AI 中文总结
该研究针对有限单群中李型单群特征标p-无理性更高的现象，证明p-互素次数的Deligne-Lusztig特征标的p-有理性水平等于对应极大环面特征标的水平，并提出Lusztig诱导对p'-次数特征标保持p-有理性的猜想。
AI 中文摘要
在有限单群中，交错群和散在群的特征标值在任意素数p下的无理性相对较低，而李型单群的特征标值可具有任意高的p-无理性。我们提供了支持该现象的具体证据。特别地，我们证明若χ:=R_T^G(θ)是有限约化群G^F的Deligne-Lusztig特征标，其中θ是极大环面T^F的不可约特征标，且χ的次数与p互素，则χ的所谓p-有理性水平恰好与θ的p-有理性水平一致。我们还给出进一步证据表明，Lusztig诱导对于p'-次数的特征标保持p-有理性。
英文摘要
Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $χ:=R_{\mathbf{T}}^{\mathbf{G}}(θ)$ is a Deligne--Lusztig character of a finite reductive group $\mathbf{G}^F$, with $θ$ an irreducible character of a maximal torus $\mathbf{T}^F$, and if $χ$ has degree prime to $p$, then the so-called $p$-rationality level of $χ$ coincides precisely with that of $θ$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p'$-degree.
Comments20 pages
