Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

Bump, Daniel; Nakasuji, Maki

doi:10.4153/cjm-2018-011-1

Cited by 9 publications

(10 citation statements)

References 19 publications

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“…if and only if the Schubert variety Y (u) is smooth at the torus fixed point e w . This is the geometric analogue of a conjecture of Bump and Nakasuji [BN11,BN19] for simply laced types, generalized to all types by Naruse [Nar14], and further analyzed by Nakasuji and Naruse [NN16]. While this paper was in preparation, Naruse informed us that he also obtained an (unpublished) proof of the implication assuming factorization.…”

Section: Introductionmentioning

confidence: 65%

“…As in the authors' previous work on CSM classes [AMSS17], the connections to Hecke algebras and K-theoretic stable envelopes yield remarkable identities among (Poincaré duals of) motivic Chern classes. We use these identities to prove two conjectures of Bump, Nakasuji and Naruse [BN11, BN19,NN16] about the coefficients in the transition matrix between the Casselman's basis and the standard basis in the Iwahori-invariant space of the principal series representation for an unramified character for a group over a non archimedean local field.…”

Section: Introductionmentioning

confidence: 99%

“…A second conjecture refers to a holomorphy property. In relation to Kazhdan-Lusztig theory, Bump and Nakasuji [BN19] defined the coefficients ru,w := u≤x≤w (−1) ℓ(x)−ℓ(u) mx,w , where the bar operator replaces q ′ by q ′−1 . Using Möbius inversion it follows that the coefficients a u,w from (1.2) satisfy āu,w = ru,w .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Aluffi¹,

Mihalcea²,

Schürmann³

et al. 2019

Preprint

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Motivic Chern classes are elements in the K-theory of an algebraic variety X, depending on an extra parameter y. They are determined by functoriality and a normalization property for smooth X. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds G/B. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of G, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Fehér, Rimányi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the Poincaré duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis. 1 The class SM Cy(Ω) := D(M Cy (Ω)) λy(T * (G/B)) may be regarded as the motivic analogue of the Segre-MacPherson class c * (1 1 Ω ) c(T (G/B)) .

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Aluffi¹,

Mihalcea²,

Schürmann³

et al. 2019

Preprint

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“…The following result was shown in [BN19] by induction on (w). We will prove it bijectively in Proposition 7.3 below.…”

Section: Arbitrary Permutations Kazhdan-lusztig Polynomials and Posit...mentioning

confidence: 85%

Symmetries of stochastic colored vertex models

Galashin¹

2020

Preprint

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We discover a new property of the stochastic colored six-vertex model called flip invariance. We use it to show that for a given collection of observables of the model, any transformation that preserves the distribution of each individual observable also preserves their joint distribution. This generalizes recent shift-invariance results of Borodin-Gorin-Wheeler. As limiting cases, we obtain similar statements for the Brownian last passage percolation, the Kardar-Parisi-Zhang equation, the Airy sheet, and directed polymers. Our proof relies on an equivalence between the stochastic colored six-vertex model and the Yang-Baxter basis of the Hecke algebra. We conclude by discussing the relationship of the model with Kazhdan-Lusztig polynomials and positroid varieties in the Grassmannian.

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“…Pulling back the stable basis to the flag variety from its cotangent bundle, we get the motivic Chern classes [BSY10] of the Schubert cells [AMSS19,FRW21]. This connection is used to prove a series of conjectures about the Casselman basis, see [AMSS19,BN11,BN19].…”

Section: Su G Zhao and C Zhongmentioning

confidence: 99%

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

Su¹,

Zhao²,

Zhong³

2021

Compositio Math.

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We compare the $K$ -theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$ -theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

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Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

Cited by 9 publications

References 19 publications

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Symmetries of stochastic colored vertex models

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

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