Weyl Group Multiple Dirichlet Series Research Articles

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to Cr and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K, but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K=Fq(T), and describe the denominators and support of the numerators.

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

We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters 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 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. 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, 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.

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta functions by an analogue of Weyl group multiple Dirichlet series in the sense of Bump et al. By combining with a result of Nakasuji, Phuksuwan and Yamasaki, our theorems yield a new method of finding functional relations among zeta-functions of root systems.

On the crystal graph description of the stable Weyl group multiple Dirichlet series

For a semisimple Lie algebra admitting a good enumeration, we prove a parametrization for the elements in its Weyl group. As an application, we give a coordinate-free comparison between the crystal graph description (when it is known) and the Lie-theoretic description of the Weyl group multiple Dirichlet series in the stable range.

On the $p$-parts of Weyl group multiple Dirichlet series

We study the structure of $p$-parts of Weyl group multiple Dirichlet series. In particular, we extend results of Chinta, Friedberg, and Gunnells (2008) and show, in the stable case, that the $p$-parts of Chinta and Gunnells (2010) agree with those constru

Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells’ work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.

A metaplectic Casselman-Shalika formula for GL r

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in the complex parameter. In addition we relate a specific spherical Whittaker function to the $p$-power part of the Weyl group multiple Dirichlet series for the root system of type $A_{r-1}$ constructed from $n$th order Gauss sums. We also show that the zonal spherical functions can be computed explicitly in terms of Hall-Littlewood polynomials as in Macdonald's formula for ${\rm GL}_r$.

Weyl group multiple Dirichlet series of typeC

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand‐Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n D 1, we prove our series are Whittaker coefficients of Eisenstein series on SO.2r C 1/.

Weyl group multiple Dirichlet series, Eisenstein series and crystal bases

We show that the Whittaker coecients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coecients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on \string previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reections. The coecients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

Constructing Weyl group multiple Dirichlet series

Let Φ \Phi be a reduced root system of rank r r . A Weyl group multiple Dirichlet series for Φ \Phi is a Dirichlet series in r r complex variables s 1 , … , s r s_1,\dots ,s_r , initially converging for R e ( s i ) \mathrm {Re}(s_i) sufficiently large, that has meromorphic continuation to C r {\mathbb C}^r and satisfies functional equations under the transformations of C r {\mathbb C}^r corresponding to the Weyl group of Φ \Phi . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.

On the p-parts of quadratic Weyl group multiple Dirichlet series

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1,…, sr, initially converging for ℜ(si) sufficiently large, which has meromorphic continuation to ℂr and satisfies functional equations under the transformations of ℂr corresponding to the Weyl group of Φ. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91–114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1–26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325–355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327–353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Φ = Ar, the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] when n = 2.

Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar

Weyl group multiple Dirichlet series were associated with a root system and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hostein [3] and Brubaker, Bump and Friedberg [4] provided n is suciently large; their coecients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when = Ar. “Twisted” Dirichet series are considered, which contain the series of [4] as a special case. These series are not Euler products, but due to the twisted multiplicativity of their coecients, they are determined by their p-parts. The p-part is given as a sum of products of Gauss sums, parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coecients of Eisenstein series on the n-fold metaplectic cover of GLr+1, and this is proved if r = 2 or n = 1. The equivalence of our definition with that of Chinta [11] when n = 2 and r 6 5 is also established.

Weyl group multiple Dirichlet series constructed from quadratic characters

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables.

Weyl group multiple Dirichlet series II: The stable case

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

On Kubota's Dirichlet series

Kubota [T. Kubota, Some results concerning reciprocity law and real analytic automorphic functions, in: 1969 Number Theory Institute (Proc. Sympos. Pure Math. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I. (1971), 382–395.] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hostein, Weyl group multiple Dirichlet series I, preprint, http://sporadic.stanford.edu/bump/wmd.pdf.]. Closely related results are in Eckhardt and Patterson [C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64(2) (1992), 225–264.].