[Paper Review] Lusztig's $a$-function for Coxeter groups with complete graphs

Nanhua Xi|arXiv (Cornell University)|Jun 13, 2011

Advanced Combinatorial Mathematics11 references16 citations

TL;DR

This paper establishes that Lusztig’s $a$-function is bounded for Coxeter groups with complete Coxeter graphs (i.e., all pairs of simple reflections non-commuting) when the finite parabolic subgroups have uniformly bounded cardinality. The key result shows that such groups possess a lowest two-sided cell, and the $a$-function is bounded by the length of the longest element in the maximal finite parabolic subgroups, generalizing known results for affine Weyl groups and crystallographic groups.

ABSTRACT

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.

Motivation & Objective

• To investigate the boundedness of Lusztig’s $a$-function in Coxeter groups with complete Coxeter graphs.
• To determine conditions under which the $a$-function is bounded by the length of the longest element in finite parabolic subgroups.
• To establish the existence of a lowest two-sided cell in such Coxeter groups.
• To generalize known results on $a$-function boundedness from affine Weyl groups and crystallographic groups to a broader class of Coxeter groups.

Proposed method

• Use of the Kazhdan-Lusztig basis and structure constants $h_{w,u,v}$ in the Hecke algebra to analyze the $a$-function via degree in $q^{1/2}$.
• Application of the equivalence $f_{w,u,v} = f_{u,v^{-1},w^{-1}}$ and polynomial degree bounds to relate $a(v)$ to the maximal degree of structure constants.
• Leveraging the Bruhat order and the existence of longest elements in finite parabolic subgroups to derive bounds on $a(w)$.
• Use of the positivity of Kazhdan-Lusztig polynomials and the structure of distinguished involutions to analyze cell decomposition.
• Proof by contradiction and cell comparison via the $LR$-preorder to show that elements with maximal $a$-value lie in the lowest two-sided cell.
• Application of results from [KL] and [L3] on cell equivalence and $ ilde{T}_w$ multiplication to derive bounds on $a(v)$.

Experimental results

Research questions

• RQ1 Under what conditions is Lusztig’s $a$-function bounded for a Coxeter group with a complete Coxeter graph?
• RQ2 Can the $a$-function be bounded by the length of the longest element in finite parabolic subgroups for such groups?
• RQ3 Does the existence of a bounded $a$-function imply the existence of a lowest two-sided cell in the group?
• RQ4 How does the structure of finite parabolic subgroups of rank 2 influence the number of two-sided cells in a complete Coxeter group?
• RQ5 Is the number of two-sided cells in a crystallographic Coxeter group with a complete graph equal to $|O| + 2$, where $|O|$ is the number of isomorphism classes of rank-2 finite parabolic subgroups?

Key findings

• The $a$-function is bounded for any Coxeter group with a complete Coxeter graph and uniformly bounded finite parabolic subgroups.
• The maximal value of the $a$-function is equal to the length of the longest element in the maximal finite parabolic subgroups.
• The group has a lowest two-sided cell, which contains all elements $w$ with $a(w)$ maximal.
• The number of two-sided cells in a crystallographic Coxeter group with a complete graph is $|O| + 2$, where $|O|$ is the number of isomorphism classes of rank-2 finite parabolic subgroups.
• For rank-3 Coxeter groups with complete graphs, the $a$-function is bounded, and the group has exactly three two-sided cells: $\{e\}$, elements with unique reduced expressions, and elements with multiple reduced expressions.
• The $a$-function is bounded by the length of the longest element in any finite parabolic subgroup, and this bound is sharp when the subgroup is maximal.