Exotic Symmetric Spaces of Higher Level: Springer Correspondence for Complex Reflection Groups

Shoji, Toshiaki

doi:10.1007/s00031-015-9350-9

Cited by 8 publications

(20 citation statements)

References 8 publications

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“…In general, the partition (8.3.1) is not compatible with closure relations, namely, the closure X λ is not a union of pieces X µ . However, we have a somewhat weaker result ([S3,Prop. 5.11]); for each λ ∈ P n,r , we have…”

Section: 11contrasting

confidence: 56%

“…The following results give the Springer correspondence for X uni , the former one is with respect to W ♮ m , the latter one is with respect to W n,r . Theorem 7.12 ( [S3,Thm. 7.12,Thm.…”

Section: 11mentioning

confidence: 99%

“…Theorem 8.2([S3, Thm. 4.5]).For each m ∈ Q n,r , π (m) * Ql [d m ]is a semisimple perverse sheaf on X m equipped with S m -action, and is decomposed asπ (m) * Ql [d m ] ≃ ρ∈S ∧ m ρ ⊗ IC(X m , L ρ )[d m ],…”

mentioning

confidence: 99%

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Springer correspndence for complex reflection groups

Shoji

2015

Preprint

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Section: 11contrasting

confidence: 56%

“…The following results give the Springer correspondence for X uni , the former one is with respect to W ♮ m , the latter one is with respect to W n,r . Theorem 7.12 ( [S3,Thm. 7.12,Thm.…”

Section: 11mentioning

confidence: 99%

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Springer correspndence for complex reflection groups

Shoji

2015

Preprint

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“…Let X λ be the closure of X λ in X uni . Then by a similar argument as in the proof of [S3,Lemma 6.17], we have…”

Section: Unipotent Varietymentioning

confidence: 68%

“…For z ∈ X F λ , we have, by Theorem 4.8 (and by [S3,Prop. 8.16]), are F -stable, and this scalar is given by q d λ .…”

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confidence: 91%

Enhanced Variety of Higher Level and Kostka Functions Associated to Complex Reflection Groups

Shoji

2017

Transformation Groups

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Abstract. Let V be an n-dimensional vector space over an algebraic closure of a finite field Fq, and G = GL(V ). A variety X = G × V r−1 is called an enhanced variety of level r. Let Xuni = Guni × V r−1 be the unipotent variety of X . We have a partition Xuni = λ X λ indexed by r-partitions λ of n. In the case where r = 1 or 2, X λ is a single G-orbit, but if r ≥ 3, X λ is, in general, a union of infinitely many G-orbits. In this paper, we prove certain orthogonality relations for the characteristic functions (over Fq) of the intersection cohomology IC(X λ ,Q l ), and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group Sn ⋉ (Z/rZ) n . IntroductionLet V be an n-dimensional vector space over an algebraic closure of a finite field F q , andhave a geometric interpretation in terms of the intersection cohomology associated to the closure of unipotent classes in G in the following sense. Let C λ be the unipotent class corresponding to a partition λ of n, and K = IC(C λ ,Q l ) be the intersection cohomology complex on the closure C λ of C λ . He proved that H i K = 0 for odd i, and that for partitions λ, µ of n,where x ∈ C µ ⊂ C λ , and n(λ) is the usual n-function. Kostka polynomials are polynomials indexed by a pair of partitions. In [S1], [S2], as a generalization of Kostka polynomials, Kostka functions K λ,µ (t) associated to the complex reflection group S n ⋉ (Z/rZ) n were introduced, which are a-priori rational functions in Q(t) indexed by r-partitions λ, µ of n (see 3.10 for the definition of r-partitions of n). It is known by [S2] that K λ,µ (t) are actually polynomials if r = 2. Although those Kostka functions are defined in a purely combinatorial way, and have no geometric background, recently various generalizations of Lusztig's result for those Kostka functions were found. Under the notation above, consider a variety X = G × V , which is called the enhanced variety, and its subvariety X uni = G uni × V is isomorphic to the enhanced nilpotent cone introduced by Achar-Henderson [AH] (here G uni is the unipotent variety of G). The set of G-orbits under the diagonal action of G on X uni is parametrized by double partitions of n ([AH], [T]). In [AH], they proved that Kostka polynomials indexed by double partitions have a geometric interpretation as in (*) in terms of the intersection cohomology associated to the closure of G-orbits in X uni .On the other hand, let V be a 2n-dimensional symplectic vector space over an algebraic closure of F q with ch F q = 2, and consider G = GL(V ) ⊃ H = Sp(V ). The variety As a generalization of the enhanced variety G × V or the exotic symmetric space G/H × V , we consider G × V r−1 or G/H × V r−1 for any r ≥ 1. G acts diagonally on G × V r−1 , and H acts diagonally on G/H × V r−1 . X = G × V r−1 is called the enhanced variety of level r, and a certain H-stable subvariety X of G/H × V r−1 is called the exotic symmetric space of level r. For those varieties X , one can consid...

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Kostka functions associated to complex reflection groups

Shoji

2017

Journal of Algebra

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Exotic Symmetric Spaces of Higher Level: Springer Correspondence for Complex Reflection Groups

Cited by 8 publications

References 8 publications

Springer correspndence for complex reflection groups

Springer correspndence for complex reflection groups

Enhanced Variety of Higher Level and Kostka Functions Associated to Complex Reflection Groups

Kostka functions associated to complex reflection groups

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