The spherical Hecke algebra for affine Kac-Moody groups I (original) (raw)
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European Congress of Mathematics Kraków, 2 – 7 July, 2012
Let G be a reductive algebraic group over a local field K or a global field F. It is well known that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the adelic group G(AF). The purpose of this talk is to give a survey of some recent constructions and results, which show that there should exist an analog of the above theories in the case when G is replaced by the corresponding affine Kac-Moody group G aff (which is essentially built from the formal loop group G((t)) of G). Specifically we discuss the following topics : affine (classical and geometric) Satake isomorphism, Iwahori-Hecke algebra of G aff , affine Eisenstein series and Tamagawa measure.
Affine Hecke algebras for Langlands parameters
2019
It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to create a similar role for affine Hecke algebras on the Galois side of the local Langlands correspondence. To every Bernstein component of enhanced Langlands parameters for G we canonically associate an affine Hecke algebra (possibly extended with a finite R-group). We prove that the irreducible representations of this algebra are naturally in bijection with the members of the Bernstein component, and that the set of central characters of the algebra is naturally in bijection with the collection of cuspidal supports of these enhanced Langlands parameters. These bijections send tempered or (essentially) square-integrable representations to the expected kind of Langlands parameters. Furthermore we check that for many reductive p-adic groups, if a Ber...
Affine Lie algebras and Hecke modular forms
Bulletin of the American Mathematical Society, 1980
The character of a highest weight representation of an affine lie algebra can be written as a finite sum of products of classical 0-functions and certain modular functions, called string functions. We find the transformation law for the string functions, which allows us to compute them explicitly in many interesting cases. Finally, we write an explicit formula for the partition function, in the simplest case A[ x \ and compute the string functions directly. After multiplication by the cube of the T?-function, they turn out to be Hecke modular forms! 1. (See [3] or [7] for details.) Let g be a complex finite-dimensional simple lie algebra, § a Cartan subalgebra of g. A the set of roots of § in g. A + a set of positive roots, II = {OL X ,..., a ; } the corresponding set of simple roots, 0 the highest root. Let (,) be an invariant symmetric bilinear form on g normalized by (6,6) = 2. For a€^* with (a, a) * 0 define H a G § by 0(# a) = 2(0, a)/(a, a) for j8 G §*. Let W be the Weyl group of § in g. Denote by M the Z-span of W6 (long roots). Let C[t, t" 1 ] be the algebra of Laurent polynomials over C in an indeterminate t. We regard g' := C[t, t~x] ® c g as an (infinite-dimensional) complex lie algebra. Define the affine Lie algebra g as follows. Let g = 'g © Cc © Cd and define the bracket by (dx \ dx for x, y G 'gf. The algebra g is an important example of a Kac-Moody algebra [5], [10]. Note that Cc is the center of the algebra g. The subalgebra % = i) <B Cc 0 Giis called the Cartan subalgebra of g. For a G §* set g a = {x G g | [ft, x] = a(h)x for ft G §}; then we have the root space decomposition 8 = ®9<r Detine a nondegenerate symmetric bilinear form (,) on § by (ft, ft') is unchanged if ft, ft' G § C 6, (ft, c) = (ft, d) = 0 for ft G fc>, (c, c) = (tf, J) = 0, (c, rf) = 1. We identify § with §* by this form; then §* is identified with a subspace in %* by a(c) = a(<2) = 0 for a G §*. For a G §* set a = aL " so that
Some examples of Hecke algebras over 2-dimensional local fields
2005
Let K be a local non-archimedian field, F=K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical (and Iwahori) Hecke algebras for representations of the group G(F) and its central extension by means of K*. For instance our spherical Hecke algebra corresponds to the subgroup G(A) where A is
Affine Hecke algebras for classical ppp-adic groups
arXiv (Cornell University), 2022
We consider four classes of classical groups over a non-archimedean local field F : symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local Langlands correspondence for any group G of this kind, via Hecke algebras. To each Bernstein block Rep(G) s in the category of smooth complex G-representations, an (extended) affine Hecke algebra H(s) can be associated with the method of Heiermann. On the other hand, to each Bernstein component Φe(G) s ∨ of the space Φe(G) of enhanced L-parameters for G, one can also associate an (extended) affine Hecke algebra, say H(s ∨). For the supercuspidal representations underlying Rep(G) s , a local Langlands correspondence is available via endoscopy, due to Moeglin and Arthur. Using that we assign to each Rep(G) s a unique Φe(G) s ∨. Our main new result is an algebra isomorphism H(s) op ∼ = H(s ∨), canonical up to inner automorphisms. In combination with earlier work, that provides an injective local Langlands correspondence Irr(G) → Φe(G) which satisfies Borel's desiderata. This parametrization map is probably surjective as well, but we could not show that in all cases. Our framework is suitable to (re)prove many results about smooth G-representations (not necessarily reducible), and to relate them to the geometry of a space of L-parameters. In particular our Langlands parametrization yields an independent way to classify discrete series G-representations in terms of Jordan blocks and supercuspidal representations of Levi subgroups. We show that it coincides with the classification of the discrete series obtained twenty years ago by Moeglin and Tadić.
Calibrated representations of affine Hecke algebras
2004
This paper introduces the notion of calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The main results are that (1) irreducible calibrated representations are indexed by placed skew shapes, (2) the dimension of an irreducible calibrated representation is the number of standard Young tableaux corresponding to the placed skew shape and (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of each generator of the affine Hecke algebra on a specific basis in the representation space. This construction is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.
Some Examples of Hecke Algebras for Two-Dimensional Local Fields
Nagoya Mathematical Journal, 2006
Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).
Double Affine Hecke Algebras and Congruence Groups
Memoirs of the American Mathematical Society, 2020
The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type A2A_{2}A2, which descends to a faithful outer action of a congruence subgroup of SL(2,mathbbZ)SL(2,\mathbb{Z})SL(2,mathbbZ) or PSL(2,mathbbZ)PSL(2,\mathbb{Z})PSL(2,mathbbZ). This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products WltimesQveeW\ltimes Q^{\vee}WltimesQvee of the Weyl group WWW with the coroot lattice QveeQ^{\vee}Qvee. We now describe our results for...
Iwahori–Hecke algebras for p-adic loop groups
Inventiones mathematicae, 2015
This paper is a continuation of [3] in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in [4]. Contents 1. Introduction 1 2. Basic Notations on Groups and Algebras 7 3. Basic Structure of p-adic Loop Groups 13 4. Generalities on Convolution Algebras 17 5. Iwahori Theory I: "Affine" Hecke Algebras and Convolution Hecke algebras 23 6. Iwahori Theory II: Intertwiners and Construction of θ µ ∨ 29 7. Spherical Theory 34 A. The Cartan Semigroup 50 B. The "affine" Root System and the Bruhat pre-order on W 52 References 55