Line bundles on the cotangent bundle of the flag variety
Let \(G\) be a reductive group defined over an algebraically closed field \(k\). Choose a Borel subgroup \(B\) containing a maximal torus \(T\). Every character \(\chi\) of \(T\) extends to a character of \(B\) and gives rise to an invertible sheaf \({\mathcal L}_{G/B} (k_ \chi)\). The main result of the present article is: The pull back \({\mathcal L}_{\mathcal T} (k_ \chi)\) of \({\mathcal L}_{G/B} (k_ \chi)\) to the cotangent bundle \({\mathcal T}\) of \(G/B\) has vanishing cohomology if and only if there is no dominant weight strictly between \(\chi\) and \(\chi^ +\) in the Chevalley order, where \(\chi^ +\) is the dominant weight in the Weyl group orbit of \(\chi\). This result corrects a vanishing statement of \textit{H. H. Andersen} and \textit{J. C. Jantzen} [Math. Ann. 269, 487-525 (1984) = Prepr. Ser., Aarhus Univ. 1983/84, No. 34 (1984; Zbl 0529.20027)], and together with the following result, generalizes results of \textit{W. H. Hesselink} [Math. Ann. 223, 249-252 (1976; Zbl 0318.14007) and 252, 179-182 (1980; Zbl 0447.17006)]. In the case of vanishing, the global sections of \({\mathcal L}_{\mathcal T} (k_ \chi)^*\) and the global sections \({\mathcal J} (\chi^ +)\) of \({\mathcal L}_{\mathcal T} (k_{\chi^ +})^*\) are isomorphic after a shift of degrees. Another main result of the article is: \({\mathcal J} (\chi^ +)\) considered as an \(k[{\mathfrak g}]\) module is generated by its elements of degree zero, forming a simple \(G\)-module of type \(V_{\chi^ +}^*\). The latter result implies a conjecture of \textit{R. K. Brylinski}. The above results have many important and interesting applications, some of which had already been pointed out by Brylinski. An application is to the study of the structure of the subregular nilpotent variety \({\mathcal S}\). When \(G\) is simple and \(\varphi\) its short dominant root, the author proves that \({\mathcal J} (\varphi)\), after a shift in degrees, is the prime ideal defining \({\mathcal S}\) in the coordinate ring \(K[{\mathcal N}]\) of the variety \({\mathcal N}\) of nilpotent elements in the Lie algebra \({\mathfrak g}\) of \(G\). Moreover, \({\mathcal S}\) is a normal Gorenstein variety with rational singularities. Further applications include refinements of results of \textit{W. A. Graham} [Invent. Math. 108, No. 1, 15-27 (1992; Zbl 0781.22010)] and some interesting connections with the theory of Kazhdan- Lusztig polynomials of the associated affine Weyl group.
scientific article; zbMATH DE number 549678
scientific article