A Splitting into the Double Cover of SL(3,$\mathbb{R}$)

TL;DR

This paper derives an explicit formula for splitting a congruence subgroup of SL(3,R) into its double cover using Plücker coordinates, facilitating Fourier coefficient computations of Eisenstein series.

Contribution

It provides a new explicit formula for the splitting in terms of Plücker coordinates and proves its twisted multiplicativity, advancing the understanding of double covers of SL(3,R).

Findings

Formula for splitting in terms of Plücker coordinates

Proof of twisted multiplicativity of the splitting

Facilitates computation of Eisenstein series Fourier coefficients

Abstract

We provide a formula for the splitting of a congruence subgroup of SL$(3,\mathbb{R})$ into the double cover of SL$(3,\mathbb{R})$ in terms of Pl\"{u}cker coordinates and prove that the splitting satisfies a twisted multiplicativity. The existence of this splitting and a formula (in terms of a different set of coordinates) was proved by S.D. Miller in an unpublished note; the formula in terms of Pl\"{u}cker coordinates is advantageous to the computation of the Fourier coefficients of an Eisenstein series on the double cover of SL$(3)$ over $\mathbb{Q}$. The computation of these Fourier coefficients will be addressed in a forthcoming work.