zbMATH — the first resource for mathematics

Matsuki correspondence for sheaves. (English) Zbl 0789.53033
Authors’ introduction: “In this paper we extend the Matsuki correspondence [T. Matsuki, J. Math. Soc. Japan 31, 331-357 (1979; Zbl 0396.53025)] to sheaves. The existence of this correspondence was conjectured by M. Kashiwara in [Open problems, Proc. Taniguchi Symp. Katata (1986]. At the same time we give a simple geometric proof of some of the results of T. Matsuki [loc. cit.; Hiroshima Math. J. 12, 307-320 (1983; Zbl 0495.53049); ibid. 18, No. 1, 59-67 (1988; Zbl 0652.53035)]. Let \(G_ R\) be a reductive Lie group, \(G\) its complexification, \(K_ C \subset G\) the complexification of a maximal compact subgroup \(K_ R \subset G_ R\) and \(X\) the flag variety of \(G\). In [J. Math. Soc. Japan (loc. cit.)] Matsuki constructed a bijection between the \(K_ C\)-orbits and \(G_ R\)-orbits on \(X\). Here we prove the following extension of this result conjectured by Kashiwara in [loc. cit.]. Theorem. The equivariant derived categories \(D_{G_ R}(X)\) and \(D_{K_ C}(X)\) are naturally equivalent.

53C35 Differential geometry of symmetric spaces
37C10 Dynamics induced by flows and semiflows
Full Text: DOI EuDML
[1] [BB1] Beilinson, A., Bernstein, J.: Localisations deg-modules. C.R. Acad. Sci., Paris292, 15–18 (1981) · Zbl 0476.14019
[2] [BB2] Beilinson A., Bernstein, J.: Proof of Jantzen’s conjecture. (Preprint)
[3] [BBD] Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque100 (1982) · Zbl 0536.14011
[4] [BL] Bernstein, J., Lunts, V.: Equivariant derived categories. (Preprint)
[5] [K] Kashiwara, M.: Open problems. Proceedings of Taniguchi symposium at Katata (1986)
[6] [KS] Kashiwara, M., Schapira, P.: Sheaves on manifolds. Berlin Heidelberg New York: Springer (1990) · Zbl 0709.18001
[7] [Ma1] Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan31, 331–357 (1979) · Zbl 0396.53025
[8] [Ma2] Matsuki, T.: Orbits on affine symmetric spaces under the action of parabolic subgroups. Hiroshima Math. J.12, 307–320 (1983)
[9] [Ma3] Matsuki, T.: Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups and Intersections of associated orbits. Hiroshima Math. J.18, 59–67 (1988) · Zbl 0652.53035
[10] [Ma4] Matsuki, T.: Orbits on flag manifolds, to appear in Proc. International Congress of Mathematicians, 1990, Kyoto
[11] [MV] Mirković, I., Vilonen, K.: Characteristic varieties of character shaves. Invent. Math.93, 405–418 (1988) · Zbl 0683.22012
[12] [N] Ness, L: A stratification of the null cone via the moment map. Am. J. Math.106, 1281–1325 (1984) · Zbl 0604.14006
[13] [So] Sorgel, W.:n-cohomology of simple highest weight modules of the walls and purity. Invent. Math.98, 565–580 (1989) · Zbl 0781.22011
[14] [SV] Scmid, W., Vilonen, K.: Characters, fixed points and Osborn’s conjecture (announcement). (Preprint)
[15] [Sp] Springer, A.: A purity result for fixed point varieties in flag manifolds. J. Fac. Sci., Univ. Tokyo, Sect. IA31, 271–282 (1984) · Zbl 0581.20048
[16] [U] Uzawa, T.: Invariant hyperfunction sections of line bundles, Thesis, 1990, Yale University, New Haven
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.