Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2

Xue, Ting

doi:10.1016/j.aim.2011.11.010

Cited by 8 publications

(5 citation statements)

References 23 publications

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“…2) There are other Springer correspondences (see e.g. Xue [Xue12]). It might be interesting to see whether they give rise to Kostka systems, and how they are related with those in this paper.…”

Section: Kostka Systems Arising From Reductive Groupsmentioning

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: Kostka Systems Arising From Reductive Groupsmentioning

confidence: 99%

A homological study of Green polynomials*

Kato

2015

Ann. Sci. École Norm. Sup.

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“…. By [L1, Prop.1.2] (actually a similar argument works also for the Lie algebra case, see [X2,Prop. 3…”

Section: 4mentioning

confidence: 84%

Symmetric spaces associated to classical groups with even characteristic

Dong¹,

Shoji²,

Gao³

2018

Preprint

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Let G = GL(V ) for an N -dimensional vector space V over an algebraically closed field k, and G θ the fixed point subgroup of G under an involution θ on G. In the case where G θ = O(V ), the generalized Springer correpsondence for the unipotent variety of the symmetric space G/G θ was described in [SY], assuming that ch k = 2. The definition of θ given there, and of the symmetric space arising from θ, make sense even if ch k = 2. In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if N is even, that the Springer correspondence is reduced to that of symplectic Lie algebras in ch k = 2, which was determined by Xue. While if N is odd, the number of G θ -orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level r = 3.

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“…By the result of [Spa82] (see also [Xue12a, Section 8]), the Springer correspondence of Lie Sp 2n (F 2 ) is governed by combinatorics of Z2n+2,n+1 . (Note that the convention of [Xue12a] is slightly different from ours in a sense that (r, s) = (n + 1, n + 1) therein is equivalent to (r, s) = (2n+2, n+1) in our setting.) Then one can follow the argument of [Lus86, Section 24] or equivalently [Ach11, Section 6].…”

Section: Equality Of Total Springer Representationsmentioning

confidence: 97%

“…(This follows from the existence of the isogeny SO 2n+1 → Sp 2n in characteristic 2.) The Springer correspondences for the cases above are already known; see [Lus84] for classical types in good characteristic, [LS85] for classical Lie groups in characteristic 2, and [Spa82] and [Xue12a] for classical Lie algebras and their duals in characteristic 2.…”

Section: Introductionmentioning

confidence: 99%

On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

Kim¹

2019

Preprint

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Let W be the Weyl group of type BCn. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of W which is of type BC n−1 . Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.

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Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2

Abstract: We describe the Springer correspondence explicitly for exceptional Lie algebras of type G 2 and F 4 and their duals in bad characteristics, i.e. in characteristics 2 and 3.

Cited by 8 publications

References 23 publications

A homological study of Green polynomials*

A homological study of Green polynomials*

Symmetric spaces associated to classical groups with even characteristic

On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

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