Cohomology of $p$-adic Chevalley groups

TL;DR

This paper computes the cohomology algebra of certain $p$-adic Chevalley groups and their subgroups, using new Lie algebra presentations and geometric methods, with implications for algebraic and homotopy theory.

Contribution

It introduces a novel presentation of graded Lie algebras in Lazard's theory and applies it to compute cohomology of $p$-adic groups and their subgroups.

Findings

Computed cohomology algebra of $G(\\mathcal{O}_K)$ and Iwahori subgroups

Established connections between Lie algebra presentations and cohomology

Extended results to Morava stabilizer groups in homotopy theory

Abstract

Let $G$ be a split connected reductive group over the ring of integers of a finite unramified extension $K$ of $\mathbf{Q}_p$. Under a standard assumption on the Coxeter number of $G$, we compute the cohomology algebra of $G(\mathcal{O}_K)$ and its Iwahori subgroups, with coefficients in the residue field of $K$. Our methods involve a new presentation of some graded Lie algebras appearing in Lazard's theory of saturated $p$-valued groups, and a reduction to coherent cohomology of the flag variety in positive characteristic. We also consider the case of those inner forms of $\mathrm{GL}_n(K)$ that give rise to the Morava stabilizer groups in stable homotopy theory.