12 problems

Depth-$r$ stable-center conjecture. The image of $\xi^r$ is contained in the depth-$r$ stable center:

Let $G$ be the group under consideration, let $r$ be an integral depth, and let $\mathcal Z^{st,r}(G)=\mathcal Z^r(G)\cap\mathcal Z^{st}(G)$ denote the subspace of stable elements…

Let ${\rm G}$ be a classical group, let $\mathfrak{Z}_{{\rm G},\overline{\mathbb{Z}}[1/p]}^{\operatorname{st}}$ denote the stable center, and let $\mathcal{EP}({\rm G})$ be the set…

Assume $G$ is split, semisimple, and simply connected over $F=\mathbb{F}_q((t))$. Let $LG$ be the loop group, $\mathfrak A^{\operatorname{Fr}}(LG)$ the Frobenius-equivariant Grothe…

Let $G$ be a reductive group over the local field $E$, let $W_E$ be its Weil group, and let $\mathcal Z^{\mathrm{spec}}(G,\mathbb Q)$ be the natural rational form of the spectral B…

Let $F$ be a non-archimedean local field, let $G_n=\operatorname{GL}_n(F)$, let $Z_n$ be the Bernstein center, and let $e$ be the primitive idempotent associated with an inertial t…

Let $F$ be a non-archimedean local field, let $G_n=\operatorname{GL}_n(F)$, let $Z_n$ be the Bernstein center of the category of smooth representations of $G_n$, and let $\nu$ be a…

With the notation above, let $R_\nu^{\operatorname{inv}}$ denote the subalgebra of $\mathcal G_\nu$-invariant elements of $R_\nu$. Strong local Langlands in families. There is an i…

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 $G$ be the reductive group in the Shimura datum, let $K_p\subset G(\mathbb Q_p)$ be the prescribed level subgroup, and let $K_{p^r}\subset G(\mathbb Q_{p^r})$ be its base chang…

Strong local Langlands interpolation conjecture. If $\overline{\rho}$ is semisimple, there is a unique isomorphism

Weak local Langlands interpolation conjecture. For any $\overline{\rho}$, there is a map