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Register a new userMinimal reduction type and the Kazhdan-Lusztig map
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Zhiwei Yun
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We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig, and that the Kazhdan-Lusztig map is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the ``skeleta of affine Springer fibers.
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We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and super duality conjecture.
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $imath$Schur duality between an $imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wangs quasi-split $imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $imath$Schur duality.
Let $F$ be a non-archimedean local field with residue field $mathbb{F}_q$ and let $G = GL_2/F$. Let $mathbf{q}$ be an indeterminate and let $H^{(1)}(mathbf{q})$ be Vigneras generic pro-p Iwahori-Hecke algebra of the p-adic group $G(F)$. Let $V_{widehat{G}}$ be the Vinberg monoid of the dual group of $G$. We establish a generic version for $H^{(1)}(mathbf{q})$ of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{widehat{G}}$ and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization $mathbf{q} = q in mathbb{C}$. At $mathbf{q} = q = 0 inoverline{mathbb{F}}_q$, the antispherical map provides a dual parametrization of all the irreducible $H^{(1)}_{overline{mathbb{F}}_q}(0)$-modules. This work supersedes our earlier work arXiv:1907.08808. We explain the relationship between the two articles in the introduction.
In this paper we show that the leading coefficients $mu(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in the affine Weyl group of type $widetilde{B_n}$ can be $n$; in the cases of types $widetilde{C_n}$ and $widetilde{D_n}$ they can be $n+1.$ Consequently, for the corresponding simply connected simple algebraic groups, the dimensions of the first extension groups between certain irreducible modules will go to infinity when $n$ increases.
Let T_n be the maximal torus of diagonal matrices in GL_n, t_n be the Lie algebra of T_n and let N_n=N_{GL_n}(T_n) be the normalizer of T_n in GL_n. Consider then the quotient stacks [t_n/N_n] and [gl_n/GL_n] for the conjugation actions. In this paper we construct an equivalence I:D_c^b([t_n/N_n]) --> D_c^b([gl_n/GL_n]) between the bounded derived categories of constructible l-adic sheaves.
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