43 problems

Let $G_0$ be any connected unipotent group over ${\mathbb F}_q$, let $G=G_0\otimes_{{\mathbb F}_q}{\mathbb F}$ with Frobenius $\operatorname{Fr}$, and let…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, and let its functional dimension be $d_e=(\dim G-n_e)/2$. The integrality…

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, let ${\mathcal M}_e^{perv}$ be the full subcategory of perverse objects in…

The character-sheaf duality and perversity conjecture. For every minimal idempotent $e$, there exists $n_e\in\\{0,1,\dotsc,\dim G\\}$ such that

Let ${\mathcal C}_\zeta^{{\underline c}_f}$ be the monoidal category associated with a finite two-sided cell ${\underline c}_f\subset W_f(\zeta)$, and let…

Lusztig's tensor-category conjecture. One should have $K\circ K'\in\mathcal{C}_{\mathcal{U}}$, and the assignment $(K,K')\mapsto K\circ K'$ should define a tensor-category structur…

Lusztig's character-sheaf conjecture. If $\cl$ is regular, then $a_!b^*\cl$ is an intersection cohomology complex on $G$.

Let $G$ be the algebraic group in the paper, let $\bG^1$ be its component, and let $\tSS$ be the simple perverse sheaf on $\bG^1$ whose restriction to $G^1$ is the distinguished si…

Let $M\subset G$ be a Levi subgroup contained in a parabolic subgroup, and let … be the proposed Grothendieck–Springer functor. The existing intertwiner isomorphisms are constructe…

Boyarchenko–Drinfeld's conjecture. $G$ is easy if and only if its $cL$-packets of character sheaves are trivial.

Let $G$ be a reductive group, let $\xi=(M,\lambda,L,\mathcal{O},\mathcal{L})$ be a polar-cuspidal datum for $G$, and let $\mathbf{Q}\subset LM$ be a parahoric subgroup containing…

Let $\theta:G\to G$ be an automorphism of order $m\in\mathbb{Z}_{>0}$, let $G_0$ be the neutral component of $G^\theta$, and let $\mathfrak{g}_1$ be the $\zeta_m$-eigenspace of…

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…

Principal-series character-sheaf equivalence conjecture. For $\ell$ sufficiently large, this principal-series category is equivalent to the derived category of differential graded…

Let $G$ be the connected reductive group with involution $\theta$, let $K=G^\theta$, and let $\mathfrak g_1$ be the degree-one component of the associated graded Lie algebra. Write…

Let $G$ be the connected reductive group with involution $\theta$, let $K=G^\theta$, and let $\mathfrak g_1$ be the degree-one component of the associated graded Lie algebra. For a…

Vilonen–Xue conjecture. The character sheaves constructed in this paper, together with those constructed in the earlier work, constitute all cuspidal character sheaves for graded c…

Let $W$ be a twisted finite Coxeter group, and let $w$ be an elliptic element of minimal length. Write $\mathrm{Tran}(w)$ for its centraliser under the relevant transport structure…

Let ${\mathfrak{g}}_1$ be the degree-one component of one of the stably graded Lie algebras considered here, let $K$ act on ${\mathfrak{g}}_1$, and let ${\mathfrak{g}}_1^{rs}$ be i…

Let ${\mathfrak{g}}_1$ be the degree-one component of a stably graded Lie algebra, let $K$ act on ${\mathfrak{g}}_1$, and let $\chi$ be a character indexing the nearby-cycle sheaf…

Let $\mathbf G$ be an adjoint group of type $E_6$ over a field of good characteristic, and let $\mathbf G(p)$ denote its group of points over the prime field. Consider the characte…

Let $\Sigma$ be a compact Riemann surface, let $G$ be a reductive group, and let $\check{G}$ be its Langlands dual group. Write $\operatorname{Bun}_G(\Sigma)$ for the moduli stack…

Let $G$ be a complex reductive group, let $E$ be an elliptic curve, and let $\mathcal H_G$ be the affine Hecke category. Let $\mathcal Shv_{\mathcal N}(Bun_G(E))$ denote the catego…

General character-sheaf conjecture. Analogues of Theorems 1, 2 and 3 should hold for any algebraic group $G$ over $\mathtt{k}$.