. Gaussent and G. Rousseau, ”Kac-Moody groups, hovels and Littelmann paths,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 58, iss. 7, pp. 2605-2657, 2008. · Zbl 1161.22007 · doi:10.5802/aif.2423
[7] G. Rousseau, Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées, 2010. · Zbl 1401.20055
[8] G. Rousseau, Almost split Kac-Moody groups over ultrametric fields, 2012. · Zbl 1423.20058
[9] G. Rousseau, ”Masures affines,” Pure Appl. Math. Q., vol. 7, pp. 859-921, 2011. · Zbl 1255.51009 · doi:10.4310/PAMQ.2011.v7.n3.a10
[10] M. Patnaik, The Satake map for \(p\)-adic loop groups and the analogue of Mac Donald’s formula for spherical functions.
[11] A. Braverman, D. Kazhdan, and M. Patnaik, Iwahori-Hecke algebras for \(p\)-adic loop groups. · Zbl 1345.22011 · doi:10.1007/s00222-015-0612-x
[12] A. Braverman, H. Garland, D. Kazhdan, and M. Patnaik, ”An affine Gindikin-Karpelevich formula,” in Perspectives in Representation Theory, Etingof, P., Khovanov, M., and Savage, A., Eds., Providence, RI: Amer. Math. Soc., 2014, vol. 610, pp. 43-64. · Zbl 1302.20048 · doi:10.1090/conm/610/12193
[13] R. V. Moody and A. Pianzola, ”On infinite root systems,” Trans. Amer. Math. Soc., vol. 315, iss. 2, pp. 661-696, 1989. · Zbl 0676.17011 · doi:10.2307/2001300
[14] R. V. Moody and A. Pianzola, Lie Algebras with Triangular Decompositions, New York: John Wiley & Sons, 1995. · Zbl 0874.17026
[15] N. Bardy, Systèms de Racines Infinis, Paris: Math. Soc. France, 1996, vol. 65. · Zbl 0880.17019
[16] V. G. Kac, Infinite-Dimensional Lie Algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990. · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[17] C. Charignon, Structures immobilières pour un groupe de Kac-Moody sur un corps local, 2010.
[18] C. Charignon, Immeubles affines et groupes de Kac-Moody, masures bordées.
[19] M. Kapovich and J. J. Millson, ”A path model for geodesics in Euclidean buildings and its applications to representation theory,” Groups Geom. Dyn., vol. 2, iss. 3, pp. 405-480, 2008. · Zbl 1147.22011 · doi:10.4171/GGD/46
[20] S. Gaussent and P. Littelmann, ”LS galleries, the path model, and MV cycles,” Duke Math. J., vol. 127, iss. 1, pp. 35-88, 2005. · Zbl 1078.22007 · doi:10.1215/S0012-7094-04-12712-5
[21] N. Bardy-Panse, C. Charignon, S. Gaussent, and G. Rousseau, ”Une preuve plus immobilière du théorème de “saturation” de Kapovich-Leeb-Millson,” Enseign. Math., vol. 59, iss. 1-2, pp. 3-37, 2013. · Zbl 1298.51012 · doi:10.4171/LEM/59-1-1
[22] M. Kapovich, B. Leeb, and J. J. Millson, ”The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra,” Mem. Amer. Math. Soc., vol. 192, iss. 896, p. viii, 2008. · Zbl 1140.22009 · doi:10.1090/memo/0896
[23] J. Tits, ”Uniqueness and presentation of Kac-Moody groups over fields,” J. Algebra, vol. 105, iss. 2, pp. 542-573, 1987. · Zbl 0626.22013 · doi:10.1016/0021-8693(87)90214-6
[24] B. Rémy, Groupes de Kac-Moody déployés et Presque déployés, Paris: Math. Soc. France, 2002, vol. 277. · Zbl 1001.22018
[25] F. Kellil and G. Rousseau, ”Opérateurs invariants sur certains immeubles affines de rang 2,” Ann. Fac. Sci. Toulouse Math., vol. 16, iss. 3, pp. 591-610, 2007. · Zbl 1213.43013 · doi:10.5802/afst.1160
[26] S. Gaussent and P. Littelmann, ”One-skeleton galleries, the path model, and a generalization of Macdonald’s formula for Hall-Littlewood polynomials,” Int. Math. Res. Not., vol. (2012), p. no. 12, 2649-2707. · Zbl 1244.05226 · doi:10.1093/imrn/rnr108
[27] E. Looijenga, ”Invariant theory for generalized root systems,” Invent. Math., vol. 61, iss. 1, pp. 1-32, 1980. · Zbl 0436.17005 · doi:10.1007/BF01389892
[28] P. Cartier, ”Representations of \(p\)-adic groups: a survey,” in Automorphic Forms, Representations and \(L\)-Functions, Part 1, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 111-155.
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