A quotient of the Lubin-Tate tower

In this talk we construct the quotient of an infinite-level Lubin-Tate space by the Borel subgroup
$B(Q_p)$ of upper triangular matrices in $GL(2,Q_p)$ as a perfectoid space. The motivation for
this is as follows. As I will explain in the talk, Scholze recently constructed a candidate for the
mod $p$ Jacquet-Langlands correspondence and the mod $p$ local Langlands correspondence
for $GL(n,F)$, $F/Q_p$ finite. Given a smooth admissible representation $\pi$ of $GL(n,F)$,
the candidate for these correspondences is given by the etale cohomology groups of the adic
projective space $P^{n-1}$ with coefficients in a sheaf $F_\pi$ that one constructs from $\pi$.
The finer properties of this candidate remain mysterious.
As an application of the quotient construction one can show that in the case of $n=2,F=Q_p$,
and $\pi$ an irreducible principal series representation the cohomology $H^i_et(P^1,F_\pi)$
is concentrated in degree one.