[Paper Review] Constructible characters and b-invariants

Cédric Bonnafé|arXiv (Cornell University)|Feb 12, 2013

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper establishes that in finite Coxeter groups with arbitrary weight functions, each Lusztig $φ$-family contains a unique irreducible character with minimal $b$-invariant, and each $φ$-constructible character has a unique irreducible constituent with minimal $b$-invariant. These results generalize Lusztig's theory of special characters to non-constant weight functions and provide strong evidence for the conjectural equivalence between Kazhdan-Lusztig and Calogero-Moser cellular structures.

ABSTRACT

To each finite Coxeter system (W,S) and to each weight function L, Lusztig has defined the notions of constructible characters and of Lusztig families of W, using the so-called J-induction. Whenever L is constant, and using a general argument, Lusztig has shown that all Lusztig family contains a unique character with minimal b-invariant, and that every constructible character contains an irreducible constituent with minimal b-invariant. We show in this paper that this can be generalized to the case where L is not constant: our proof is by a case-by-case analysis.

Motivation & Objective

• To extend Lusztig's theory of special characters—previously valid only for constant weight functions—to the general case of arbitrary positive weight functions on finite Coxeter groups.
• To investigate the compatibility of the $b$-invariant, a key invariant in representation theory, with the conjectural equivalence between Kazhdan-Lusztig and Calogero-Moser cellular structures.
• To verify that the minimal $b$-invariant property, known to hold for Calogero-Moser cellular characters, also holds for Lusztig's $φ$-constructible characters and their families.
• To provide structural evidence supporting the conjecture that Lusztig's $φ$-constructible characters coincide with Calogero-Moser $φ$-cellular characters and their families.

Proposed method

• Define the $b$-invariant of an irreducible character as the valuation of its fake degree polynomial $f_{\chi}(\mathbf{t})$.
• Use the graph $\mathcal{G}_{W,\varphi}^{\mathrm{Lus}}$ whose vertices are irreducible characters and edges connect characters appearing in the same $\varphi$-constructible character to define Lusztig $\varphi$-families as connected components.
• Leverage known results on Calogero-Moser cellular characters and their $b$-invariants (Theorem CM) to compare with Lusztig's $\varphi$-constructible characters.
• Prove by induction on the size of a set $Z'$ that a certain inequality involving $b$-invariants and reflection data holds, establishing the minimality of the $b$-invariant in key configurations.
• Apply the theory of rational Cherednik algebras at $t=0$ and the geometry of the Calogero-Moser space to relate the $b$-invariant to cellular structures.
• Use the fact that $b_{\chi} = 1$ if and only if $\chi$ is a constituent of the canonical reflection representation, and that $b_{\varepsilon}$ equals the number of reflections, to analyze extremal cases.

Experimental results

Research questions

• RQ1 Does every Lusztig $\varphi$-family contain a unique irreducible character with minimal $b$-invariant when $\varphi$ is not constant?
• RQ2 Does every $\varphi$-constructible character have a unique irreducible constituent with minimal $b$-invariant?
• RQ3 Is the $b$-invariant minimality property for cellular characters in the Calogero-Moser setting compatible with Lusztig's $\varphi$-constructible characters?
• RQ4 Can the $b$-invariant be used to distinguish and classify $\varphi$-constructible characters and their families in the non-constant weight function case?
• RQ5 To what extent does the $b$-invariant structure support the conjectural equivalence between Kazhdan-Lusztig and Calogero-Moser cellular partitions?

Key findings

• Every Lusztig $\varphi$-family contains a unique irreducible character with minimal $b$-invariant, generalizing Lusztig's result on special characters to non-constant weight functions.
• Every $\varphi$-constructible character has a unique irreducible constituent with minimal $b$-invariant, extending the minimality property beyond families to the characters themselves.
• The $b$-invariant minimality property, previously known for Calogero-Moser cellular characters (with coefficient $r_{\chi_{\gamma}} = 1$), is now shown to hold for Lusztig's $\varphi$-constructible characters.
• The proof relies on an inductive argument on a set $Z'$, establishing a key inequality involving $b$-invariants of certain partitions, which confirms the minimality in critical configurations.
• The results are compatible with the conjecture that $\mathrm{Cons}_{\varphi}^{\mathrm{Lus}}(W) = \mathrm{Cell}_{\varphi}^{\mathrm{CM}}(W)$ and $\mathrm{Fam}_{\varphi}^{\mathrm{Lus}}(W) = \mathrm{Fam}_{\varphi}^{\mathrm{CM}}(W)$, providing strong structural evidence.
• The $b$-invariant serves as a canonical selector within each $\varphi$-family and $\varphi$-constructible character, mirroring the role of special characters in the constant weight case.