45 problems

Let $G$ be a finite reductive group, and let $(M,\mu)$ be a $\Phi_m$-cuspidal pair with relative Weyl group $W_{M,\mu}^G$. Let $H_{M,\mu}^G(x)$ be the Broué–Malle algebra, and let…

Let $W$ be the Weyl group of an $F$-split reductive group, let $B_W$ be its braid group, and let $\boldsymbol{w}$ denote the positive braid lift of $w\in W$. Let $w_0$ be the longe…

Let $\mathfrak{g}^F$ be a finite Lie algebra, and call two elements $X,Y\in\mathfrak{g}^F$ of the same type when their centralisers $C_G(X)$ and $C_G(Y)$ are conjugate. Polynomiali…

Let $\{\mathfrak{g}_n\}_{n=1}^{\infty}$ be a sequence of reductive Lie algebras, with $\mathfrak{g}_n=\operatorname{Lie}(G_n)$, defined over algebraically closed fields $k_n$ of po…

Level-rank conjecture. The two image sets are unions of $H_{L,\lambda}^G(\zeta_m)$- and $H_{M,\mu}^G(\zeta_l)$-blocks, respectively; the bijection between them induced by…

Let ${\mathbf{G}}$ be a connected reductive group with Frobenius map $F$, let $\ell$ be a prime, and set $e:=e_\ell(q)$, the multiplicative order of $q$ modulo $\ell$ (or modulo…

Let $W$ be a Weyl group, let $\mathcal{J}$ be its set of near involutions, and write $\dim(M_w)=P_w(q)$, where $P_w(t)\in\mathbb{Q}[t]$. For $w\in\mathcal{J}$, let…

Let $F$ be a nonarchimedean local field with inertia subgroup $I_F$, Frobenius element $Fr$, and Langlands dual group $\mathbb{G}^{\vee}(\mathbb{C})$. Let $\overline{\mathbf{G}}$ b…

Let $G$ be a connected reductive group with Frobenius map $F$, and let $C$ be an $F$-stable unipotent class in $G$. For $u\in C^F$, write $A_G(u)$ for the component group associate…

Let $G$ be a connected reductive group over a finite field, and let $G^F$ be its finite group of fixed points under a Frobenius map. Deligne–Lusztig define Green functions on the u…

Let $\mathbb{G}$ be a generic finite reductive group, let $e>0$, and let $(\mathbb{L},\lambda)$ be a $\Phi_e$-cuspidal pair. Let $H_{W_{\mathbb{G},\mathbb{L},\lambda}}(\vec{u})$ be…

Let $\mathbb{G}$ be a generic finite reductive group, let $e>0$ be an integer, and let $(\mathbb{L},\lambda)$ be a $\Phi_e$-cuspidal pair. The associated Deligne–Lusztig representa…

Let $({\mathbf G},F)$ be as above, let $e=e_\ell(q)$, and for an $\ell$-block $B$ of ${\mathbf G}^F$ let $\mathcal{L}^d_e(B)_\pm/{\mathbf G}^F$ denote the positive- and negative-pa…

Weak e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$,

Cabanes--Enguehard conjecture. The relation $\leq_e$ is transitive and therefore coincides with $\ll_e$. The conjecture concerns the poset of $e$-pairs and would make the directly…

The block-isomorphism character-triple conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$, there is an ${\rm Aut}_{\mathbb F}({\mathbf{G}}^F)_B$-equivariant b…

The e-local counting conjecture. For every $\ell$-block $B$ of ${\mathbf{G}}^F$ and $d\geq 0$,

Unipotent-family conjecture. There exists a unique bijection

Broué–Malle–Michel conjecture. There exists such a parabolic subgroup $\mathbf P$ for which the cohomology modules in distinct degrees have no common irreducible constituent, and f…

Let $G$ be a twisted reductive group over an algebraically closed field, with Borel subgroup $B$. Let $W$ be its corresponding Weyl group, let $w\in W$, and let $s_i$ be a simple r…

Let $W$ be the Weyl group of a maximal torus contained in a Borel subgroup $B$ of a twisted reductive group over an algebraically closed field. Let $w$ be a minimal-length element…

Let $G$ be a finite reductive group, let $L_0$ be a Levi subgroup, and let $X_0$ be a $kL_0^F$-module. Consider the Harish–Chandra series of $(L_0,X_0)$ and a $kG^F$-module in that…

Let $G$ be a finite reductive group with connected center, let $L_0$ be a Levi subgroup, and let $X_0$ be a $kL_0^F$-module defining a Harish–Chandra series. For a proper split Lev…

Let $G$ be a finite reductive group with Frobenius endomorphism $F$, let $k$ be the underlying field of characteristic $p$, and let $d\in\Delta$ index the corresponding grading of…

Counting conjecture. Suppose that $p$ is good for $\mathbf{G}$. Then $\alpha_\ell$ is the number of unipotent irreducible Brauer characters of $G$.