Exotic Symmetric Space Over a Finite Field, Ii

Shoji, Toshiaki; Sorlin, Karine

doi:10.1007/s00031-014-9272-y

Cited by 10 publications

(12 citation statements)

References 15 publications

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“…By comparing (2.1.2*) with (1.3.1) in [SS2], we obtain a formula which is a replacement of (2.1.5) in [SS2].…”

Section: Appendixmentioning

confidence: 99%

“…We consider the variety X = G ιθ × V on which H acts diagonally. In [SS], [SS2], the intersection cohomology complexes associated to H-orbits on X were studied. In particular, the set of character sheaves X on X was defined in [SS] as a certain set of H-equivariant simple perverse sheaves on X .…”

Section: Introductionmentioning

confidence: 99%

“…such that F * A ≃ A). It was shown in [SS2] that the set of characteristic functions of character sheaves in X F forms a basis of the space C q (X ) of H F -invariant functions on X F , as far as q is large enough.…”

Section: Introductionmentioning

confidence: 99%

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Exotic Symmetric Space Over a Finite Field, Iii

Shoji

Sorlin

2014

Transformation Groups

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Let X = G/H × V , where V is a symplectic space such that G = GL(V ) and H = Sp(V ). In previous papers, the authors constructed character sheaves on X , based on the explicit data. On the other hand, there exists a conceptual definition of character sheaves on X based on the idea of Ginzburg in the case of symmetric spaces. Our character sheaves form a subset of Ginzburg type character sheaves. In this paper we show that these two definitions actually coincide, which implies a classification of Ginzburg type character sheaves on X . * supported by ANR JCJC REPRED, ANR-09-JCJC-0102-01.But by comparing (1.3.1) and (3.5.2) in Appendix (or in [SS]), such A is exactly the same A appearing in the decomposition of (ψ m ) * α * 0 E. Then by the discussion in the proof of (3.6.1*) in Appendix, we see that A| Y m−1 gives no contribution to the former factors in (3.6.1*) for (ψ m−1 ) * α * 0 E. Hence it gives no contribution for the former factors of (1.3.3) as asserted. This proves (1.3.3) for m.By considering the case where m = n ′ , we obtain the following result, which is an analogue of Proposition 3.6 in [SS] (see Appendix).is a semisimple complex on Y ′ , equipped with W ′ Eaction, and is decomposed asWe consider θ-stable parabolic subgroups P ⊂ Q of G. We follow the notation in 4.2. In particular,Then V L is a quotient of V ′′ as P θ -module. The following transitivity holds.Proposition 5.2. Let K ∈ D H (X ). Then there is an isomorphism in D P θ (X L ) Res X X L ,V ′′ ,P K ≃ (Res X M X L ,V ′ M ,P M • Res X X M ,V ′ ,Q )K.

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“…By comparing (2.1.2*) with (1.3.1) in [SS2], we obtain a formula which is a replacement of (2.1.5) in [SS2].…”

Section: Appendixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exotic Symmetric Space Over a Finite Field, Iii

Shoji

Sorlin

2014

Transformation Groups

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“…Following Kato's foundational papers on the exotic nilpotent cone, there has been subsequent work extending various results about the nilpotent cone to the exotic setting. In , Achar and Henderson conjecturally describe the intersection cohomology of orbit closures in the exotic nilpotent cone; these have since been proven independently by Shoji–Sorlin (see [, Theorem 5.7]); and by Kato (see [, Theorem A.1.8; , Theorem A; , Remark 5.8]). Achar, Henderson and Sommers make an explicit connection between special pieces for

frakturN

and those for the ordinary nilpotent cone in .…”

Section: Introductionmentioning

confidence: 99%

Irreducible components of exotic Springer fibres

Nandakumar

Rosso

Saunders

2018

J. London Math. Soc.

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Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. Contents

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“…This paper is a survey on a joint work with K. Sorlin [SS1], [SS2] concerning the theory of character sheaves on the exotic symmetric space. The theory of character sheaves on reductive groups was established by Lusztig [L2] in 1980's for computing irreducible characters of finite reductive groups in a uniform way.…”

Section: §1 Introductionmentioning

confidence: 99%

Character sheaves on exotic symmetric spaces and Kostka polynomials

Shoji¹

Schubert Calculus — Osaka 2012

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This paper is a survey on a joint work with K. Sorlin concerning the theory of character sheaves on the exotic symmetric space. After explaining the historical background, we introduce character sheaves on this variety. By using those character sheaves, we show that modified Kostka polynomials, indexed by a pair of double partitions, can be interpreted in terms of the intersection cohomology of the orbits in the exotic nilpotent cone, as conjectured in Achar-Henderson.

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Exotic Symmetric Space Over a Finite Field, Ii

Cited by 10 publications

References 15 publications

Exotic Symmetric Space Over a Finite Field, Iii

Exotic Symmetric Space Over a Finite Field, Iii

Irreducible components of exotic Springer fibres

Character sheaves on exotic symmetric spaces and Kostka polynomials

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