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ON THE UNRAMIFIED SPECTRUM OF SPHERICAL VARIETIES OVER pADIC FIELDS
, 2008
"... The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G × G by left and right multiplication. For a split padic reductive group G over a local nonarchime ..."
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The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G × G by left and right multiplication. For a split padic reductive group G over a local nonarchimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the “Langlands dual ” group. We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space C ∞ c (X) are in natural “almost bijection ” with a number of copies of A ∗ X /WX, the quotient of a complex torus by the “little Weyl group ” of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations “distinguished ” by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a
Spherical functions on spherical varieties
, 2009
"... Let X = H\G be a spherical variety for a split reductive group G over the integers o of a padic field k, and K = G(o) a hyperspecial maximal compact subgroup of G = G(k). We compute eigenfunctions (“spherical functions”) on X = X(k) under the action of the unramified (or spherical) Hecke algebra ..."
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Let X = H\G be a spherical variety for a split reductive group G over the integers o of a padic field k, and K = G(o) a hyperspecial maximal compact subgroup of G = G(k). We compute eigenfunctions (“spherical functions”) on X = X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of “CasselmanShalika" also prove a variant of the formula which involves a certain Lvalue, and we present several applications such as: (1) a statement on “good test vectors” (namely, that an Hinvariant functional on an irreducible unramified representation π is always nonzero on π K), (2) the unramified Plancherel formula for X, including a formula for the “Tamagawa measure ” of X(o), and (3) a computation of the most continuous
HECKE OPERATORS ON QUASIMAPS INTO HOROSPHERICAL VARIETIES
, 2006
"... Abstract. Let G be a connected reductive complex algebraic group. This paper and its companion [GN06] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we deve ..."
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Abstract. Let G be a connected reductive complex algebraic group. This paper and its companion [GN06] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup ˇ H of the dual group ˇ G. The construction of ˇ H is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finitedimensional representations of ˇ H. In this paper, we focus on horospherical varieties, a class of varieties closely related
Documenta Math. 19 Hecke Operators on Quasimaps into Horospherical Varieties
, 2006
"... Abstract. Let G be a connected reductive complex algebraic group. This paper and its companion [GN] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develo ..."
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Abstract. Let G be a connected reductive complex algebraic group. This paper and its companion [GN] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup ˇ H of the dual group ˇG. The construction of ˇ H is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finitedimensional representations of ˇ H. In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical Gvariety
A CONJECTURE OF SAKELLARIDISVENKATESH ON THE UNITARY SPECTRUM OF SPHERICAL VARIETIES. WEE TECK GAN AND
"... with admiration and appreciation 1. ..."
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