14 problems

Let $G$ be the reductive algebraic group, $G_1$ its first Frobenius kernel, $h$ the Coxeter number, and $W_p^+$ the set of minimum-length right coset representatives in…

Character map conjecture. If $i:K(\mathcal O_{m|n}^\Delta)\to\mathscr T^{m|n}_{\mathbb Z}$ is the map defined in the cited theorem, then

Tensor-ideal equality conjecture. For any $G$ and $p$, all of the following hold:

Naturality conjecture. The conditions in Proposition are satisfied for an arbitrary $\mathfrak{g}$, with the assumptions of Section; equivalently, support of tilting modules is nat…

Scheme-theoretic traditional Humphreys conjecture. For every $w\in W_{\mathrm{ext}}^\varnothing$, the coherent sheaf

Relative Humphreys conjecture.

Traditional Humphreys conjecture.

Let $G$ be a semisimple algebraic group over $k$, let $r\geq 1$, let $M$ be a rational $G$-module, and let $\operatorname{St}_r$ denote the $r$-th Steinberg representation. A good…

Let $G$ be a reductive algebraic group over a field $k$ of characteristic $p$, let $G_rT$ be the corresponding Frobenius kernel together with a maximal torus, and let $X_r$ denote…

Let $\mathbb{X}^+$ be the dominant weights and let $\lambda,mu,mu'\inmathbb{X}^+$. Write … where $s(\mu)$ is the Weyl-orbit sum. Use the order relation $\uparrow$ on weights. Monoto…

Suppose we are in the reductive case. Let $boldsymbol{ u}in-boldsymbol{X}^+$, and let $({mathscr{O}}{boldsymbol{ u}},boldsymbol{\rho}{boldsymbol{ u}})inXi$ be the pair corresponding…

Let $\mathbf{G}$ be a reductive algebraic group whose derived subgroup is simply connected, let $\mathbf{B}\subset\mathbf{G}$ be a Borel subgroup, and let…

Andersen's conjecture. Suppose Lusztig's conjecture holds for $G$ in characteristic $p$. For $\lambda \in A_0$ and dominant $x \in \prescript{p}{}{W}$, one has

Let $G$, $G_1$, $W$, ${}^{\mathrm{f}}W^{\mathrm{f}}$, $\mathcal{C}_p$, $\iota_{\Bbbk}$, and the corresponding nilpotent orbit $\mathscr{O}^{\mathbb{C}}_{\mathbf{c}(w)}$ be as in th…