Representations of algebraic groups and their Lie algebras
Representations of algebraic groups and their Lie algebras
复制标题
代数群及其李代数的表示
DOI：
--
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
J. Jantzen
中科院分区：
文献类型：
--
作者：
J. Jantzen
陷害设K是代数闭域，G是K上的连通和单连通半单群。回忆一下第一讲中的重要记号：· X = Homalg.gr（T，K），T · X的特征标群= Homalg.gr（K，T），T · X的余特征标群，X ×X → Z使得λ（φ（t））= t <$λ，φ <$对于所有t ∈ K× · Φ <$X，G关于T · Φ <$= {α <$|α ∈ Φ}，对应的对偶根· Φ，正根的选择系· B，包含T的G的Borel子群对应于−Φ · U，B的幂幺根。所以我们有一个半直分解B = TU作为代数群。· X+，关于Φ · L（λ）的支配特征标的集合，indBKλ的唯一单子模，对于任何λ ∈ X+这里有两个额外的符号，很快就会发挥重要作用：· W，G关于T · ρ的Weyl群，正根之和的一半。在我们的假设下，我们有ρ ∈ X。
Set-up. Let K be again an algebraically closed field and G a connected and simply connected semi-simple group over K. Recall the crucial notations from Lecture I: • X = Homalg.gr(T,K), the character group of T • X∨ = Homalg.gr(K, T ), the cocharacter group of T • 〈 , 〉, the pairing X ×X∨ → Z such that λ(φ(t)) = t〈λ,φ〉 for all t ∈ K× • Φ ⊂ X, the root system of G with respect to T • Φ∨ = {α∨ | α ∈ Φ} ⊂ X∨, the corresponding dual roots • Φ, a chosen system of positive roots • B, the Borel subgroup of G containing T that corresponds to −Φ • U , the unipotent radical of B. So we have a semi-direct decomposition B = T U as algebraic group. • X+, the set of dominant characters with respect to Φ • L(λ), the unique simple submodule of indBKλ, for any λ ∈ X+ Here are two additional notations that will soon play an important role: • W , the Weyl group of G with respect to T • ρ, half the sum of the positive roots. Under our assumption we have ρ ∈ X.