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**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.

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.

Reviewer: Jürgen Elstrodt (Münster)

### 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) |