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...