  Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta,
Friedberg and Hoffstein, are expected to be Whittaker coefficients of
Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these
multiple Dirichlet series for all the finite root systems using the method of
averaging a Weyl group action on the field of rational functions. In this
paper, we generalize Chinta and Gunnells' work and construct Weyl group
multiple Dirichlet series for the root systems associated with symmetrizable
Kac-Moody algebras, and establish their functional equations and meromorphic
continuation.
