The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra

TL;DR

This paper establishes a triangulated equivalence between the derived category of finitely generated unipotent representations of -adic G and perfect complexes over a dg Schur algebra, revealing new algebraic structures in representation theory.

Contribution

It introduces a dg Schur algebra that models the derived category of unipotent -adic G representations in the non-banal case, providing a new algebraic framework.

Findings

Triangulated equivalence between categories established

The dg Schur algebra is the endomorphism algebra of a specific projective resolution

The module V is a classical generator of the derived category

Abstract

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$.