## Weyl group multiple Dirichlet series. III: Eisenstein series and twisted unstable $$A_r$$.(English)Zbl 1154.11016

The work under review is Part III of an extensive research project which was (and presumably still is) undertaken by the authors and several co-authors. For Parts I, II see C. Gautam, S. Friedberg, P. E. Gunnells [J. Reine Angew. Math. 623, 1–23 (2008; Zbl 1138.11017), Proc. Symp. Pure Math. 75, 91–114 (2006; Zbl 1112.11025), in addition see Proc. Symp. Pure Math. 75, 115–134 (2006; Zbl 1117.11029)].
Recall the basic situation: Given a reduced root system $$\Phi$$ of rank $$r$$ with Weyl group $$W$$ and an algebraic number field $$F$$ containing the $$n$$-th roots of unity, a Weyl group multiple Dirichlet series was defined in the works quoted above provided that $$n$$ is sufficiently large (depending on $$\Phi)$$. The coefficients of these series can be given (basically) in terms of certain $$n$$-th order Gauß sums. It was conjectured in the previous works quoted above that the main results of these papers will continue to hold for small $$n$$ as well but that the proofs are much harder. In the paper under review the case where $$n$$ is small is addressed in the special (but nevertheless highly technical) case $$\Phi=A_r$$. The authors consider so-called “twisted” Dirichlet series which contain the series of Part II as a special case. The series don’t decompose into Euler products, but their coefficients are nevertheless determined by their $$p$$-parts. These $$p$$-parts are given as a sum of products of Gauß sums, and they are parametrized in terms of strict Gelfand-Tsetlin patterns. This yields a single formula valid for all $$n$$ for the coefficients which reduces to the previous result when $$n$$ is large. The authors conjecture that the multiple Dirichlet series under consideration are Whittaker coefficients of Eisenstein series on an $$n$$-fold metaplectic cover of an algebraic group defined over $$F$$ whose root system is the dual root system of $$\Phi$$. This conjecture is proved if $$\Phi=A_2$$ for all $$n$$, and if $$\Phi=A_r$$ for $$n=1$$.
Since it is impossible to describe the full content of this weighty work in a few lines, the interested reader is advised to read the authors’ 8-page informative introduction.

### MSC:

 11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series 11M41 Other Dirichlet series and zeta functions 11F55 Other groups and their modular and automorphic forms (several variables)

### Citations:

Zbl 1138.11017; Zbl 1112.11025; Zbl 1117.11029
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