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Singular category O multiplicities are parabolic Kazhdan–Lusztig values

Statement

Assume the Axiom of Choice (The Axiom of Choice) and work in the type- $A$ root datum $W = S_{n}$ of the page's Hecke module and ordinary multiplicity theorem. Let $(λ, μ)$ be as in The facet tensor-weight exclusion lemma with $J = J (μ)$ , so $O_{μ}$ is the singular integral block of The singular integral block and its dot stabilizer with labels normalized by the minimal right coset representatives $W^{J}$ , and let $w_{0}$ be the longest element of $W$ . Then for all $y, w \in W^{J}$ , $[M (y \cdot μ) : L (w \cdot μ)] = (P (w \cdot μ) : Δ (y \cdot μ)) = P_{y w_{0}, w w_{0}} (1),$ the value at $q = 1$ of the ordinary Kazhdan--Lusztig polynomial attached to the pair $(y w_{0}, w w_{0})$ in the classical normalization of Kazhdan–Lusztig polynomials in the classical $q$ -normalization. Equivalently, put $k (w) = w_{0}^{J} w^{- 1} w_{0} \in^{J} W$ . Then $[M (y \cdot μ) : L (w \cdot μ)] = m_{k (y), k (w)} (1) .$ This is the spherical family. The antispherical family instead governs the genuine regular parabolic locally finite category with dominant labels; changing to an opposite positive system does not identify the two families. In particular all singular multiplicities are computable from the ordinary Kazhdan--Lusztig coefficients of $S_{n}$ , and no individual polynomial coefficients beyond the value at one occur.

Facts & Assumptions

Given: The facet data $(λ, μ)$ with $J = J (μ)$ , $W_{μ} = W_{J}$ , the regular block $O_{λ}$ , the singular block $O_{μ}$ with labels $W^{J}$ (The singular integral block and its dot stabilizer, Translation to the wall on Verma, simple and projective objects), and the translation functor $T = T_{λ}^{μ}$ .

[F1]

The block conventions: labels $w \cdot μ$ depend only on the coset $w W_{J}$ and $W^{J}$ is a set of minimal representatives; BGG reciprocity $(P (w \cdot μ) : Δ (y \cdot μ)) = [M (y \cdot μ) : L (w \cdot μ)]$ holds; $Δ (w \cdot μ) = M (w \cdot μ)$ ; restricted duality fixes label pairs; all objects have finite length (The singular integral block and its dot stabilizer, BGG reciprocity, Standard and costandard objects).

[F2]

Wall translation: $T$ is exact, $T Δ (z \cdot λ) ≅ Δ (z \cdot μ)$ for all $z$ , and $T L (z \cdot λ) ≅ L (z \cdot μ)$ for $z \in W^{J}$ while $T L (z \cdot λ) = 0$ for $z \notin W^{J}$ (Translation to the wall on Verma, simple and projective objects (1),(2)); Translation functors are exact and biadjoint supplies exactness and the biadjunction.

[F3]

Ordinary regular multiplicities: in a regular integral block of type $A$ with dominant regular base $λ_{d o m}$ , $[M (y \cdot λ_{d o m}) : L (w \cdot λ_{d o m})] = P_{y, w} (1)$ for $y, w \in W$ , with vanishing unless $y \leq w$ (Simple Verma composition multiplicities are Kazhdan–Lusztig values, The regular integral dot-orbit labelling convention).

[F4]

Ordinary Kazhdan--Lusztig polynomial data: $P_{a, b} (1)$ is defined by the classical normalization, and simultaneous inversion gives $P_{a, b} = P_{a^{- 1}, b^{- 1}}$ (Kazhdan–Lusztig polynomials in the classical $q$ -normalization, Existence and uniqueness of the Kazhdan–Lusztig basis). The latter supplier also gives uniqueness of the bar-fixed triangular basis; the quadratic/braid presentation and normalized standard elements are those of The normalized type-A Hecke algebra and its bar involution, and the conjugation-by- $w_{0}$ symmetry needed below is proved from that uniqueness in step 1.3.

[F5]

Spherical reduction: for $y, x \in^{J} W$ , $m_{y, x} = p_{w_{0}^{J} y, w_{0}^{J} x}$ and $p_{a, b} = v^{ℓ (b) - ℓ (a)} P_{a, b} (v^{- 2})$ with $P_{a, b} (1)$ the value at $q = 1$ (Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones (2), The parabolic Kazhdan–Lusztig bases and polynomials); the labelling $W^{J} = (^{J} W)^{- 1}$ is Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification.

Proof

technique · direct: push a composition series of the regular standard through the exact translation functor, count only the surviving minimal-representative simple, and translate the labels to the dominant base

1.1F1F2algebra

Composition multiplicities on the wall. Fix $y, w \in W^{J}$ and a composition series of $Δ (y \cdot λ)$ with factors $L (z \cdot λ)$ , $z \in W$ . Exactness of $T$ [F2] turns it into a series of $T Δ (y \cdot λ) ≅ Δ (y \cdot μ)$ with factors $T L (z \cdot λ)$ ; by [F2] the only nonzero factors occur for $z \in W^{J}$ , and then the factor is $L (z \cdot μ)$ , which can equal the simple $L (w \cdot μ)$ only for $z = w$ since distinct $W^{J}$ -labels have distinct dot translates [F1]. Hence $[Δ (y \cdot μ) : L (w \cdot μ)] = [Δ (y \cdot λ) : L (w \cdot λ)]$ .

1.2F3algebra

Translate to the dominant base. Let $λ_{d o m} : = w_{0} \cdot λ$ ; since $λ$ is dot-antidominant regular, $λ_{d o m}$ is dominant regular and $λ = w_{0} \cdot λ_{d o m}$ , so for every $z \in W$ , $z \cdot λ = (z w_{0}) \cdot λ_{d o m}$ . Applying [F3] in the regular block of $λ_{d o m}$ gives $[Δ (y \cdot λ) : L (w \cdot λ)] = [M (y w_{0} \cdot λ_{d o m}) : L (w w_{0} \cdot λ_{d o m})] = P_{y w_{0}, w w_{0}} (1)$ , the ordinary value supported on $y w_{0} \leq w w_{0}$ .

1.3F4F5algebra

The spherical dictionary. Conjugation by $w_{0}$ permutes the type- $A$ simple reflections, so the assignment $H_{z} \mapsto H_{w_{0} z w_{0}}$ , $v \mapsto v$ is an algebra automorphism: it preserves the quadratic and braid presentations, the bar, length and Bruhat order. It sends a bar-fixed triangular element with top $H_{b}$ to one with top $H_{w_{0} b w_{0}}$ ; uniqueness in [F4] therefore gives $p_{a, b} = p_{w_{0} a w_{0}, w_{0} b w_{0}}$ . Now set $k (w) = w_{0}^{J} w^{- 1} w_{0}$ . Inversion sends $W^{J}$ to $^{J} W$ [F5]. In type $A$ , left-minimality means that the values in each consecutive $J$ -block occur in increasing order: the left descent $s_{i}$ occurs exactly when $i + 1$ precedes $i$ in one-line notation. Right multiplication by $w_{0}$ reverses all positions, making those values decrease; left multiplication by $w_{0}^{J}$ reverses the values inside each block, restoring increasing order. Thus $k (w) \in^{J} W$ . The spherical reduction [F5] gives $m_{k (y), k (w)} = p_{w_{0}^{J} k (y), w_{0}^{J} k (w)} = p_{y^{- 1} w_{0}, w^{- 1} w_{0}} = p_{w_{0} y, w_{0} w} = p_{y w_{0}, w w_{0}}$ , using simultaneous inversion from [F4] and the conjugation symmetry just proved. Evaluating at $v = 1$ and applying the classical normalization yields $m_{k (y), k (w)} (1) = P_{y w_{0}, w w_{0}} (1)$ .

2.1F1step 1.2algebra

The projective version. By BGG reciprocity [F1], $(P (w \cdot μ) : Δ (y \cdot μ)) = [M (y \cdot μ) : L (w \cdot μ)] = P_{y w_{0}, w w_{0}} (1)$ ; this is the second displayed identity.

3.1F5step 1.2step 2.1step 1.3∎

Conclusion. Combining the preceding steps, $[M (y \cdot μ) : L (w \cdot μ)] = (P (w \cdot μ) : Δ (y \cdot μ)) = P_{y w_{0}, w w_{0}} (1) = m_{k (y), k (w)} (1)$ for all $y, w \in W^{J}$ , so the singular multiplicities are spherical parabolic values at one. The antispherical family is a different family (it is the sign-module normalization, used for the genuine regular parabolic category with dominant labels), and no change of positive system identifies the two; this is exactly the normalization distinction recorded in The parabolic Kazhdan–Lusztig bases and polynomials.

Remarks

Open obligation. The proof uses the batch-14 interfaces thm-simple-verma-composition-multiplicities-are-kazhdan-lusztig-values and def-regular-integral-dot-orbit-labelling-convention through fact [F3], cited in step 1.2 for the single-term reduction $[M (y w_{0} \cdot λ_{d o m}) : L (w w_{0} \cdot λ_{d o m})] = P_{y w_{0}, w w_{0}} (1)$ ; both were scaffolded but unwritten when this item was authored, and both are now written in items/ with statements matching this use. The Step-3 decision remains escalated because only the owner may refresh an escalation receipt; the exact supplier IDs and the consuming step are recorded in the pair report. The labelling convention is used only through its explicitly quoted clauses (regular block data, Bruhat reversal and the Kazhdan--Lusztig normalization), not through its Soergel dictionary.

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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)
Nicolas Libedinsky and Geordie Williamson, The anti-spherical category (author-hosted complete PDF of the article) (standard reference, not scraped)
James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, AMS Graduate Studies in Mathematics 94 (standard reference, not scraped)