Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable $A_r$

Ben Brubaker¹ ; Daniel Bump² ; Solomon Friedberg³ ; Jeffrey Hoffstein⁴

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
² Department of Mathematics, Stanford University, Stanford, CA 94305, United States
³ Department of Mathematics, Boston College, Chestnut Hill, MA 02467, United States
⁴ Department of Mathematics, Brown University, Providence, RI 02912, United States

Annals of mathematics, Tome 166 (2007) no. 1, pp. 293-316.

Résumé

Weyl group multiple Dirichlet series were associated with a root system

$\Phi$

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

$\Phi = A_r$

. “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 metaplectic cover of

$\mathrm{GL}_{r + 1}$

, and this is proved if

$r = 2$

or

$n = 1$

. The equivalence of our definition with that of Chinta [11] when

$n = 2$

and

$r \leqslant 5$

is also established.

Détail

DOI : 10.4007/annals.2007.166.293