Weyl group multiple Dirichlet series of type A 2

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

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.

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.

Constructing Affine Weyl Group Multiple Dirichlet Series

Chapter One. Type A Weyl Group Multiple Dirichlet Series

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order L-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the L-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their p-parts.

The lower order and linear order of multiple dirichlet series

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Let \(\mathfrak{g}_{0}\) be a real semisimple Lie algebra. Let \(\mathfrak{g}_{0} = \mathfrak{k}_{0} \oplus \mathfrak{p}_{0}\) be the corresponding Cartan decomposition and \(\mathfrak{h}_{0} = \mathfrak{t}_{0} \oplus \mathfrak{a}_{0}\) be a maximally compact Cartan subalgebra of \(\mathfrak{g}_{0}\). Let \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) and \(\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}\) be the corresponding complexifications. The set \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) consists of all the linear forms on \(\mathfrak{t}\) which are the restriction to \(\mathfrak{t}\) of the roots in the root system \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\) of \(\mathfrak{g}\) with respect to \(\mathfrak{h}\). The main result of the paper is to prove that \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) is also a (maybe non-reduced) root system and its Weyl group can be identified with a subgroup of the Weyl group of \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\). Let \(Spin\,\nu :\mathfrak{k} \to End\,S\) be the composition of the isotropy representation \(\nu :\mathfrak{k} \to \mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)}\) with the spin representation \(Spin:\mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)} \to End\,S\). Finally as an application, we give a nice description of the \(\mathfrak{k}\)-module structure on S in terms of the restricted root system \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) and its Weyl group.

General multiple Dirichlet series from perverse sheaves

This is equivalent to the coefficients apn(G) satisfing a linear recurrence relation with constant coefficients for sufficiently large n. These Poincare series were first studied in the case when G is a finitely generated, torsion-free nilpotent group in a paper by Grunewald, Segal and Smith [GSSm]. There the authors established that, for such a group, (G,p(s) is a rational function in p8s. In that setting the functions (G,p(S) are the local factors associated with the Dirichlet series

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by simple reflections and Coxeter relations. In a broad sense this answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto’s theorem. Therefore the word problem is solved for the groupoid.