Abstract
We define the spherical Hecke algebra $\mathcal{H}$ for an almost split Kac-Moody group $G$ over a local non-archimedean field. We use the hovel $\mathscr I$ associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer $K$ of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define $\mathcal{H}$ as the algebra of $K$-bi-invariant functions on $G$ with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup $G^+$ of $G$. We prove that the structure constants of $\mathcal{H}$ are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between $\mathcal{H}$ and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, $\mathcal{H}$ is always commutative. Actually, our results apply to abstract “locally finite” hovels, so that we can define the spherical algebra with unequal parameters.
-
[Sa63]
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.
@article{Sa63,
author = {Satake, Ichir{ô}},
journal = {Inst. Hautes Études Sci. Publ. Math.},
pages = {5--69},
title = {Theory of spherical functions on reductive algebraic groups over {${\mathfrak p}$}-adic fields},
volume = {18},
year = {1963},
issn = {0073-8301},
url = {http://www.numdam.org/item?id=PMIHES_1963__18__5_0},
} -
[BrT72]
F. Bruhat and J. Tits, "Groupes réductifs sur un corps local," Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972.
@article{BrT72,
author = {Bruhat, F. and Tits, J.},
journal = {Inst. Hautes Études Sci. Publ. Math.},
pages = {5--251},
title = {Groupes réductifs sur un corps local},
volume = {41},
year = {1972},
issn = {0073-8301},
url = {http://www.numdam.org/item?id=PMIHES_1972__41__5_0},
} -
@article{P06,
author = {Parkinson, James},
journal = {J. Algebra},
number = {1},
pages = {1--49},
title = {Buildings and {H}ecke algebras},
volume = {297},
year = {2006},
doi = {10.1016/j.jalgebra.2005.08.036},
issn = {0021-8693},
} -
[BrK10]
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.
@article{BrK10,
author = {Braverman, Alexander and Kazhdan, David},
journal = {Ann. of Math.},
number = {3},
pages = {1603--1642},
title = {The spherical {H}ecke algebra for affine {K}ac-{M}oody groups {I}},
volume = {174},
year = {2011},
doi = {10.4007/annals.2011.174.3.5},
issn = {0003-486X},
} -
[BrK12]
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.
@incollection{BrK12, address = {Zürich},
author = {Braverman, Alexander and Kazhdan, David},
booktitle = {European Congress of Mathematics},
pages = {91--117},
publisher = {Eur. Math. Soc.},
title = {Representation of affine {K}ac-{M}oody groups over local and global fields: a survey of some recent results},
year = {2014},
doi = {10.4171/120-1/6},
} -
[GR08]
S. Gaussent and G. Rousseau, "Kac-Moody groups, hovels and Littelmann paths," Ann. Inst. Fourier $($Grenoble$)$, vol. 58, iss. 7, pp. 2605-2657, 2008.
@article{GR08,
author = {Gaussent, St{é}phane and Rousseau, Guy},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
number = {7},
pages = {2605--2657},
title = {Kac-{M}oody groups, hovels and {L}ittelmann paths},
volume = {58},
year = {2008},
doi = {10.1093/imrn/rnr108},
issn = {0373-0956},
} -
[R12] G. Rousseau, Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées, 2010.
@misc{R12,
author = {Rousseau, Guy},
title = {Groupes de {K}ac-{M}oody déployés sur un corps local, 2 {M}asures ordonnées},
year = {2010},
} -
[R13] G. Rousseau, Almost split Kac-Moody groups over ultrametric fields, 2012.
@misc{R13,
author = {Rousseau, Guy},
title = {Almost split {K}ac-{M}oody groups over ultrametric fields},
year = {2012},
} -
@article{R11,
author = {Rousseau, Guy},
journal = {Pure Appl. Math. Q.},
note = {special issue in honor of Jacques Tits},
pages = {859--921},
title = {Masures affines},
volume = {7},
year = {2011},
doi = {10.4310/PAMQ.2011.v7.n3.a10},
issn = {1558-8599},
} -
[P10] M. Patnaik, The Satake map for $p$-adic loop groups and the analogue of Mac Donald’s formula for spherical functions.
@misc{P10,
author = {Patnaik, Manish},
key = {P10},
note = {lecture Nancy, December 10 2010},
title = {The {S}atake map for $p$-adic loop groups and the analogue of {M}ac {D}onald's formula for spherical functions},
} -
[BrKP12] A. Braverman, D. Kazhdan, and M. Patnaik, Iwahori-Hecke algebras for $p$-adic loop groups.
@misc{BrKP12,
author = {Braverman, Alexander and Kazhdan, David and Patnaik, Manish},
title = {Iwahori-{H}ecke algebras for $p$-adic loop groups},
} -
[BrGKP13]
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.
@incollection{BrGKP13, address = {Providence, RI},
author = {Braverman, Alexander and Garland, Howard and Kazhdan, David and Patnaik, Manish},
booktitle = {Perspectives in Representation Theory},
editor = {Etingof, P. and Khovanov, M. and Savage, A.},
pages = {43--64},
publisher = {Amer. Math. Soc.},
series = {Contemp. Math.},
title = {An affine {G}indikin-{K}arpelevich formula},
volume = {610},
year = {2014},
doi = {10.1090/conm/610/12193},
} -
[MP89]
R. V. Moody and A. Pianzola, "On infinite root systems," Trans. Amer. Math. Soc., vol. 315, iss. 2, pp. 661-696, 1989.
@article{MP89,
author = {Moody, R. V. and Pianzola, A.},
journal = {Trans. Amer. Math. Soc.},
number = {2},
pages = {661--696},
title = {On infinite root systems},
volume = {315},
year = {1989},
doi = {10.2307/2001300},
issn = {0002-9947},
} -
[MP95] R. V. Moody and A. Pianzola, Lie Algebras with Triangular Decompositions, New York: John Wiley & Sons, 1995.
@book{MP95, address = {New York},
author = {Moody, Robert V. and Pianzola, Arturo},
pages = {xxii+685},
publisher = {John Wiley \& Sons},
series = {Canad. Math. Soc. Ser. Monogr. Adv. Texts},
title = {Lie Algebras with Triangular Decompositions},
year = {1995},
isbn = {0-471-63304-6},
} -
[Ba96] N. Bardy, Systèms de Racines Infinis, Paris: Math. Soc. France, 1996, vol. 65.
@book{Ba96, address = {Paris},
author = {Bardy, Nicole},
pages = {vi+188},
publisher = {Math. Soc. France},
series = {Mém. Soc. Math. Fr.},
title = {Systèms de Racines Infinis},
volume = {65},
year = {1996},
issn = {0249-633X},
} -
[K90]
V. G. Kac, Infinite-Dimensional Lie Algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990.
@book{K90, address = {Cambridge},
author = {Kac, Victor G.},
edition = {Third},
pages = {xxii+400},
publisher = {Cambridge Univ. Press},
title = {Infinite-Dimensional {L}ie Algebras},
year = {1990},
doi = {10.1017/CBO9780511626234},
isbn = {0-521-37215-1; 0-521-46693-8},
} -
[Ch10] C. Charignon, Structures immobilières pour un groupe de Kac-Moody sur un corps local, 2010.
@misc{Ch10,
author = {Charignon, Cyril},
title = {Structures immobilières pour un groupe de {K}ac-{M}oody sur un corps local},
year = {2010},
} -
[Ch11] C. Charignon, Immeubles affines et groupes de Kac-Moody, masures bordées.
@misc{Ch11,
author = {Charignon, Cyril},
note = {thèse Nancy, 2 juillet 2010, ISBN 978-613-1-58611-8 (Éditions universitaires européennes, Sarrebruck, 2011)},
title = {Immeubles affines et groupes de {K}ac-{M}oody, masures bordées},
} -
[KM08]
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.
@article{KM08,
author = {Kapovich, Michael and Millson, John J.},
journal = {Groups Geom. Dyn.},
number = {3},
pages = {405--480},
title = {A path model for geodesics in {E}uclidean buildings and its applications to representation theory},
volume = {2},
year = {2008},
doi = {10.4171/GGD/46},
issn = {1661-7207},
} -
[GL05]
S. Gaussent and P. Littelmann, "LS galleries, the path model, and MV cycles," Duke Math. J., vol. 127, iss. 1, pp. 35-88, 2005.
@article{GL05,
author = {Gaussent, S. and Littelmann, P.},
journal = {Duke Math. J.},
number = {1},
pages = {35--88},
title = {L{S} galleries, the path model, and {MV} cycles},
volume = {127},
year = {2005},
doi = {10.1215/S0012-7094-04-12712-5},
issn = {0012-7094},
} -
[BCGR11]
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.
@article{BCGR11,
author = {Bardy-Panse, Nicole and Charignon, Cyril and Gaussent, St{é}phane and Rousseau, Guy},
journal = {Enseign. Math.},
key = {BCGR},
number = {1-2},
pages = {3--37},
title = {Une preuve plus immobilière du théorème de ``saturation'' de {K}apovich-{L}eeb-{M}illson},
volume = {59},
year = {2013},
doi = {10.4171/LEM/59-1-1},
issn = {0013-8584},
} -
[KLM08]
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.
@article{KLM08,
author = {Kapovich, Michael and Leeb, Bernhard and Millson, John J.},
journal = {Mem. Amer. Math. Soc.},
number = {896},
pages = {viii+83},
title = {The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra},
volume = {192},
year = {2008},
doi = {10.1090/memo/0896},
isbn = {978-0-8218-4054-2},
issn = {0065-9266},
} -
[T87]
J. Tits, "Uniqueness and presentation of Kac-Moody groups over fields," J. Algebra, vol. 105, iss. 2, pp. 542-573, 1987.
@article{T87,
author = {Tits, Jacques},
journal = {J. Algebra},
number = {2},
pages = {542--573},
title = {Uniqueness and presentation of {K}ac-{M}oody groups over fields},
volume = {105},
year = {1987},
doi = {10.1016/0021-8693(87)90214-6},
issn = {0021-8693},
} -
[Re02] B. Rémy, Groupes de Kac-Moody déployés et Presque déployés, Paris: Math. Soc. France, 2002, vol. 277.
@book{Re02, address = {Paris},
author = {R{é}my, Bertrand},
pages = {viii+348},
publisher = {Math. Soc. France},
series = {Astérisque},
title = {Groupes de {K}ac-{M}oody déployés et Presque déployés},
volume = {277},
year = {2002},
issn = {0303-1179},
} -
[KeR07]
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.
@article{KeR07,
author = {Kellil, Ferdaous and Rousseau, Guy},
journal = {Ann. Fac. Sci. Toulouse Math.},
number = {3},
pages = {591--610},
title = {Opérateurs invariants sur certains immeubles affines de rang 2},
volume = {16},
year = {2007},
doi = {10.5802/afst.1160},
issn = {0240-2963},
} -
[GL11]
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.
@article{GL11,
author = {Gaussent, St{é}phane and Littelmann, Peter},
journal = {Int. Math. Res. Not.},
key = {GL12},
pages = {no.~12, 2649--2707},
title = {One-skeleton galleries, the path model, and a generalization of {M}acdonald's formula for {H}all-{L}ittlewood polynomials},
volume = {(2012)},
doi = {10.1093/imrn/rnr108},
issn = {1073-7928},
} -
[Loo]
E. Looijenga, "Invariant theory for generalized root systems," Invent. Math., vol. 61, iss. 1, pp. 1-32, 1980.
@article{Loo,
author = {Looijenga, Eduard},
journal = {Invent. Math.},
number = {1},
pages = {1--32},
title = {Invariant theory for generalized root systems},
volume = {61},
year = {1980},
doi = {10.1007/BF01389892},
issn = {0020-9910},
} -
[Ca79] 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.
@incollection{Ca79, address = {Providence, R.I.},
author = {Cartier, P.},
booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 1},
pages = {111--155},
publisher = {Amer. Math. Soc.},
series = {Proc. Sympos. Pure Math.},
title = {Representations of {$p$}-adic groups: a survey},
volume = {XXXIII},
year = {1979},
}