24 problems

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 $G$ be a quasisplit reductive group over $E$, let $\Psi_G:\mathcal Z^{\mathrm{spec}}(G,\Lambda)\to\mathcal Z(G(E),\Lambda)$ be the map from the spectral to the Bernstein center…

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

Stable center conjecture. The subspace $\mathcal{Z}^{st}(G)\subset\mathcal{Z}(G)$ is a unital subalgebra. This conjectural stable-center structure is related to a decomposition of…

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…

Let ${\rm G}$ be a reductive group over a non-archimedean local field, let $5chat{{\rm G}}$ be its dual group, and let…

Let $G$ be a split reductive group over a non-archimedean local field $F$, and let $\phi$ be the map constructed in the paper from the relevant moduli or parameter space to the dep…

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 split reductive group over a non-archimedean local field $F$, and let $Z^0(G(F))$ be the depth zero Bernstein center. Consider the geometric construction of elements i…

Let $G$ be a split connected reductive group over a non-archimedean local field $F$. Let $Z^{st,0}(G(F))$ denote the depth zero stable Bernstein center, and let…

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 $\overline{\rho}:G_F\to\operatorname{GL}_n(k)$ be a residual representation, let $R_{\overline{\rho}}^{\Box}$ be its universal framed deformation ring, and let…

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…

Bernstein-centre compatibility conjecture. La correspondance de Langlands induit des isomorphismes

Let $G$ be a reductive $p$-adic group, let $R(G)$ be its category of smooth representations, and let $\Pi_r\in Z_G$ be the Bernstein projector onto the subcategory of representatio…

Let $G$ be a reductive $p$-adic group and let $Z_G$ be its Bernstein center. The stable distributions in $Z_G$ form a subspace consisting of elements whose corresponding invariant…

Let $G$ be a reductive $p$-adic group, let $Z_G$ be its Bernstein center, and let $\operatorname{Irr}(G)$ be the set of irreducible smooth representations of $G$. A stable distribu…

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…

Let $G$ be a reductive $p$-adic group, let $\operatorname{H}(G)$ be its Hecke algebra, and let $1_{G_c}$ denote the compact-restriction operator. For an idempotent…

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