The spherical Hecke algebra for affine Kac-Moody groups I

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 $G_{\rm aff}$.

  • [bf] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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] Go to document 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},
      }

Authors

Alexander Braverman

Department of Mathematics
Brown University
Providence, RI 02912

David Kazhdan

Einstein Institute of Mathematics
The Hebrew University of Jerusalem,
Edmond J. Safra Campus, Givat Ram
Jerusalem 91904
Israel