Homological vanishing for the Steinberg representation

For a field $\text{k}$, we prove that the $i$th homology of the groups $\operatorname{GL}_{n}(\text{k})$, $\operatorname{SL}_{n}(\text{k})$, $\operatorname{Sp}_{2n}(\text{k})$, $\operatorname{SO}_{n,n}(\text{k})$, and $\operatorname{SO}_{n,n+1}(\text{k})$ with coefficients in their Steinberg representations vanish for $n\geqslant 2i+2$.

A QUOTIENT OF THE LUBIN–TATE TOWER

In this article we show that the quotient${\mathcal{M}}_{\infty }/B(\mathbb{Q}_{p})$of the Lubin–Tate space at infinite level${\mathcal{M}}_{\infty }$by the Borel subgroup of upper triangular matrices$B(\mathbb{Q}_{p})\subset \operatorname{GL}_{2}(\mathbb{Q}_{p})$exists as a perfectoid space. As an application we show that Scholze’s functor$H_{\acute{\text{e}}\text{t}}^{i}(\mathbb{P}_{\mathbb{C}_{p}}^{1},{\mathcal{F}}_{\unicode[STIX]{x1D70B}})$is concentrated in degree one whenever$\unicode[STIX]{x1D70B}$is an irreducible principal series representation or a twist of the Steinberg representation of$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

Distinction of the Steinberg representation II: An equality of characters

In a previous paper, Broussous and the author prove that for symmetric spaces of the form