单线性代数群的不可约表示中特征空间的最小余维数
Minimum codimension of eigenspaces in irreducible representations of simple linear algebraic groups
摘要：
设$k$是特征$p \geq 0$的代数闭域，$G$是秩为$\ell$的单连通经典线性代数群，$T$是$G$中具有有理特征群$X(T)$的极大环面.对于一个非零的$p$ -限制的支配权$λ\in X(T)$，令$V$是相关的不可约$kG$ -模.定义$ν_{G}(V)$为$V$上$G$的非中心元素所对应的特征空间的最小余维。在本文中，我们计算了$A_{\ell}$、$\ell \geq 16$和$dim(V) \leq \frac{\ell^{3}}{2}$型$G$的$ν_{G}(V)$; $B_{\ell}$型$C_{\ell}$、$\ell \geq 14$和$dim(V) \leq 4\ell^{3}$型$G$的$ν_{G}(V)$;以及$D_{\ell}$、$\ell \geq 16$和$dim(V) \leq 4\ell^{3}$型$G$的$ν_{G}(V)$。此外，对于秩较小的群及其对应的维数满足上述界的不可约模，我们确定了$ν_{G}(V)$的下界。
英文摘要：
Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $λ\in X(T)$, let $V$ be the associated irreducible $kG$-module. Define $ν_{G}(V)$ to be the minimum codimension of eigenspaces corresponding to non-central elements of $G$ on $V$. In this paper, we calculate $ν_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell \geq 16$, and $dim(V) \leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\ell}$, respectively $C_{\ell}$, $\ell \geq 14$, and $dim(V) \leq 4\ell^{3}$; and for $G$ of type $D_{\ell}$, $\ell \geq 16$, and $dim(V) \leq 4\ell^{3}$. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for $ν_{G}(V)$.
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