Stability in the category of smooth mod-p representations of SL_2(Qp)

Abstract

Let $p \geq 5$ be a prime number and let $G = SL_2(\mathbb{Q}_p)$. Let $\Xi$ = Spec$(Z)$ denote the spectrum of the centre $Z$ of the pro-$p$ Iwahori Hecke algebra of $G$ with coefficients in a field $k$ of characteristic $p$. Let $\mathcal{R} \subset \Xi \times \Xi$ denote the support of the pro-$p$ Iwahori Ext-algebra of $G$, viewed as a $(Z,Z)$-bimodule. We show that the locally ringed space $\Xi/\mathcal{R}$ is a projective algebraic curve over Spec$(k)$ with two connected components, and that each connected component is a chain of projective lines. For each Zariski open subset $U$ of $\Xi/\mathcal{R}$, we construct a stable localising subcategory $\mathcal{L}_U$ of the category of smooth $k$-linear representations of $G$.