Abstract
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for this algebra, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group $G_{\rm aff}$.
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[bf]
A. Braverman and M. Finkelberg, "Pursuing the double affine Grassmannian. I. Transversal slices via instantons on $A_k$-singularities," Duke Math. J., vol. 152, iss. 2, pp. 175-206, 2010.@article {bf, MRKEY = {2656088},
AUTHOR = {Braverman, Alexander and Finkelberg, Michael},
TITLE = {Pursuing the double affine {G}rassmannian. {I}. {T}ransversal slices via instantons on {$A\sb k$}-singularities},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {152},
YEAR = {2010},
NUMBER = {2},
PAGES = {175--206},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14D24 (14D21 14F43 14J60 20G44)},
MRNUMBER = {2656088},
DOI = {10.1215/00127094-2010-011},
ZBLNUMBER={1200.14083},
} -
[bf1] A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian II: Convolution.
@misc{bf1,
author = {Braverman, Alexander and Finkelberg, Michael},
TITLE={Pursuing the double affine {G}rassmannian {II: C}onvolution},
ARXIV={0908.3390},
} -
[BKM] A. Braverman, D. Kazhdan, and M. Patnaik, The Iwahori-Hecke algebra for $p$-adic Kac-Moody groups I.
@misc{BKM,
author = {Braverman, Alexander and Kazhdan, D. and Patnaik, M.},
TITLE={The {I}wahori-{H}ecke algebra for $p$-adic {K}ac-{M}oody groups {I}},
NOTE={in preparation},
} -
[Cart] P. Cartier, "Representations of $p$-adic groups: a survey," in Automorphic Forms, Representations and $L$-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. 33, pp. 111-155.
@incollection {Cart, MRKEY = {0546593},
AUTHOR = {Cartier, P.},
TITLE = {Representations of {$p$}-adic groups: a survey},
BOOKTITLE = {Automorphic Forms, Representations and {$L$}-Functions},
VENUE={{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {O}re., 1977},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME={33},
PAGES = {111--155},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1979},
MRCLASS = {22E50},
MRNUMBER = {0546593},
MRREVIEWER = {Allan J. Silberger},
ZBLNUMBER = {0421.22010},
} -
[CherMa] I. Cherednik and X. Ma, A new take on spherical, Whittaker and Bessel functions.
@misc{CherMa,
author={Cherednik, I. and Ma, X.},
TITLE={A new take on spherical, {W}hittaker and {B}essel functions},
ARXIV={0904.4324},
URL = {http://front.math.ucdavis.edu/0904.4324},
} -
[Fal] G. Faltings, "Algebraic loop groups and moduli spaces of bundles," J. Eur. Math. Soc., vol. 5, pp. 41-68, 2003.
@article{Fal,
author={Faltings, G.},
TITLE={Algebraic loop groups and moduli spaces of bundles},
JOURNAL={J. Eur. Math. Soc.},
VOLUME={5},
PAGES={41--68},
MRNUMBER={1961134},
YEAR={2003},
ZBLNUMBER={1020.14002},
} -
[Groth] A. Grothendieck, Fondements de la Géométrie Algébrique, Extraits du Seminarie Bourbaki 7 (1957-62), New York: Springer-Verlag, 1971.
@book{Groth,
author={Grothendieck, A.},
TITLE={Fondements de la Géométrie Algébrique, Extraits du Seminarie Bourbaki {\bf 7} (1957-62) },
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
YEAR={1971},
MRNUMBER={0146040},
ZBLNUMBER={0239.14002},
URL={http://www.numdam.org/item?id=SB_1961-1962__7__297_0},
} -
[Kac] V. G. Kac, Infinite-Dimensional Lie Algebras, Second ed., Cambridge: Cambridge Univ. Press, 1985.
@book {Kac, MRKEY = {0823672},
AUTHOR = {Kac, Victor G.},
TITLE = {Infinite-Dimensional {L}ie Algebras},
EDITION = {Second},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1985},
PAGES = {xviii+280},
ISBN = {0-521-32133-6},
MRCLASS = {17B65},
MRNUMBER = {0823672},
ZBLNUMBER = {0574.17010},
} -
[Kap-H] M. Kapranov, "Double affine Hecke algebras and 2-dimensional local fields," J. Amer. Math. Soc., vol. 14, iss. 1, pp. 239-262, 2001.
@article {Kap-H, MRKEY = {1800352},
AUTHOR = {Kapranov, M.},
TITLE = {Double affine {H}ecke algebras and 2-dimensional local fields},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {14},
YEAR = {2001},
NUMBER = {1},
PAGES = {239--262},
ISSN = {0894-0347},
MRCLASS = {20C08 (11S85)},
MRNUMBER = {1800352},
MRREVIEWER = {Robert B{é}dard},
DOI = {10.1090/S0894-0347-00-00354-4},
ZBLNUMBER = {0958.20005},
} -
[Kap-E] M. Kapranov, The elliptic curve in the ${S}$-duality theory and Eisenstein series for Kac-Moody groups.
@misc{Kap-E,
author={Kapranov, M.},
TITLE={The elliptic curve in the ${S}$-duality theory and {E}isenstein series for {K}ac-{M}oody groups},
ARXIV={math.AG/0001005},
} -
[Kum] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Boston, MA: Birkhäuser, 2002, vol. 204.
@book {Kum, MRKEY = {1923198},
AUTHOR = {Kumar, Shrawan},
TITLE = {Kac-{M}oody Groups, their Flag Varieties and Representation Theory},
SERIES = {Progr. Math.},
VOLUME = {204},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {2002},
PAGES = {xvi+606},
ISBN = {0-8176-4227-7},
MRCLASS = {22E46 (14M15 17B67 22E65)},
MRNUMBER = {1923198},
MRREVIEWER = {Guy Rousseau},
ZBLNUMBER = {1026.17030},
} -
[Loi] E. Looijenga, "Root systems and elliptic curves," Invent. Math., vol. 38, iss. 1, pp. 17-32, 1976/77.
@article {Loi, MRKEY = {0466134},
AUTHOR = {Looijenga, Eduard},
TITLE = {Root systems and elliptic curves},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {38},
YEAR = {1976/77},
NUMBER = {1},
PAGES = {17--32},
ISSN = {0020-9910},
MRCLASS = {14B05},
MRNUMBER = {0466134},
MRREVIEWER = {Henry C. Pinkham},
DOI = {10.1007/BF01390167},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0358.17016},
} -
[Lu-qan] G. Lusztig, "Singularities, character formulas, and a $q$-analog of weight multiplicities," in Analysis and Topology on Singular Spaces, II, III, Paris: Soc. Math. France, 1983, vol. 101, pp. 208-229.
@incollection {Lu-qan, MRKEY = {0737932},
AUTHOR = {Lusztig, George},
TITLE = {Singularities, character formulas, and a {$q$}-analog of weight multiplicities},
BOOKTITLE = {Analysis and Topology on Singular Spaces, {II},
{III}},
VENUE={{L}uminy, 1981},
SERIES = {Astérisque},
VOLUME = {101},
PAGES = {208--229},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1983},
MRCLASS = {17B10 (05A30 20G05 22E47)},
MRNUMBER = {0737932},
MRREVIEWER = {James E. Humphreys},
ZBLNUMBER = {0561.22013},
} -
[mac] I. G. Macdonald, "Spherical functions on a $p$-adic Chevalley group," Bull. Amer. Math. Soc., vol. 74, pp. 520-525, 1968.
@article {mac, MRKEY = {0222089},
AUTHOR = {Macdonald, I. G.},
TITLE = {Spherical functions on a {$p$}-adic {C}hevalley group},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical Society},
VOLUME = {74},
YEAR = {1968},
PAGES = {520--525},
ISSN = {0002-9904},
MRCLASS = {14.50},
MRNUMBER = {0222089},
MRREVIEWER = {E. Abe},
DOI = {10.1090/S0002-9904-1968-11989-5},
ZBLNUMBER = {0273.22012},
} -
[MV] I. Mirković and K. Vilonen, "Perverse sheaves on affine Grassmannians and Langlands duality," Math. Res. Lett., vol. 7, iss. 1, pp. 13-24, 2000.
@article {MV, MRKEY = {1748284},
AUTHOR = {Mirkovi{ć},
Ivan and Vilonen, Kari},
TITLE = {Perverse sheaves on affine {G}rassmannians and {L}anglands duality},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {7},
YEAR = {2000},
NUMBER = {1},
PAGES = {13--24},
ISSN = {1073-2780},
MRCLASS = {14F43 (20G05 32S60)},
MRNUMBER = {1748284},
MRREVIEWER = {Alexander E. Polishchuk},
ZBLNUMBER = {0987.14015},
} -
[Tits] J. Tits, "Groups and group functors attached to Kac-Moody data," in Workshop Bonn 1984, New York: Springer-Verlag, 1985, vol. 1111, pp. 193-223.
@incollection {Tits, MRKEY = {0797422},
AUTHOR = {Tits, Jacques},
TITLE = {Groups and group functors attached to {K}ac-{M}oody data},
BOOKTITLE = {Workshop {B}onn 1984},
VENUE={{B}onn, 1984},
SERIES = {Lecture Notes in Math.},
VOLUME = {1111},
PAGES = {193--223},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1985},
MRCLASS = {22E65 (17B67)},
MRNUMBER = {0797422},
MRREVIEWER = {Andrew Pressley},
DOI = {10.1007/BFb0084591},
ZBLNUMBER = {0572.17010},
}
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