Steinberg quotients, Weyl characters and Kazhdan-Lusztig polynomials (Q6567174)

This paper considers the characters of indecomposable tilting modules $T((p-1)\rho + \lambda)$ for a reductive algebraic group $G$ in characteristic $p > 0$ and a dominant weight $\lambda$. Such a character is divisible by the Steinberg character $\chi((p-1)\rho)$ and the corresponding \emph{Steinberg quotient} is denoted $t(\lambda)$. When expressed as a non-negative sum of $W$-orbits of weights, the linkage principle places a condition on which $W$-orbits can occur and prior work of the same author shows that all $W$-orbits satisfying this condition do in fact occur when $\lambda$ is $p$-restricted [\textit{P. Sobaje}, Math. Z. 297, No. 3--4, 1733--1747 (2021; Zbl 1493.20013)].\N\NThe paper under review extends these results to include quotients $t(\lambda)$ for all dominant weights $\lambda$, as well as quantum Steinberg quotients $t_\zeta(\lambda)$ for a quantum group specialised at a primitive $p$-th root of unity $\zeta$. Specifically, let $\eta = \sum c_\mu s(\mu)$ be a $W$-stable formal character, the sum being over dominant characters $\mu$ and $s(\mu)$ denoting the $W$-orbit; moreover for each dominant $\lambda$, let $\chi(\lambda)$ denote the character of the Weyl module of highest weight $\lambda$. Then Theorem 3.3.1(2) states that if for all dominant $\lambda$, the product $\chi((p-1)\rho + p\lambda)\eta$ is a non-negative sum of other Weyl characters $\chi(\lambda')$, then $c_\mu \ge c_\mu'$ whenever $\mu - \rho \uparrow \mu' - \rho$. This directly generalises the case $\eta = t(\lambda)$ from the author's prior work. The particular case $\eta = t_\zeta(\lambda)$ then gives the analogue in the quantum setting (Theorem 3.5.1).\N\NIn Section 6 of the paper, the author introduces $\mathcal{M}_p(\lambda)$, the smallest $W$-stable formal character such that $\chi((p-1)\rho + p\mu)\mathcal{M}_p(\lambda)$ is a non-negative sum of Weyl characters, for all dominant weights $\mu$. They then prove that it suffices to check this condition for a certain finite set of weights $\mu$ and hence obtain an algorithm for calculating $\mathcal{M}_p(\lambda)$ explicitly. Noting that $t_\zeta(\lambda)$ and $t(\lambda)$ each satisfy this condition and that $t_\zeta(\lambda)$ is a lower bound for $t(\lambda)$ in a precise sense, it is natural to check whether $\mathcal{M}_p(\lambda)$ and $t_\zeta(\lambda)$ are close and this turns out to be the case. Indeed, in many cases in type $A_n$ $(n \le 5)$ and $p$-restricted weights $\lambda$, the characters $\mathcal{M}_p(\lambda)$ and $t_\zeta(\lambda)$ turn out to be equal, or to differ only by $1$ in the multiplicity of the least weight orbit. Further interesting observations and calculations are made along the way.