论文标题
普拉萨德关于偶发性的猜想
Prasad's Conjecture about dualizing involutions
论文作者
论文摘要
令$ g $为一个有限字段$ \ mathbb {f} _q $，带有相应的frobenius $ f $的连接还原组。令D. Prasad在假设$ 2 \ mathrm {h}^1（f，z（g））= 0 $下定义的双重性相关性，其中$ z（g）$表示$ g $的中心。我们表明，对于$ g^f $的每个不可约的字符$ρ$ $ g^f $，$ 〜b $ tak $ρ$在其双$ρ^{\ vee} $上，并且仅当适合约旦的角色分解时，相关的单位单位$u_ρ$具有Frobenius eigenius eigenius eignius eignius eignius eigrare $ $ $ $ $ $ $ \ $ $ \ g g g g g g g g g g g g g g g g g y he and。满足$ 2 \ mathrm {h}^1（f，z（g））= 0 $，然后二重性差异$ b $ tak $ρ$ to dual $ρ^{\ vee} $ for $ g^f $ $ g^f $。我们的结果解决了D. prasad的猜想的有限组。
Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ with corresponding Frobenius $F$. Let $ι_G$ denote the duality involution defined by D. Prasad under the hypothesis $2\mathrm{H}^1(F,Z(G))=0$, where $Z(G)$ denotes the center of $G$. We show that for each irreducible character $ρ$ of $G^F$, the involution $ι_G$ takes $ρ$ to its dual $ρ^{\vee}$ if and only if for a suitable Jordan decomposition of characters, an associated unipotent character $u_ρ$ has Frobenius eigenvalues $\pm$ 1. As a corollary, we obtain that if $G$ has no exceptional factors and satisfies $2\mathrm{H}^1(F,Z(G))=0$, then the duality involution $ι_G$ takes $ρ$ to its dual $ρ^{\vee}$ for each irreducible character $ρ$ of $G^F$. Our results resolve a finite group counterpart of a conjecture of D.~Prasad.
