7 problems

Let $\Gamma^\bullet$ denote the graph obtained from the Brauer tree of the principal $\ell$-block by removing the exceptional node and all edges incident to it. For each root of un…

Let $G=F_4(q)$ and consider the principal $\Phi_{12}$-block, whose simple modules and ordinary characters are arranged in a Brauer tree with an exceptional vertex. Let $M$ be a sim…

Let $\Gamma$ be the Brauer tree of the principal $\Phi_h$-block. The characters $\chi_{m_\zeta}$ arise from the Frobenius eigenspaces in the rational $\ell$-adic cohomology, and th…

Let $\Gamma$ be the Brauer tree of the principal $\Phi_h$-block. Let $\chi_{m_\zeta}$ denote the characters indexed by the integers $m_\zeta$ arising from the Frobenius eigenspaces…

Let $E_7$ and $E_8$ denote the indicated exceptional finite reductive groups, and consider respectively their principal $\Phi_{14}$- and $\Phi_{24}$-blocks. Conjecture on the E7 an…

Let $G=\mathbf{G}^F$ be a finite reductive group, let $h$ be the Coxeter number, and let $\Phi_h$ be the $h$th cyclotomic polynomial. Consider the principal $\Phi_h$-block and the…

Let $G$ be a finite reductive group whose principal $\ell$-block has cyclic defect, and let $h_0$ be the relevant order of the roots of unity occurring in the block. Let $q^\delta$…