[Paper Review] Lusztig's $a$-function in type $B_n$ in the asymptotic case

Meinolf Geck, Lăcrimioara Iancu|ArXiv.org|Apr 11, 2005

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper explicitly determines Lusztig’s $\mathbf{a}$-function in the asymptotic case for the Weyl group of type $B_n$ using leading matrix coefficients and the generalized Robinson–Schensted correspondence. It confirms all of Lusztig’s conjectural properties (P1)–(P15), except possibly (P9), (P10), and (P15), and establishes a precise correspondence between the two-sided cell ideal structure and the Dipper–James–Murphy basis in the Iwahori–Hecke algebra $\mathcal{H}_n$.

ABSTRACT

In this paper, we study Lusztig's $a$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the ``asymptotic case'' in type $B_n$, where the left cells have been determined in terms of a generalized Robinson--Schensted correspondence by Bonnafé and the second author. As a consequence, we can also show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the ``leading matrix coefficients'' introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori--Hecke algebra $\bH_n$ in terms of the Dipper--james--Murphy basis of $\bH_n$.

Motivation & Objective

• Address the lack of explicit determination of Lusztig’s $\mathbf{a}$-function in the unequal parameter setting for type $B_n$.
• Confirm Lusztig’s conjectural properties (P1)–(P15) for the asymptotic case in type $B_n$, except possibly (P9), (P10), and (P15).
• Establish a precise correspondence between the two-sided cell ideals in the Iwahori–Hecke algebra $\mathcal{H}_n$ and the Dipper–James–Murphy basis.
• Provide a structural description of the ring $J$ associated with the asymptotic algebra in type $B_n$.
• Utilize leading matrix coefficients to analyze cell structures and $\mathbf{a}$-function values in the asymptotic regime.

Proposed method

• Apply the theory of leading matrix coefficients introduced by Geck to analyze the structure of the Iwahori–Hecke algebra $\mathcal{H}_n$ in the asymptotic case.
• Use the generalized Robinson–Schensted correspondence for type $B_n$ to classify left cells and determine the $\mathbf{a}$-function values via cell decomposition.
• Construct the asymptotic algebra $J$ as a quotient of $\mathcal{H}_n$ and analyze its two-sided cell ideals using the Dipper–James–Murphy basis.
• Establish a ring isomorphism $J_\lambda \cong M_2(\mathbb{Z})$ for the two-cell case by identifying basis elements with matrix units under the orthogonal representation.
• Prove that the two-sided cell ideals $N^\lambda$ in $\mathcal{H}_n$ coincide with the ideals $M^\lambda$ generated by Kazhdan–Lusztig basis elements corresponding to RS-cells of shape $\nu$ with $\lambda \trianglelefteq \nu$.
• Use rank comparison and freeness of modules over $A$ to conclude equality of ideals $N^\lambda = M^\lambda$ after tensoring with the field of fractions $K_0$.

Experimental results

Research questions

• RQ1 What is the explicit value of Lusztig’s $\mathbf{a}$-function for all elements in the Weyl group of type $B_n$ under the asymptotic parameter conditions?
• RQ2 Which of Lusztig’s conjectural properties (P1)–(P15) hold in the asymptotic case for type $B_n$, and which remain open?
• RQ3 How do the two-sided cell ideals in the Iwahori–Hecke algebra $\mathcal{H}_n$ relate to the Dipper–James–Murphy basis?
• RQ4 What is the structure of the ring $J$ associated with the asymptotic algebra in type $B_n$?
• RQ5 Can the leading matrix coefficient method be used to verify the cell decomposition and $\mathbf{a}$-function values in the asymptotic regime?

Key findings

• Lusztig’s $\mathbf{a}$-function is explicitly determined for all elements in the Weyl group of type $B_n$ under the asymptotic parameter conditions.
• All of Lusztig’s conjectural properties (P1)–(P15) hold in the asymptotic case for type $B_n$, except possibly (P9), (P10), and (P15), as confirmed by subsequent work.
• The two-sided cell ideals $N^\lambda$ in the Iwahori–Hecke algebra $\mathcal{H}_n$ coincide exactly with the ideals $M^\lambda$ generated by Kazhdan–Lusztig basis elements corresponding to RS-cells of shape $\nu$ with $\lambda \trianglelefteq \nu$.
• The ring $J_\lambda$ associated with a two-cell corresponds isomorphically to $M_2(\mathbb{Z})$, with basis elements mapping to matrix units under the orthogonal representation.
• The ideal structure defined by two-sided cells in the asymptotic case matches precisely the ideal structure given by the Dipper–James–Murphy basis in $\mathcal{H}_n$.
• The rank of the ideal $N^\lambda$ as a free $A$-module equals $\sum_{\lambda \trianglelefteq \nu} d_\nu^2$, matching the rank of the ideal $M^\lambda$, which implies $N^\lambda = M^\lambda$ after tensoring with the field of fractions $K_0$.