Integral structures in the $p$-adic holomorphic discrete series

TL;DR

This paper constructs integral structures within p-adic holomorphic discrete series representations of GL_{d+1}(K) using equivariant sheaves on formal schemes, proving their cohomological vanishing properties and relating to vector bundle cohomology on Deligne-Lusztig varieties.

Contribution

It introduces new integral lattice sheaves in p-adic discrete series representations and establishes their cohomological properties, linking p-adic representation theory with algebraic geometry over finite fields.

Findings

Cohomology H^t vanishes for t>0 in certain sheaves.

Constructs GL_{d+1}(K)-equivariant sheaves that are lattices in discrete series.

Connects p-adic representation theory with vector bundle cohomology on Deligne-Lusztig varieties.

Abstract

For a local non-Archimedean field $K$ we construct ${\rm GL}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are ${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete series representations (in $K$-vector spaces) of ${\rm GL}_{d+1}(K)$ as defined by P. Schneider \cite{schn}. We prove that the cohomology $H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t>0$, for ${\mathcal V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other main topic of this paper: over a finite field $k$, the study of the cohomology of vector bundles on the natural normal crossings compactification $Y$ of the Deligne-Lusztig variety $Y^0$ for ${\rm GL}_{d+1}/k$ (so $Y^0$ is the open subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational hyperplanes).