On the Spectrum of the Equivariant Cohomology Ring

Goresky, Mark; MacPherson, Robert

doi:10.4153/cjm-2010-016-4

Cited by 24 publications

(27 citation statements)

References 25 publications

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“…Corollary E does not hold for the usual cohomologies in general, and it has been regarded as a mysterious aspect of Springer fibers (cf. [DP81,Tan82,Car85,GM10,KP12]). Hence, our framework provides one reasonable answer to this mystery.…”

Section: Introductionmentioning

confidence: 99%

A homological study of Green polynomials*

Kato

2015

Ann. Sci. École Norm. Sup.

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We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups ([Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence ([Lusztig, Invent. Math. 75 (1984)]) in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type BC. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] §3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type A and asymptotic type BC cases, and therefore one can skip geometric sections §3-5 to see the key ideas and basic examples/techniques. * The word "green" means 'midori' in Japanese. † French translation: Uneétude homologique des polynômes de Green

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Section: Introductionmentioning

confidence: 99%

A homological study of Green polynomials*

Kato

2015

Ann. Sci. École Norm. Sup.

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“…It turns out that the semiorthogonal decomposition in Theorem C can be computed in terms of the equivariant cohomology of Springer fibers. To deduce from this the explicit form of the functors in Theorem B we use the computation of these equivariant cohomology algebras (in type A) in terms of certain linear arrangements, esttablished in [30] and [46]. Using Theorem B we are able to prove Conjecture A for some other group actions.…”

Section: (C[λ]mentioning

confidence: 99%

“…The main result of this section, Theorem 3.3.3, gives a semiorthogonal decomposition of D b W (t) into subcategories indexed by nilpotent orbits and describes these subcategories as derived categories of modules over certain algebras. In §3.4 we discuss the description (in some cases) of modules generating the subcategories of our semiorthogonal decomposition, in terms of some natural linear arrangements, which follows from a similar description of the equivariant cohomology of Springer fibers in [30] and [46]. In Sections 3.5 and 3.6 we discuss in more details the A W -modules giving our semiorthogonal decomposition in the case of type A, i.e., for the standard action of S n on A n .…”

Section: (C[λ]mentioning

confidence: 99%

“…3.1]) and Kumar-Procesi (see [46]). Note that [30] and [46] assume G to be semisimple, while we only assume it to be connected reductive, however, one can check that the constructions and arguments of [30] and [46] still go through in this case. In fact, we are mainly interested in the case G = GL n (due to the assumption ( †) below).…”

Section: Equivariant Cohomology Of the Springer Fibersmentioning

confidence: 99%

“…For example, this assumption is satisfied in the case of type A, due to the work of Spaltenstein [65]. Under the assumption ( †) there is an isomorphism of graded algebras (see [30,Thm. 3.1], [46])…”

Section: Equivariant Cohomology Of the Springer Fibersmentioning

confidence: 99%

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Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

Polishchuk

Bergh

2019

J. Eur. Math. Soc.

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We consider the derived category D b G (V ) of coherent sheaves on a complex vector space V equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G 2 , F 4 , as well as the groups G(m, 1, n) = (µm) n ⋊ Sn, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces V g /C(g), where C(g) is the centralizer subgroup of g ∈ G. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne in [23]. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on C n , where C is a smooth curve.

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Commutative Algebra of Subspace and Hyperplane Arrangements

Schenck

Sidman

2012

Commutative Algebra

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On the Spectrum of the Equivariant Cohomology Ring

Cited by 24 publications

References 25 publications

A homological study of Green polynomials*

A homological study of Green polynomials*

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

Commutative Algebra of Subspace and Hyperplane Arrangements

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