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A generalization of Springer theory using nearby cycles. (English) Zbl 0938.22011
Let \(f: {\mathbb C}^d \rightarrow {\mathbb C}\) be a non-constant polynomial. Then the complex cohomology groups of the Milnor fibre at each point fit together to give a constructible bounded complex, \(P\), of sheaves, which is called the sheaf of nearby cycles on \(f\). It is a perverse sheaf in the sense of intersection homology [M. Goresky and R. MacPherson, Astérisque 101-102, 135-192 (1983; Zbl 0524.57022)]. The definition of the sheaf of nearby cycles, \(P\), generalises to the case of a dominant map, \(f: {\mathbb C}^d \rightarrow {\mathbb C}^r\) and the author studies the structure of \(P\) when the components of \(f\) are given by homogeneous polynomials. The motivation comes from Springer theory, which studies the singularities of the adjoint quotient map, \({\mathcal G} \rightarrow G \backslash\backslash {\mathcal G}\) of a complex semisimple Lie algebra, \({\mathcal G}\). For the class of polar representations, \(V\), of \(G\) (satisfying an extra orbit finiteness condition called visibility) the author’s main result states that the nearby cycles complex for the quotient map, \(V \rightarrow G \backslash\backslash V\), satisfies \[ {\mathcal F}P \cong {\mathcal IC}((V^{*})^{rs} , {\mathcal L}) . \] Here \({\mathcal F}\) is the Fourier transform functor of R. Hotta and M. Kashiwara [Invent. Math. 75, 327-358 (1984; Zbl 0538.22013)]. Using this description the author establishes a number of other interesting results, including a description of the holonomy of the local system, \({\mathcal L}\), and the monodromy action on \(P\).

MSC:
22E46 Semisimple Lie groups and their representations
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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