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Line bundles on the cotangent bundle of the flag variety. (English) Zbl 0807.14043

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 H. H. Andersen and 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 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 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 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.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

[1] Andersen H.H., Jantzen, J.C.: Cohomology of induced representations for algebraic groups. Math. Ann.269, 487-525 (1984) · Zbl 0529.20027 · doi:10.1007/BF01450762
[2] Beilinson A., Bernstein, J.: Localisation de g-modules C.R. Acad. Sci., Paris292, 15-18 (1981) · Zbl 0476.14019
[3] Borho, W., Kraft, H.: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv.54, 61-104 (1979) · Zbl 0395.14013 · doi:10.1007/BF02566256
[4] Broer, A.: Hilbert series in invariant theory. Thesis, Rijksuniversiteit Utrecht (1990)
[5] Brylinski, R.K.: Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits. J. Am. Math. Soc.2, 517-533 (1989) · Zbl 0729.17005
[6] Brylinski, R.K.: Twisted ideals of the nullcone. In: Connes, A., Duflo, M., Joseph, A., Rentschler, R. (eds.) Operator algebras, unitary representations, enveloping algebras and invariant theory. Boston: Birkhäuser 1990
[7] Buchsbaum, D.A., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math.99, 447-485 (1977) · Zbl 0373.13006 · doi:10.2307/2373926
[8] Graham, W.A.: Functions on the universal cover of the principal nilpotent orbit. Invent. Math.108, 15-27 (1992) · Zbl 0781.22010 · doi:10.1007/BF02100596
[9] Gulliksen, T.H. and Negård, O.G.: Un complexe résolvant pour certains idéaux déterminantiels. C.R. Acad. Sci. Paris, Sér, A274, 16-18 (1972)
[10] Gupta, R.K.: Generalized exponents via Hall-Littlewood symmetric functions. Bull. Am. Math. Soc.16, 287-291 (1987) · Zbl 0648.22011 · doi:10.1090/S0273-0979-1987-15519-4
[11] Gupta, R.K.: Characters and the q-analog of weight multiplicity. J. Lond. Math. Soc.36, 68-76 (1987) · Zbl 0649.17009 · doi:10.1112/jlms/s2-36.1.68
[12] Hartshorne, R.: Algebraic Geometry. (Grad. Texts Math., vol. 52) Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[13] Hesselink, W.H.: Cohomology and the resolution of the nilpotent variety. Math. Ann.223, 249-252 (1976) · Zbl 0325.14007 · doi:10.1007/BF01360956
[14] Hesselink, W.H.: Characters of the nullcone. Math. Ann.252, 179-182 (1980) · Zbl 0447.17006 · doi:10.1007/BF01420081
[15] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math.74, 329-387 (1961) · Zbl 0134.03501 · doi:10.2307/1970237
[16] Kostant, B.: Lie group representations on polynomial rings. Am. J. Math.85, 327-404 (1963) · Zbl 0124.26802 · doi:10.2307/2373130
[17] Kraft, H. Procesi, C.: Closures of conjugacy classes of matrices are normal. Invent. Math.53, 227-247 (1979) · Zbl 0434.14026 · doi:10.1007/BF01389764
[18] Kraft, H. Procesi, C.: On the geometry of conjugacy classes in classical groups. Comment. Math. Helv.57, 539-602 (1982) · Zbl 0511.14023 · doi:10.1007/BF02565876
[19] Kraft, H.: Geometrische Methoden in der Invariantentheorie. Braunschweig: Vieweg 1984 · Zbl 0569.14003
[20] Kraft, H.: Closures of conjugacy classes in G2. J. Algebra126, 454-465 (1989) · Zbl 0693.17004 · doi:10.1016/0021-8693(89)90313-X
[21] Slodowy, P.: Simple singularities and simple algebraic groups. (Lect. Notes Math., vol. 815) Berlin Heidelberg, New York: Springer 1980 · Zbl 0441.14002
[22] Springer, T.A.: The unipotent variety of a semisimple group. Proc. Colloq. Alg. Geo. Tata Institute, 373-391 (1968)
[23] Steinberg, R.: Lectures on conjugacy classes in algebraic groups. (Lect. Notes Math., vol.366) Berlin Heidelberg New York: Springer 1974 · Zbl 0281.20037
[24] Weyman, J.: The equations of conjugacy classes of nilpotent matrices. Invent. Math.98, 229-245 (1989) · Zbl 0717.20033 · doi:10.1007/BF01388851
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