Weyl Group Multiple Dirichlet Series, Eisenstein Series and Crystal Bases

If F is a local field containing the group μn of n-th roots of unity, and if G is a split semisimple simply connected algebraic group, then Matsumoto [27] defined an n-fold covering group of G(F ), that is, a central extension of G(F ) by μn. Similarly if F is a global field with adele ring AF containing μn there is a cover G̃(AF ) of G(AF ) that splits over G(F ). The construction is built on ideas of Kubota [23] and makes use of the reciprocity laws of class field theory. It can be extended to reductive and non-simply connected groups, sometimes at the expense of requiring more roots of unity in F . We will refer to such an extension as a metaplectic group. The special case n = 1 is contained in this situation, but is simpler and we will refer to this as the nonmetaplectic case. Fourier-Whittaker coefficients of Eisenstein series play a central role in the theory of automorphic forms. In the nonmetaplectic case one has uniqueness of Whittaker models ([31, 33, 18]). Over a global field, this implies that the Whittaker functional is Eulerian, i.e. factors as a product over primes. And at almost all places, the local contribution to the Whittaker coefficient may be computed using the CasselmanShalika formula, which expresses a value of the spherical Whittaker function as a character of a finite-dimensional representation of the Langlands dual group LG◦. In the metaplectic case, one may again define Whittaker functionals, but with the fundamental difference that these are now usually not unique. As a consequence, the Whittaker coefficients of metaplectic automorphic forms are not in general Eulerian. The lack of uniqueness of Whittaker models may also be the reason that the Whittaker coefficients of metaplectic Eisenstein series and the extension of the Casselman-Shalika formula to metaplectic groups have not been investigated extensively.