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.
General
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