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Spanning homogeneous vector bundles. (English) Zbl 0705.14014
Let B be a Borel subgroup generated in the complex Lie group G by the negative roots of G. Let $$G_{\alpha}$$ denote the rank one simple subgroup of G generated by a positive root $$\alpha$$, and let $$B_{\alpha}$$ be the intersection of $$G_{\alpha}$$ with B. Then $$B_{\alpha}=T_{\alpha}U_{-\alpha}$$ where $$T_{\alpha}$$ is a maximal torus of $$G_{\alpha}$$ and $$U_{-\alpha}$$ the unipotent subgroup generated by -$$\alpha$$. Let E be a B-module. Considered as a $$U_{-\alpha}$$-module, E extends to a $$G_{\alpha}$$-module and decomposes uniquely (up to rearrangement) $$E=E_ 1\oplus \cdot \cdot \cdot \oplus E_ k$$ where $$E_ i=m_{i,\alpha}\lambda_{\alpha}| G_{\alpha}$$ is the $$G_{\alpha}$$-module induced from a nonnegative multiple of the fundamental dominant weight $$\lambda_{\alpha}$$. Each summand $$E_ i$$ is invariant under $$T_{\alpha}$$ with highest weight $$t_{i,\alpha}\lambda_{\alpha}$$, $$1\leq i\leq k$$. Thus, as a $$B_{\alpha}$$-module, $$E_ i=m_{i,\alpha}\lambda_{\alpha}|^{G_{\alpha}}\otimes n_{i,\alpha}\lambda_{\alpha}$$ with $$n_{i,\alpha}=t_{i,\alpha}- m_{i,\alpha}$$. The elements of the sequence of integers $$n_{i,\alpha}$$, $$i=1,...,k$$, are called the $$\alpha$$-indices of E.
By $$E|^ G$$ we denote the induced G-module of all B-equivariant algebraic maps $$G\to E$$, and the evaluation map $$\epsilon$$ is the function $$v\mapsto v(1):\;E|^ G\to E.$$ For a B-module E, the associated homogeneous vector bundle $${\mathcal E}=G\times_ BE$$ is spanned by the global sections if and only if $$\epsilon$$ is surjective. For an element w in the Weyl group with reduced expression $$s_ 1\cdot \cdot \cdot s_{i_ n}$$ let $$X_ w$$ denote the closure of BwB in G/B. Call $${\mathcal E}_ w$$ the restriction of $${\mathcal E}$$ to X.
Theorem. The vector bundle $${\mathcal E}_ w$$ is spanned by global sections if and only if the $$\alpha$$-indices of E are non-negative for all simple roots $$\alpha_ j$$, $$j=i_ 1,...,i_ n$$, corresponding to the sequence of reflections $$s_ k.$$
Corollary. A homogeneous vector bundle $${\mathcal E}$$ is spanned by global sections if and only if the $$\alpha$$-indices of E are nonnegative for all simple roots $$\alpha$$.
It is further shown that $${\mathcal E}$$ is spanned by global sections if and only if for some n, the n-th symmetric power $$S^ n({\mathcal E})$$ is spanned by global sections if and only if $$\xi^ n_ E$$ is spanned by global sections for some n, where $$\xi_ E$$ is the tautological line bundle over the projectivization $${\mathbb{P}}({\mathcal E})$$ of $${\mathcal E}$$ whose restriction to the fiber $${\mathbb{P}}(E)$$ is $${\mathcal O}(1)$$.
Reviewer: K.H.Hofmann

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 22E10 General properties and structure of complex Lie groups
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