12 problems

Let $\theta$ be a stable grading on the dual Lie algebra $\widehat{\mathfrak g}$, with associated parahoric subgroup $P$. Let $p:\widehat{\mathfrak g}_1\to\widehat{\mathfrak g}(1)$…

Let $G$ be a connected reductive group over the maximal unramified extension of a non-archimedean local field, let $G_r$ be the truncated parahoric group scheme associated to a poi…

Let $R$ satisfy (N), and let ${\mathcal G}$ be a tame parahoric group scheme with minuscule coweight $\mu$. Write ${\rm M}^{\rm loc}={\rm M}^{\rm loc}({\mathcal G},\{\mu\})$ for th…

Open-and-closed immersion conjecture. The maps $\iota^0$ and $\iota^\pm$ are always open and closed immersions.

Let $\mathcal G$ be the Bruhat–Tits parahoric group scheme over $\mathbb Z_{p^r}$ with generic fiber $G_{\mathbb Q_{p^r}}$ and $\mathcal G(\mathbb Z_{p^r})=K_{p^r}$. Let…

Assume ${\bf G}_{\mathbb Q_{p^r}}$ is unramified and $K_p\subset{\bf G}(\mathbb Q_p)$ is parahoric. Let $d={\rm dim}(Sh_{K_p})$, and let $z_{-\mu,j}$ be the Bernstein function asso…

Let $\underline{\mathbb{G}}$ be a group of parahoric type over a perfect field $k$, and suppose that it satisfies the required condition: the rings $R_\alpha$ associated with its a…

Vertexwise admissibility conjecture. The inclusion

Assume that $\operatorname{char}(k)=0$, that $\mathcal G_\eta$ is semisimple, simply connected, and absolutely simple, and that $\mathcal G_{\eta_x}$ splits over a tamely ramified…

Assume that $\mathcal G_\eta$ is semisimple, simply connected, and absolutely simple, and that $\mathcal G_{\eta_x}$ splits over a tamely ramified extension of $K_x$ for every…

Assume the setup of a parahoric group scheme $\mathcal G$ over a curve $X$ and let $\mathcal M_{\mathcal G/X}$ denote the moduli stack of $\mathcal G$-torsors. Let $x\in X(k)$, let…

Assume that $\operatorname{char}(k)\neq 2$, that $\pi:\widetilde X\rightarrow X$ is a possibly ramified double cover with involution $\sigma$, and set…