Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolution

Su, Changjian; Zhao, Gufang; Zhong, Changlong

doi:10.48550/arxiv.1904.03769

Cited by 3 publications

(5 citation statements)

References 26 publications

(62 reference statements)

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“…For the case of X = G/B the natural Weyl group action translates simultaneously the slope and the coweight chamber C, and the resulting invariance of the stable envelopes is given by [AMSS19, Lemma 8.2a]. A detailed picture of the action of the Weyl group on the set of stable envelopes corresponding to various slopes is given in [SZZ19]. There is also another effect -the Grothendieck duality, see [AMSS19, Lemma 8.2b], which involves the change of polarization to the opposite onw.…”

Section: Stable Envelopes For the Cotangent Bundlementioning

confidence: 99%

Twisted motivic Chern class and stable envelopes

Jakub¹,

Weber²

2021

Preprint

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We present a definition of twisted motivic Chern classes for singulars pairs (X, ∆) consisting of a singular space X and a Q-Cartier divisor containing the singularities of X. The definition is a mixture of the construction of motivic Chern classes previously defined by Brasselet-Schürmann-Yokura with the construction of multiplier ideals. The twisted motivic Chern classes are the limits of the the elliptic classes defined by Borisov-Libgober. We show that with a suitable choice of the divisor ∆ the twisted motivic Chern classes satisfy the axioms of the stable envelopes in K-theory. Our construction is an extension of the results proven by the first author for the fundamental slope.Both authors are supported by NCN grant 2016/23/G/ST1/04282 (Beethoven 2).

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Section: Stable Envelopes For the Cotangent Bundlementioning

confidence: 99%

Twisted motivic Chern class and stable envelopes

Jakub¹,

Weber²

2021

Preprint

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“…The wall R-matrices are important objects of geometric representation theory. They were investigated for X given by Springer resolutions in [31]. It was shown that the action of wall R-matrices generate the action of the affine Hecke algebra on K T (X) is this case.…”

Section: 1mentioning

confidence: 99%

Pursuing quantum difference equations II: 3D-mirror symmetry

Yakov¹,

Smirnov²

2020

Preprint

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Let X and X ! be a pair of symplectic varieties dual with respect to 3D-mirror symmetry. The K-theoretic limit of the elliptic duality interface is an equivariant K-theory class m ∈ K(X × X ! ). We show that this class provides correspondencesmapping the K-theoretic stable envelopes to the K-theoretic stable envelopes. This construction allows us to extend the action of various representation theoretic objects on K(X), such as action of quantum groups, quantum Weyl groups, R-matrices etc., to their action on K(X ! ). In particular, we relate the wall R-matrices of X to the Rmatrices of the dual variety X ! .As an example, we apply our results to X = Hilb n (C 2 ) -the Hilbert scheme of n points in the complex plane. In this case we arrive at the conjectures of E.Gorsky and A.Negut from [13].

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“…In this section, we study the wall crossing R-matrices for the Springer resolution, which will be used for the categorification in the next section. The main reference for these two sections is [SZZ19].…”

Section: Unramified Principle Series Of P-adic Langlands Dual Groupmentioning

confidence: 99%

“…Changing alcoves defines the so-called wall R-matrices ( [OS16]). We first recall these wall R-matrices for the Springer resolution, which are computed in [SZZ19]. The formulae can be nicely packed using the Hecke algebra actions.…”

Section: Introductionmentioning

confidence: 99%