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Simple Verma composition multiplicities are Kazhdan–Lusztig values

Statement

Assume the Axiom of Choice. In a regular integral type- $A$ block $O_{λ}$ , for all $y, w \in W$ $[M (y \cdot λ) : L (w \cdot λ)] = (P (w \cdot λ) : Δ (y \cdot λ)) = P_{y, w} (1),$ with the pair ordered so that the multiplicity vanishes unless $y \leq w$ in the Bruhat order; in particular the Kazhdan–Lusztig conjecture's multiplicity statement holds for every dominant regular integral weight, with the same combinatorial answer.

Facts & Assumptions

Given: The Axiom of Choice (inherited from the BGG reciprocity and block theory cited below), a regular integral type- $A$ block $O_{λ}$ with $W = S_{n}$ , and labels $y, w \in W$ .

[F1]

BGG reciprocity (BGG reciprocity): for all weights $λ, μ$ one has $(P (λ) : Δ (μ)) = [Δ (μ) : L (λ)] = [M (μ) : L (λ)]$ , where the left multiplicity is the Verma-flag multiplicity and the right one the simple composition multiplicity.

[F2]

The projective-Verma multiplicity theorem (Projective Verma multiplicities are Kazhdan–Lusztig values at one): in the regular integral type- $A$ block, $(P (w \cdot λ) : Δ (y \cdot λ)) = P_{y, w} (1)$ for all $y, w \in W$ , and this vanishes unless $y \leq w$ .

[F3]

Standards are Vermas (Standard and costandard objects, Finite Verma flags and their multiplicities): $Δ (ν) = M (ν)$ for every weight $ν$ , and the Verma-flag multiplicity $(X : Δ (ν))$ is the well-defined multiplicity of that standard in a finite flag.

[F4]

Block conventions (The regular integral dot-orbit labelling convention): the simples $L (w \cdot λ)$ , standards $Δ (w \cdot λ) = M (w \cdot λ)$ and indecomposable projectives $P (w \cdot λ)$ indexed by $w \in W$ , the composition multiplicity notation $[M (y \cdot λ) : L (w \cdot λ)]$ , and the Bruhat direction $[M (y \cdot λ) : L (w \cdot λ)] \neq 0 \Rightarrow y \leq w$ .

Proof

technique · direct

1.1F1F3F4

Apply BGG reciprocity [F1] with the two weights $w \cdot λ$ and $y \cdot λ$ : because $P (w \cdot λ)$ is the projective cover of $L (w \cdot λ)$ and $Δ (y \cdot λ) = M (y \cdot λ)$ by [F3], reciprocity reads $(P (w \cdot λ) : Δ (y \cdot λ)) = [Δ (y \cdot λ) : L (w \cdot λ)] = [M (y \cdot λ) : L (w \cdot λ)]$ .

1.2F2F4

The projective-Verma theorem [F2] evaluates the projective-flag multiplicity in the regular integral block: $(P (w \cdot λ) : Δ (y \cdot λ)) = P_{y, w} (1)$ , the value at $q = 1$ of the classical Kazhdan–Lusztig polynomial, and this value is zero unless $y \leq w$ in the Bruhat order.

2.1F2F4step 1.1step 1.2∎

Substituting the value of step 1.2 into the chain of equalities of step 1.1 gives $[M (y \cdot λ) : L (w \cdot λ)] = (P (w \cdot λ) : Δ (y \cdot λ)) = P_{y, w} (1)$ for all $y, w \in W$ , and the vanishing clause transfers along the same chain: the multiplicity is zero unless $y \leq w$ , in agreement with the Bruhat direction recorded in [F4]. Since $P_{y, w} (1)$ is a combinatorial quantity of the symmetric group alone, this multiplicity statement — the Kazhdan–Lusztig conjecture's multiplicity assertion — holds with the same answer for every dominant regular integral weight.

Depends on

Used by

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Sources

Lin Chen, lecture notes (Spring 2024), Lecture 9: standard filtrations, BGG reciprocity and translation functors (complete 7-page lecture) (standard reference, not scraped)
Lin Chen, lecture notes (Spring 2024), Lecture 8: Hecke algebras and the Kazhdan-Lusztig conjecture (complete 10-page lecture) (standard reference, not scraped)