Double Steinberg coinvariants for special linear groups

Abstract

For a field $F$ we study the coinvariants for the $S{L}_n(F)$-action on the double Steinberg module $S{t}_n(F)\otimes S{t}_n(F)$ and show they have a rich algebraic structure: for $n=2$ it is the Grothendieck-Witt group of $F$, and for all $n$ they assemble to a graded nonunital $\mathbb{Z}[F^{\times }]$-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups $S{L}_n(F)$ assemble to an $E_{\infty }$-algebra in a suitable functor category, whose $E_2$-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.

20 pages