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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

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