TAILIEUCHUNG - Đề tài "Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar "

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = Ar . “Twisted” Dirichet series are considered, which contain the series of [4] as a special case. | Annals of Mathematics Weyl group multiple Dirichlet series III Eisenstein series and twisted unstable Ar By B. Brubaker D. Bump S. Friedberg and J. Hoffstein Annals of Mathematics 166 2007 293 316 Weyl group multiple Dirichlet series III Eisenstein series and twisted unstable Ar By B. Brubaker D. Bump S. Friedberg and J. Hoffstein Abstract Weyl group multiple Dirichlet series were associated with a root system and a number field F containing the n-th roots of unity by Brubaker Bump Chinta Friedberg and Hoffstein 3 and Brubaker Bump and Friedberg 4 provided n is sufficiently large their coefficients involve n-th order Gauss sums. The case where n is small is harder and is addressed in this paper when Ar. Twisted Dirichet series are considered which contain the series of 4 as a special case. These series are not Euler products but due to the twisted multiplicativity of their coefficients they are determined by their p-parts. The p-part is given as a sum of products of Gauss sums parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coefficients of Eisenstein series on the n-fold metaplec-tic cover of GLr 1 and this is proved if r 2 or n 1. The equivalence of our definition with that of Chinta 11 when n 2 and r 5 is also established. Let F be a totally complex algebraic number field containing the group I2n of 2n-th roots of unity. Thus 1 is an n-th power in F. Let c Rr be a reduced root system. It has been shown in Brubaker Bump Chinta Friedberg and Hoffstein 3 and Brubaker Bump and Friedberg 4 how one can associate a multiple Dirichlet series with its coefficients involve n-th order Gauss sums. A condition of stability is imposed in this definition which amounts to n being sufficiently large depending on . In this paper we will propose a description of the Weyl group multiple Dirichlet series in the unstable case when has Cartan type Ar and present the evidence in support of this description. We will .

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