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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.

MSC:
53C35 Differential geometry of symmetric spaces
37C10 Dynamics induced by flows and semiflows
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