arXiv 1402.55042014-02-22math.RT

正根和的一半，Coxeter元和Kostant定理

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant

Dipendra Prasad

摘要：

在域$k$上交换环面$T$的特征标群和余特征标群引入了一个逆变函子$T \rightarrow \widehat{T}$。利用这种对偶性解释$ρ:T\rightarrow {\mathbb C}^\times$，即$T$是单连通半单群$G$（在${\mathbb C}$上）的极大环面的正根之和的一半，我们得到一个余特征标$\widehatρ: {\mathbb C}^\times \rightarrow \widehat{T}$，它在$e^{\frac{2 πi}{h}}$（$h$是Coxeter数）处的值是对偶群$\widehat{G}$的Coxeter共轭类.这一观点给出了一个相当透明的证明一个定理的科斯坦特的特征值不可约有限维表示的$G$在Coxeter元素：证明金额的事实，即在$\widehat{G}_{sc}$，简单连接覆盖的$\widehat{G}$，有一个独特的经常共轭类的形象在$\widehat{G}$有秩序$h$（这是Coxeter共轭类）。

英文摘要：

Interchanging character and co-character groups of a torus $T$ over a field $k$ introduces a contravariant functor $T \rightarrow \widehat{T}$. Interpreting $ρ:T\rightarrow {\mathbb C}^\times$, half the sum of positive roots for $T$ a maximal torus in a simply connected semi-simple group $G$ (over ${\mathbb C}$) using this duality, we get a co-character $\widehatρ: {\mathbb C}^\times \rightarrow \widehat{T}$ whose value at $e^{\frac{2 πi}{h}}$ ($h$ the Coxeter number) is the Coxeter conjugacy class of the dual group $\widehat{G}$. This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite dimensional representations of $G$ at the Coxeter element: the proof amounting to the fact that in $\widehat{G}_{sc}$, the simply connected cover of $\widehat{G}$, there is a unique regular conjugacy class whose image in $\widehat{G}$ has order $h$ (which is the Coxeter conjugacy class).

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英文摘要：

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