Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory: 175 (Annals of Mathematics Studies, 175) - Softcover

Brubaker, Ben

Softcover

ISBN 10: 0691150664 ISBN 13: 9780691150666

Verlag: Princeton University Press, 2011

Inhaltsangabe

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Über diesen Titel

Über die Autorin bzw. den Autor

Ben Brubaker is assistant professor of mathematics at Massachusetts Institute of Technology. Daniel Bump is professor of mathematics at Stanford University. Solomon Friedberg is professor of mathematics at Boston College.

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Weyl Group Multiple Dirichlet Series

Type A Combinatorial TheoryBy Ben Brubaker Daniel Bump Solomon Friedberg

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-15066-6

Contents

Preface..............................................................................................vii1. Type A Weyl Group Multiple Dirichlet Series.......................................................12. Crystals and Gelfand-Tsetlin Patterns.............................................................103. Duality...........................................................................................224. Whittaker Functions...............................................................................265. Tokuyama's Theorem................................................................................316. Outline of the Proof..............................................................................367. Statement B Implies Statement A...................................................................518. Cartoons..........................................................................................549. Snakes............................................................................................5810. Noncritical Resonances...........................................................................6411. Types............................................................................................6712. Knowability......................................................................................7413. The Reduction to Statement D.....................................................................7714. Statement E Implies Statement D..................................................................8715. Evaluation of ?G and ?? and Statement G.....................8916. Concurrence......................................................................................9617. Conclusion of the Proof..........................................................................10418. Statement B and Crystal Graphs...................................................................10819. Statement B and the Yang-Baxter Equation.........................................................11520. Crystals and p-adic Integration..................................................................132Bibliography.........................................................................................143Notation.............................................................................................149Index................................................................................................155

Chapter One

Type A Weyl Group Multiple Dirichlet Series

We begin by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, we choose the following parameters.

? F, a reduced root system. Let r denote the rank of F.

? n, a positive integer,

? F, an algebraic number field containing the group µ_2n of 2n-th roots of unity,

? S, a finite set of places of F containing all the archimedean places, all places ramified over Q, and large enough so that the ring

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of S-integers is a principal ideal domain,

? m = (m₁, ..., m_r), an r-tuple of nonzero S-integers.

We may embed F and o_S into F_S = ?_{v[member of]S} F_v along the diagonal. Let (d, c)_n,S denote the S-Hilbert symbol, the product of local Hilbert symbols (d, c)_n,v [member of] µ_n at each place v [member of] S, defined for c, d [member of] F^x_S. Let ? : (F^x_S)^r [right arrow] C be any function satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any e₁, ..., e_r [member of] o^x_S (F^x,n_S) and c₁, ..., c_r [member of] F^x_S. Here (F^x,n_S) denotes the set of n-th powers in F^x_S. It is proved in that the set M of such functions is a finite-dimensional (nonzero) vector space.

To any such function ? and data chosen as above, Weyl group multiple Dirichlet series are functions of r complex variables s = (s₁, ..., s_r) [member of] C^r of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where Nc is the cardinality of o_S/co_S, and it remains to define the coefficients H(c;m) in the Dirichlet series. In particular, the function ? is not independent of the choice of representatives in o_S/o^X_S, so the function H must possess complementary transformation properties for the sum to be well-defined.

Indeed, the function H satisfies a "twisted multiplicativity" in c, expressed in terms of n-th power residue symbols and depending on the root system F, which specializes to the usual multiplicativity when n = 1. Recall that the n-th power residue symbol (c/d)_n is defined when c and d are coprime elements of o_S and gcd(n; d) = 1. It depends only on c modulo d, and satisfies the reciprocity law

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(The properties of the power residue symbol and associated S-Hilbert symbols in our notation are set out in.) Then given c = (c₁, ..., c_r) and c' = (c'₁, ..., c'_r) in o^t_S with gcd(c₁ ... c_r, c'₁ ... c'_r) = 1, the function H satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where a_i, i = 1, ..., r denote the simple roots of F and we have chosen a Weyl group invariant inner product <·, ·> for our root system embedded into a real vector space of dimension r. The inner product should be normalized so that for any a, ß [member of] F both ||a||² = and 2 are integers. We will devote the majority of our attention to F of Type A, in which case we will assume the inner product is chosen so that all roots have length 1.

The function H possesses a further twisted multiplicativity with respect to the parameter m. Given any

c = (c₁, ..., c_r), m = (m₁, ..., m_r), m' = (m'₁, ... m'_r)

with gcd(m'₁ ... m'_r, c₁ ... c_r) = 1, H satisfies the twisted multiplicativity relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

As a consequence of properties (1.3) and (1.4) the specification of H reduces to the case where the components of c and m are all powers of the same prime. Given a fixed prime p of o_S and any m = (m₁, ... m_r), let l_i = ord_p(m_i). Then we must specify [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any r-tuple of nonnegative integers k = (k₁, ..., k_r). For brevity, we will refer to these coefficients as the "p- part" of H. To summarize, specifying a multiple Dirichlet series Z⁽ⁿ⁾_? (s; m; F) with chosen data is equivalent to specifying the p-parts of H.

Remark. Both the transformation property of ? in (1.1) and the definition of twisted multiplicativity in (1.3) depend on an enumeration of the simple roots of F. However, the product H · ? is independent of this enumeration of roots and furthermore well-defined modulo units, according to the reciprocity law. The p-parts of H are also independent of this enumeration of roots.

The definitions given above apply to any root system F. In most of this text, we will take F to be of Type A. In this case we will give two combinatorial definitions of the p-part of H. These two definitions of H will be referred to as H_G and H_?, and eventually shown to be equal. Thus either may be used to define the multiple Dirichlet series Z(s;m;A_r). Both definitions will be given in terms of Gelfand-Tsetlin patterns.

By a Gelfand-Tsetlin pattern of rank r we mean an array of integers

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where the rows interleave; that is, a_i-1,j-1 = a_i,j = a_{i- 1,j}. Let ? = (?₁, ..., ?_r) be a dominant integral element for SL_r+1, so that ?₁ = ?₂ = ... = ?_r. In the next chapter, we will explain why Gelfand-Tsetlin patterns with top row (?₁, ..., ?_r, 0) are in bijection with basis vectors for the highest weight module for SL_r+1(C) with highest weight ?.

The coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in both definitions H_G and H_? will be described in terms of Gelfand-Tsetlin patterns with top row (or equivalently, highest weight vector)

? + ? = (l₁ + l₂ + ··· + l_r + r, ···, l_r-1 + l_r + 2, l_r + 1, 0). (1.6)

We denote by GT(?+?) the set of all Gelfand-Tsetlin patterns having this top row. Here

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To any Gelfand-Tsetlin pattern T, we associate the following pair of functions with image in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern T can be read from differences of consecutive row sums in the pattern, so both k_G and k_? are expressions of the weight of the pattern up to an affine linear transformation.

Then given a fixed r-tuple of nonnegative integers (l₁, ···, l_r), we make the following two definitions for p-parts of the multiple Dirichlet series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where the functions G_G and G_? on Gelfand-Tsetlin patterns will now be defined.

We will associate with T two arrays G(T) and ?(T). The entries in these arrays are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

with 1 = i = j = r, and we often think of attaching each entry of the array G(T) (resp. ?(T)) with an entry of the pattern a_i,j lying below the fixed top row. Thus we think of G(T) as applying a kind of right-hand rule to T, since G_i,j involves entries above and to the right of a_i,j as in (1.11); in ? we use a left-hand rule where ?_i;j involves entries above and to the left of a_i,j as in (1.11). When we represent these arrays graphically, we will right-justify the G array and left-justify the ? array. For example, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To provide the definitions of G_G and G_? corresponding to each array, it is convenient to decorate the entries of the G and ? arrays by boxing or circling certain of them. Using the right-hand rule with the G array, if a_i,j = a_{i-1,j_1} then we say G_i,j is boxed, and indicate this when we write the array by putting a box around it, while if a_i,j = a_i-1,j we say it is circled (and we circle it). Using the left-hand rule to obtain the ? array, we box ?_i,j if a_i,j = a_{i- 1,j} and we circle it if a_i,j = a_{i-1, j-1}. For example, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

then the decorated arrays are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

We sometimes use the terms right-hand rule and left-hand rule to refer to both the direction of accumulation of the row differences, and to the convention for decorating these accumulated differences.

If m, c [member of] o_S with c [not equal to] 0 define the Gauss sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

where [??] is a character of F_S that is trivial on o_S and no larger fractional ideal. With p now fixed, for brevity let

g(a) = g(p^a-1, p^a) and h(a) = g(p^a, p^a). (1.15)

These functions will only occur with a > 0. The reader may check that g(a) is nonzero for any value of a, while h(a) is nonzero only if n|a, in which case h(a) = (q - 1)q^a-1, where q is the cardinality of o_S/po_S. Thus if n|a then h(a) = f(p^a), the Euler phi function for p^ao_S.

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We say that the pattern T is strict if a_i,j > a_i,j+1 for every 0 = i = j < r. Thus the rows of the pattern are a strictly decreasing sequence. Otherwise we say that the pattern is non-strict. It is clear from the definitions that T is non-strict if and only if G(T) has an entry that is both boxed and circled, so GG(T) = 0 for non-strict patterns. Similarly let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, we may use the decorated arrays as in (1.13) to write down GG(T) and G?(T) for the pattern T appearing in (1.12) as follows:

GG(T) = q⁴h(4)h(3)h(4)g(2)h(1), and G?(T) = g(2)h(7)h(8)h(3)q³h(1).

Inserting the respective definitions for G_G and G_? into the formulas (1.9) and (1.10) completes the two definitions of the p-parts of H_G and H_?, and with it two definitions for a multiple Dirichlet series Z_?(s;m). For example, the pattern T in (1.12) would appear in the p-part of Z_?(s;m) if

(ord_p(m₁), ···, ordp(m_r)) = (1, 5, 3)

according to the top row of T. Moreover, this pattern would contribute GG(T) to the term H_G(p^k, p^l) = H_G(p⁴, p⁸, p⁶, p¹, p⁵, p³) according to the definitions in (1.8) and (1.9).

In [17], the definition HG was used to define the series, and so we will state our theorem on functional equations and analytic continuation of Z?(s;m) using this choice. Before stating the result precisely, we need to define certain normalizing factors for the multiple Dirichlet series. These have a uniform description for all root systems (see Section 3.3 of [12]), but for simplicity we state them only for Type A here.

(Continues...)

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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 0691150664
• ISBN 13: 9780691150666
• Einband: Tapa blanda
• Anzahl der Seiten: 174
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