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Study of perverse sheaves arising from graded Lie algebras. (English) Zbl 1001.17033

From the introduction: We consider a connected complex reductive group \(G\) acting on a complex algebraic variety \(X\). Some examples of such \((G,X)\) are:
(a) \(X\) is the product of two copies of the flag manifold of \(G\) (the action of \(G\) is diagonal);
(b) \(G\) is the identity component of the centralizer of an involution in a larger reductive group \(\widetilde{G}\) and \(X\) is the flag manifold of \(\widetilde{G}\);
(c) \(X\) is the moduli space of representations of prescribed dimension of a quiver of type \(A\), \(D\) or \(E\) and \(G\) is a product of general linear groups acting naturally on \(X\), so that its orbits are the isomorphism classes of representations of the quiver;
(d) \(G\) is the centralizer of a 1-parameter subgroup \(\iota: \mathbb{C}^*\to \widetilde{G}\) where \(\widetilde{G}\) is a larger reductive group, and \(X\) is the subspace of the Lie algebra of \(\widetilde{G}\) on which \(\text{Ad}(\iota(t))\) acts as \(t^2\) times identity for any \(t\in \mathbb{C}^*\);
(e) \(X\) is the variety of nilpotent elements in the Lie algebra of \(G\);
(f) \(X=G\) and \(G\) acts by conjugation.
In each of these examples, there is a natural subclass \({\mathcal M}'\) of the class \({\mathcal M}\) of all simple \(G\)-equivariant perverse sheaves on \(X\): in examples (a)–(e), \(G\) acts with finitely many orbits, and we take \({\mathcal M}'={\mathcal M}\); in example (f), \({\mathcal M}'\) is a proper subset of \({\mathcal M}\) (unless \(G=\{1\}\)) consisting of character sheaves.
In each case, the problem of determining the structure of the cohomology sheaves of the perverse sheaves in \({\mathcal M}'\) is of considerable interest since there is much representation theoretic information encoded in this structure. (We refer to this problem in case (a),(b),… as \(P(a), P(b),\dots\).)
For example, in the same way as examples (a) (resp. (b)) are related in the well-known way to representations of complex (resp. real) groups, the example (d) is related (at least conjecturally) to representations of \(p\)-adic groups. On the other hand, examples (e), (f) are significant for the computation of complex characters of finite reductive groups and example (c) is significant for the study of canonical bases.
The problem stated above has been already solved in all examples above, except for \(P(d)\). (See the references in the paper for details.)
This paper is concerned with the problem \(P(d)\) in the general case. We note that for \(G\) of type \(A\), \(P(d)\) is equivalent to \(P(c)\) for quivers of type \(A\). We therefore take the point of view that \(P(d)\) should be regarded as a generalization of \(P(c)\) and that we should try to imitate the methods used in the definition of canonical bases in earlier papers.
The contents of the paper is as follows: Preliminaries on perverse sheaves, Preliminaries on nilpotent orbits, The module \({\mathcal K}({\mathfrak g}_n)\), The \(n\)-rigid case, The parabolic subalgebra associated to \(x\in {\mathfrak g}_n\), Induction, Normal complexes, Restriction, Adjunction, Fourier-Deligne transform, Primitive pairs in \({\mathcal I} ({\mathfrak g}_n)\), Primitive pairs and Fourier transform, Generation by quasi-monomial objects, Property \(\star\), Inner product formula, The module \(\mathbb{K}({\mathfrak g}_{\pm n})\), The isomorphism \(\mathbb{K}({\mathfrak g}_{\pm n})\cong {\mathcal K}({\mathfrak g}_n)\otimes \mathbb{Q}(v)\), Purity, Computation of an inner product, Blocks in \({\mathcal I}({\mathfrak g}_n)\), Parity, A Lagrangian variety.

MSC:

17B70 Graded Lie (super)algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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