On Iwahori–Whittaker functions for metaplectic groups

Patnaik, Manish M.; Puskás, Anna

doi:10.1016/j.aim.2017.04.005

Cited by 17 publications

(51 citation statements)

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“…As for computing the Whittaker function, the same strategy as in [36], [37] can be used here (N.B. a version of this approach for spherical functions, which motivated the Whittaker story, appeared earlier in [4]): first one breaks up the Whittaker function into "Iwahori-Whittaker" pieces, then one shows that each of these pieces can be expressed via certain Demazure-Lusztig-type operators, and finally one reassembles this sum using some combinatorial identities.…”

Section: M1 Frommentioning

confidence: 99%

“…Our approach is similar to theirs, though we adopt the "metaplectic root datum" framework which perhaps clarifies somewhat the role of imaginary roots (they are just the imaginary roots in the new metaplectic Kac-Moody root system). Next, we note that the same decomposition argument as in [36], [37] expresses the Kac-Moody Whittaker function as a sum of Iwahori-Whittaker pieces. A recursion argument using intertwiners ( [37,Corollary 5.4]) shows that each of these Iwahori-Whittaker pieces is expressed via the application of a metaplectic Demazure-Lusztig operator.…”

Section: M1 Frommentioning

confidence: 99%

“…Next, we note that the same decomposition argument as in [36], [37] expresses the Kac-Moody Whittaker function as a sum of Iwahori-Whittaker pieces. A recursion argument using intertwiners ( [37,Corollary 5.4]) shows that each of these Iwahori-Whittaker pieces is expressed via the application of a metaplectic Demazure-Lusztig operator. However, in reassembling this sum of Demazure-Lusztig operators a technical complexity arises, as the sum is now over an infinite set (the Kac-Moody Weyl group) and one must contend with certain "convergence" issues.…”

Section: M1 Frommentioning

confidence: 99%

“…(ii) We explain how to generalize the Casselman-Shalika formula [8] for unramified Whittaker functions to our metaplectic covers. The strategy here is as in our previous work [36], [37], but some new combinatorial work is necessary. Our results are sharpest for affine Kac-Moody groups in a manner that will be explained later in this introduction.…”

Section: Introductionmentioning

confidence: 99%

“…Whereas the construction of the covers of a general Kac-Moody group G does not pose any real technical difficulty, some new work (i.e. not contained in [37], [36]) of a Kac-Moody, not metaplectic, nature is required to obtain the Casselman-Shalika formula. Before describing this, we first note that to define Whittaker functions in the finite-dimensional case, one uses certain Whittaker functionals defined as integrals over unipotent subgroups.…”

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confidence: 99%

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Metaplectic covers of Kac–Moody groups and Whittaker functions

Patnaik¹,

Puskás²

2019

Duke Math. J.

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Starting from some linear algebraic data (a Weyl-group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a cover of a Kac-Moody group generalizing the work of Matsumoto. Specializing our construction over non-archimedean local fields, for each positive integer n we obtain the notion of n-fold metaplectic covers of Kac-Moody groups. In this setting, we prove a Casselman-Shalika type formula for Whittaker functions.In generalizing the above to a Kac-Moody context, we could perhaps have chosen any of the existing constructions (not all of which are equivalent) for Kac-Moody groups. We choose here the functorial approach of Tits [43] 3 , whose input is a "root datum" consisting of a quadruple (Y, {y i }, X , {x i }) where Y, X are to the play the role of the cocharacter and character lattice respectively of the group, and y i and x i the coroots and roots respectively. In this setup the "simply connected" groups will roughly be the ones for which Y is spanned by the {y i }, though some further care is required when A is not of full rank. See §2.1.6.To carry over Matsumoto's strategy, we first need to fix an integral Weyl group invariant form Q on Y. In the "simply-connected" case just described, these are again easily classified (cf. Proposition 2.2.2). Starting from the choice of such Q and a positive integer n, we can also form a "metaplectic root datum" as in [33] or [46]. We use this new root datum to describe the Casselman-Shalika formula and the metaplectic Demazure-Lusztig operators, but we do not actually need the construction of the metaplectic dual group. In fact, our restrictions on the cardinality of the residue field mask almost all of the subtleties of this dual group which Weissman has constructed in [46]. In the affine setting, we tabulate the possibilities for metaplectic dual root systems in Table 2.3.2. Fixing Q and choosing some Steinberg symbol, Matsumoto's strategy carries over with little change to the Kac-Moody context. KM1. The construction of the cover of the torus and its family of automorphisms follows as in [29]. The possible complications involving non-degeneracy of the Cartan matrix (e.g. the loop rotation in the affine case) are conveniently taken care of using the "Q-formalism." KM2. The next step is to obtain a presentation for a group N that plays a role analogous to the normalizer of the torus in the finite-dimensional context, and then refine this to a presentation of the "integral" version of this normalizer. We follow the classical arguments of Tits here [41] adapted with almost no change to the Kac-Moody context. Here we use the simply-connected assumption to obtain a simple presentation for N, though it should be possible to remove this assumption. KM3. Finally, the construction of a cover E as a group of operators on a set S (again constructed as a fiber product using the cover of N and the Bruhat decompostion) follows as in Matsumoto. Recall also that Matsumoto's argument involves a rank two check, and at first glance it might seem...

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Section: M1 Frommentioning

confidence: 99%

Section: M1 Frommentioning

confidence: 99%

Section: M1 Frommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Metaplectic covers of Kac–Moody groups and Whittaker functions

Patnaik¹,

Puskás²

2019

Duke Math. J.

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Crystal Constructions in Number Theory

Puskás

2019

Association for Women in Mathematics Series

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Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics.Date: October 16, 2018.

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Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

Sahi

Stokman

Venkateswaran

2021

Sel. Math. New Ser.

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We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters $$g_i$$ g i , and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters $$g_i$$ g i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters $$g_i$$ g i . It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the $$g_i$$ g i , and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

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On Iwahori–Whittaker functions for metaplectic groups

Cited by 17 publications

References 20 publications

Metaplectic covers of Kac–Moody groups and Whittaker functions

Metaplectic covers of Kac–Moody groups and Whittaker functions

Crystal Constructions in Number Theory

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

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