标题： $\mathbb{Z}_\ell$-spetses的部分特征表 显示英文标题

标题： Partial character tables for $\mathbb{Z}_\ell$-spetses

摘要： 设${\mathbb{G}}$为一个单连通的${\mathbb{Z}}_\ell$-spets，设$q$为一个与$\ell$互质的素数幂，并设$S$为底层西罗$\ell$-子群。 首先，受有限拟单群的Deligne-Lusztig特征标已知公式的启发，我们提出了${\mathbb{G}}(q)$的单位特征标在$S$元素上的值的公式。 利用这个公式，我们明确列出了与Benson-Solomon融合系统Sol$(q)$相关的${\mathbb{Z}}_2$-spets$G_{24}(q)$的单位特征标值。 其次，当 $\ell > 2$ 是 ${\mathbb{G}}$ 的一个很好的素数时， ${\mathbb{G}}$ 的 Weyl 群 $W$ 的阶与 $\ell$ 互质，并且在 $q\equiv1\pmod\ell$ 我们引入了一个公式，用于计算 ${\mathbb{G}}(q)$ 主块中特征值的值，该公式扩展了李型群的 Curtis-Schewe 类型公式，并且我们证明它满足块正交性的某种版本。在两种情况下，我们提出了几个关于所提议值的猜想，并提供了证据。 显示英文摘要

摘要： Let ${\mathbb{G}}$ be a simply connected ${\mathbb{Z}}_\ell$-spets, let $q$ be a prime power, prime to $\ell$ and let $S$ be the underlying Sylow $\ell$-subgroup. Firstly, motivated by known formulae for values of Deligne-Lusztig characters of finite reductive groups, we propose a formula for the values of the unipotent characters of ${\mathbb{G}}(q)$ on the elements of $S$. Using this, we explicitly list the unipotent character values of the ${\mathbb{Z}}_2$-spets $G_{24}(q)$ related to the Benson-Solomon fusion system Sol$(q)$. Secondly, when $\ell > 2$ is a very good prime for ${\mathbb{G}}$, the Weyl group $W$ of ${\mathbb{G}}$ has order coprime with $\ell$, and $q\equiv1\pmod\ell$ we introduce a formula for the values of characters in the principal block of ${\mathbb{G}}(q)$ which extends the Curtis-Schewe type formulae for groups of Lie type, and which we show to satisfy a version of block orthogonality. In both cases we formulate and provide evidence for several conjectures concerning the proposed values.