Block decompositions in the p-adic Langlands correspondence

Let $K\mathrm{/}\text{Qp}$ be a finite extension, $\text{bG}$ a connected reductive group over $K$, and set $G=\text{bG}(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center ${\text{cC}}_G$ of the category of solid locally $\text{Qp}$-analytic representations of $G$ on $E$-vector spaces is isomorphic to the center of the locally analytic distribution algebra $D^{\text{la}}(G,E)$. We then consider the Emerton-Gee stack ${\text{frX}}_{n,K}$ of rank-$n$ $(\text{vphi},\mathrm{\Gamma })$-modules over the Robba ring for $K$ and determine its connected components. The latter are in canonical bijection with the primitive idempotents of the ring ${\text{cC}}_{{\text{GL}}_n(K)}$. Moreover, under the assumption that the ring of global functions on ${\text{frX}}_{n,K}$ has no non-zero locally nilpotent elements, we show that this ring is isomorphic to a Fréchet completion of ${\text{cC}}_{{\text{GL}}_n(K)}$.