arXiv subjects

Nanhua Xi

Publications and source records attributed to Nanhua Xi.

16 recordsLinked to original sources

arXiv2022

The based rings of two-sided cells in an affine Weyl group of type $\tilde B_3$, II

We compute the based rings of two-sided cells corresponding to the unipotent classes in $Sp_6(\mathbb C)$ with Jordan blocks (33), (411), (222) respectively. The results for the first two two-sided cells also verify Lusztig's conjecture on the structure of the based rings of two-sided cells of an affine Weyl group. The result for the last two-sided cell partially suggests a modification of Lusztig's conjecture on the structure of the based rings of two-sided cells of an affine Weyl group.

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arXiv2021

The based rings of two-sided cells in an affine weyl group of type $\tilde B_3$, I

For type $\tilde B_3$ we show that Lusztig's conjecture on the structure of the based ring of two-sided cell corresponding to the unipotent class in $Sp_6(\mathbb C)$ with 3 equal Jordan blocks needs modified.

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arXiv2016

Canonical Left Cells and the Lowest Two-sided Cell in an Affine Weyl Group

We give some discussions to the relations between canonical left cells and the lowest two-sided cell of an affine Weyl group. In particular, we use the relations to construct irreducible modules attached to the lowest two-sided cell and some one dimensional representations of an affine Hecke algebra.

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arXiv2015

The based ring of the lowest two-sided cell of an affine Weyl group, III

We show that Lusztig's homomorphism from an affine Hecke algebra to the direct summand of its asymptotic Hecke algebra corresponding to the lowest two-sided cell is related to the homomorphism constructed by Chriss and Ginzburg using equivariant K-theory by a matrix over the representation ring of the associated algebraic group.

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arXiv2014

Some Infinite Dimensional Representations of Reductive Groups with Frobenius Maps

In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.

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arXiv2012

Module Structure on Invariant Jacobians

In this paper we show that a conjecture of Stephen Yau on highest weights of invariant Jacobians is true for arbitrary connected semisimple algebraic groups.

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arXiv2011

Iwahori's Question for Affine Hecke Algebras

In this paper we show that an affine Hecke algebra $H_q$ over complex numbers field with parameter $q\ne 1$ is not isomorphic to the group algebra over complex numbers field of the corresponding extend affine Weyl group if the corresponding root system has no factors of type $A_1$ and the order of $q$ is different from 11 and 13 if the root system has factors of type $E_8$.

math.QA↗

arXiv2011

Lusztig's $a$-function for Coxeter groups with complete graphs

We show that Lusztig's $a$-function of a Coxeter group is bounded if the Coxeter group has a complete graph (i.e. any two vertices are joined) and the cardinalities of finite parabolic subgroups of the Coxeter group have a common upper bound.

math.QA↗

arXiv2010

Some Non-Trivial Kazhdan-Lusztig Coefficients of an Affine Weyl Group of Type $\tilde A_n$

In this paper we show that the leading coefficient $μ(y,w)$ of some Kazhdan-Lusztig polynomials $P_{y,w}$ with $y,w$ in an affine Weyl group of type $\tilde A_n $ is $n+2$. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.

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arXiv2010

Isomorphisms between Two Quantized Enveloping Algebras of the Same Type

The paper has been withdrawn.

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arXiv2008

Maximaland Primitive Elements in Baby Verma Modules for Type $B_2$

The purpose of this paper is to find maximal and primitive elements of baby Verma modules for a quantum group of type $B_2$. As a consequence the composition factors of the baby Verma modules are determined. Similar approach can be used to find find maximal and primitive elements of Weyl modules for type $B_2$.

math.QA↗

arXiv2008

A New Proof for Classification of Irreducible Modules of a Hecke Algebra of Type $ A_n$

In this paper we give a new proof for the classification of irreducible modules of an affine Hecke algebra of type $A_n$, which was obtained by G. E. Murphy in 1995.

math.QA↗

arXiv2008

Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II

An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by rational numbers field. This proves a weak form of a conjecture of Ginzburg proposed in 1987.

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arXiv2004

Kazhdan-Lusztig basis and a geometric filtration of an affine Hecke algebra

According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. In this paper we will show in the case of type $A$ that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

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arXiv2004

The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group $S_n$

In this paper we show that the leading coefficient $μ(y,w)$ of certain Kazhdan-Lusztig polynomials $P_{y,w}$ of the permutation group $\mathfrak S_n$ of 1,2,...,n are not greater than 1. More precisely, we show that the leading coefficients $μ(y,w)$ are not greater than 1 whenever $a(y)< a(w)$, where $a: \mathfrak S_n\to\mathbf N$ is the function defined by Lusztig.

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arXiv1998

Monomial Bases of Quantized Enveloping Algebras

We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of the positive part.

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