** **

**Abstract:**

Multiple Dirichlet series
are generalizations of L-functions

involving several complex
variables. While the functional equation

of a usual L-series is an
involution s -> 1-s, a multiple

Dirichlet series satisfies
a group of functional equations that

intermixes all the
variables.

In this talk we give
examples of multiple Dirichlet series and their

applications. We will
show how consideration of these series arises

naturally in number
theoretic applications. Then we describe a

construction of Weyl group
multiple Dirichlet series. These are

series attached to root systems
plus some extra data where the

resulting group of
functional equations is the associated Weyl group.

In general these series are
expected to be Whitaker-Fourier

coefficients of metaplectic
Eisenstein series, although that is not

how the series are defined.
An essential part of the construction is

a deformation of the Weyl
character formula for the associated

semisimple complex Lie
algebra.