The Fourier Coefficients of a Metaplectic Eisenstein Distribution on the Double Cover of SL$(3)$ over $\mathbb{Q}$

TL;DR

This paper computes explicit Fourier coefficients of a minimal parabolic Eisenstein distribution on the double cover of SL(3) over rationals, including constant terms and ramified place formulas, linking to Weyl group multiple Dirichlet series.

Contribution

It provides explicit formulas for Fourier coefficients of a metaplectic Eisenstein distribution on the double cover of SL(3), including ramified and unramified cases, advancing understanding of automorphic forms on covering groups.

Findings

Explicit formula for the constant term of the Eisenstein distribution.

Formulas for Fourier coefficients at ramified place p=2.

Unramified non-degenerate Fourier coefficients match combinatorial descriptions.

Abstract

We compute the Fourier coefficients of a minimal parabolic Eisenstein distribution on the double cover of SL$(3)$ over $\mathbb{Q}$. Two key aspects of the paper are an explicit formula for the constant term, and formulas for the Fourier coefficients at the ramified place $p=2$. Additionally, the unramified non-degenerate Fourier coefficients of this Eisenstein distribution fit into the combinatorial description provided by Brubaker-Bump-Friedberg-Hoffstein in `Weyl group multiple Dirichlet series. III. Eisenstein series and twisted unstable $A_r$'.