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Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values

Statement

Assume the Axiom of Choice (The Axiom of Choice). Fix a dominant regular integral weight $λ$ of the complex type- $A$ Lie algebra and the standard parabolic $p_{J}$ , with reductive Levi $l_{J}$ and longest Levi Weyl element $w_{J}$ (Standard parabolic and Levi subalgebras from a subset of simple roots). The $p_{J}$ -dominant weights in this regular block are exactly $y \cdot λ$ for $y \in^{J} W$ . Write $Δ^{J} (y \cdot λ) = M_{p} (y \cdot λ)$ , $L (x \cdot λ)$ and $P^{J} (x \cdot λ)$ for its generalized standards, simples and projective covers (Parabolic category O for a standard parabolic). Then for $x, y \in^{J} W$ , $[Δ^{J} (y \cdot λ) : L (x \cdot λ)] = (P^{J} (x \cdot λ) : Δ^{J} (y \cdot λ)) = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} P_{u y, x} (1) = n_{y, x} (1) .$

These are the antispherical sign coefficients, with support $y \leq x$ and diagonal $1$ . The precise character/Grothendieck identity supplying the formula is $ch Δ^{J} (y \cdot λ) = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} ch Δ (u y \cdot λ) .$

For the same highest weights in the antidominant base $λ_{a n t i} = w_{0} \cdot λ$ , the actual longest-left label is $z = y w_{0}$ , since $y \cdot λ = z \cdot λ_{a n t i}$ . Its shortest-left encoding is $a = l f (y) : = w_{J} y w_{0} \in^{J} W$ , so the weight in that encoding is $(w_{J} a) \cdot λ_{a n t i}$ , not $a \cdot λ_{a n t i}$ . With $b = l f (x)$ , the reindexed matrix is ${\tilde{N}}_{a, b} : = n_{{l f}^{- 1} (a), {l f}^{- 1} (b)} (1) = n_{w_{J} a w_{0}, w_{J} b w_{0}} (1) .$ It is the same multiplicity matrix with both indices permuted and has reversed support in the transported order. It is not in general $n_{a, b} (1)$ . Every label retains its explicit dot-baseweight. The highest-weight category theorem applies also to singular integral parabolic blocks, but this ordinary regular KL matrix formula assumes the ambient central character is regular. No singular/parabolic category equivalence, grading or Koszul duality is asserted.

Facts & Assumptions

Given: A dominant regular integral weight $λ$ of type $A$ , the standard parabolic $p_{J} = l_{J} \oplus u_{J}$ with Levi Weyl group $W_{J}$ and longest element $w_{J}$ , the parabolic category $O^{p}$ with generalized Vermas $Δ^{J} (ξ) = M_{p} (ξ)$ (Parabolic category O for a standard parabolic, Parabolic category O is a highest weight category), and the completed character ring with its geometric-series inverses (The completed formal character ring, Geometric series are invertible in the completed character ring).

[F1]

Parabolic highest-weight structure: $M_{p} (ξ)$ lies in $O^{p}$ with head $L (ξ)$ and finite length, its composition factors are parabolic simples, and in a fixed integral block every projective has a finite generalized-standard flag with BGG reciprocity $(P^{p} (ξ) : M_{p} (η)) = [M_{p} (η) : L (ξ)]$ ; the parabolic simple/standard classes form Grothendieck bases and characters are injective on the block (Parabolic category O is a highest weight category (1)-(4), Parabolic category O for a standard parabolic).

[F2]

The local PBW model: $ch M_{p} (ξ) = ch E (ξ) \cdot \prod_{β \in Φ^{+} ∖ Φ_{J}^{+}} (1 - e^{- β})^{- 1}$ and, by the PBW theorem, the identity holds in the completed character ring; the outside denominator is the PBW denominator of the negative outside radical $v_{J}$ (The PBW model of a Verma module, PBW gives an ordered monomial basis for the enveloping algebra, Standard parabolic and Levi subalgebras from a subset of simple roots (5)).

[F3]

Weyl character formula for $s_{J}$ : for a $p$ -dominant integral $ξ$ , $ch E (ξ) = \frac{\sum_{u \in W_{J}} (- 1)^{ℓ (u)} e^{u (ξ + ρ_{J})}}{\prod_{α \in Φ_{J}^{+}} (e^{α / 2} - e^{- α / 2})}$ with $ρ_{J}$ half the sum of the positive Levi roots (The Weyl character formula); here $W_{J}$ fixes $ρ - ρ_{J}$ because $(ρ - ρ_{J}) (H_{j}) = 0$ for every $j \in J$ , and $Φ_{J}^{+} = Φ_{J} \cap Φ^{+}$ .

[F4]

Ordinary regular multiplicities: for a dominant regular integral base, $[M (z \cdot λ) : L (x \cdot λ)] = P_{z, x} (1)$ with vanishing unless $z \leq x$ (Simple Verma composition multiplicities are Kazhdan–Lusztig values, The regular integral dot-orbit labelling convention), and the alternating sum identity $n_{y, x} (1) = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} P_{u y, x} (1)$ (Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones (1) at $v = 1$ ); the labelling dictionary $W^{J} = (^{J} W)^{- 1}$ and the coset minima are Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification.

[F5]

The regular-to-regular translations of Translation identifies the regular integral blocks identify labelled ordinary standards, simples and projective covers between dominant regular integral blocks of the same type- $A$ root datum, so [F4] holds in every such block, not only the principal one.

Proof

technique · direct: prove the finite-Levi character identity, compare ordinary simple coefficients using the regular Kazhdan-Lusztig multiplicity theorem, and read off the antispherical matrix and its projective version

1.1F2F3algebra

Levi character identity. Multiply the Weyl character formula [F3] for $E (ξ)$ by the outside PBW denominator of [F2]. For each $u \in W_{J}$ , $u (ρ - ρ_{J}) = ρ - ρ_{J}$ by [F3], so $e^{u (ξ + ρ_{J}) + (ρ - ρ_{J})} = e^{u (ξ + ρ)}$ ; dividing by the full PBW denominator $\prod_{α \in Φ^{+}} (e^{α / 2} - e^{- α / 2}) = \prod_{α \in Φ_{J}^{+}} (\dots) \prod_{β \notin Φ_{J}^{+}} (\dots)$ turns each summand into $ch Δ (u \cdot ξ)$ (The PBW model of a Verma module). Hence, for $p$ -dominant integral $ξ$ , $ch Δ^{J} (ξ) = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} ch Δ (u \cdot ξ)$ in the completed character ring; this is the displayed identity at $ξ = y \cdot λ$ .

1.2F4algebra

The dominant labels. Put $A = λ + ρ$ , strictly dominant and integral. The weight $y \cdot λ$ is $p_{J}$ -dominant iff $⟨ y A, α_{j}^{\lor} ⟩ - 1 \geq 0$ for every $j \in J$ . The integer $⟨ A, y^{- 1} α_{j}^{\lor} ⟩$ is positive exactly when $y^{- 1} α_{j}$ is a positive root: strict dominance makes positive-root coroot pairings positive integers and negative-root pairings negative. Thus the condition is $y^{- 1} α_{j} > 0$ for all $j \in J$ , equivalently no LEFT descent $s_{j} y < y$ : in type $A$ , $y^{- 1} α_{j} > 0$ means that the position of value $j$ precedes that of $j + 1$ , exactly the inversion-length ascent condition for swapping those values. These are precisely the shortest-left representatives $^{J} W$ in the coset-minimum convention of [F4]; no inversion of the weight label is performed.

2.1F1F4F5step 1.1step 1.2algebra

Comparing simple coefficients. By [F1] the parabolic simples are the $L (x \cdot λ)$ , $x \in^{J} W$ . Taking the coefficient of $L (x \cdot λ)$ in the identity of step 1.1 (with $ξ = y \cdot λ$ , $y \in^{J} W$ ) and using [F4] and [F5] for the ordinary Vermas in the given dominant regular integral block gives $[Δ^{J} (y \cdot λ) : L (x \cdot λ)] = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} [M (u y \cdot λ) : L (x \cdot λ)] = \sum_{u \in W_{J}} (- 1)^{ℓ (u)} P_{u y, x} (1) = n_{y, x} (1)$ . The sum is finite and the coefficients with $u y \leq̸ x$ vanish; in particular the support is $y \leq x$ with diagonal $n_{x, x} (1) = 1$ .

3.1F1step 2.1algebra

Projective multiplicities. By the parabolic BGG reciprocity of [F1], $(P^{J} (x \cdot λ) : Δ^{J} (y \cdot λ)) = [Δ^{J} (y \cdot λ) : L (x \cdot λ)] = n_{y, x} (1)$ ; the multiplicities are finite by the finite generalized-standard flag of [F1]. This proves the second identity.

3.2F1F4step 2.1algebra

Native longest-left labels. Let $λ_{a n t i} = w_{0} \cdot λ$ . Dot-action composition gives $y \cdot λ = (y w_{0}) \cdot λ_{a n t i}$ and $x \cdot λ = (x w_{0}) \cdot λ_{a n t i}$ . If $y$ has no left descent in $J$ , each $y^{- 1} α_{j}$ is positive, so $(y w_{0})^{- 1} α_{j} = w_{0} y^{- 1} α_{j}$ is negative; hence $z = y w_{0}$ is the longest representative in its left $W_{J}$ -coset. Its shortest-left encoding is $a = w_{J} z = w_{J} y w_{0}$ , with inverse $y = w_{J} a w_{0}$ ; similarly $b = w_{J} x w_{0}$ has inverse $x = w_{J} b w_{0}$ . Consequently the same multiplicity of step 2.1 is $[Δ^{J} ((w_{J} a) \cdot λ_{a n t i}) : L ((w_{J} b) \cdot λ_{a n t i})] = n_{w_{J} a w_{0}, w_{J} b w_{0}} (1) = {\tilde{N}}_{a, b}$ . This is an honest permutation of rows and columns, with support transported from $w_{J} a w_{0} \leq w_{J} b w_{0}$ , not a claim that ${\tilde{N}}_{a, b} = n_{a, b} (1)$ . Choosing the transported reversed order reads the same matrix as lower triangular.

4.1step 1.1step 2.1step 3.1step 3.2∎

The preceding steps prove the multiplicity formula, its projective form, the character identity and the reindexed version; the statement that only the regular ambient central character is assumed, and that no singular/parabolic equivalence or grading is used, is exactly the scope of the quoted suppliers [F1]--[F5].

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 [F4], cited in steps 1.2, 2.1 and 3.2 for the ordinary multiplicity value $P_{u y, x} (1)$ , and prop-translation-identifies-regular-integral-blocks through fact [F5], cited in step 2.1 for the labelled transport of the ordinary multiplicities across dominant regular integral blocks. All three were scaffolded but unwritten when this item was authored and are now written in items/ with statements matching these uses. The Step-3 decision remains escalated because only the owner may refresh an escalation receipt; the exact supplier IDs and consuming steps are recorded in the pair report. The ordinary KL formula is asserted only for the regular ambient central character; the parabolic highest-weight structure itself retains arbitrary integral blocks. No singular/parabolic equivalence, Koszul duality or grading is imported.

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Sources

Volodymyr Mazorchuk and Catharina Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Trans. Amer. Math. Soc. 357 (2005) 2939-2973 (author-hosted published version) (standard reference, not scraped)
Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrden, Some homological properties of category O VI, Documenta Mathematica 26 (2021) 1237-1269 (complete article) (standard reference, not scraped)
Anne Moreau, M2 Representation Theory 2025–2026 (current complete 129-page author notes) (standard reference, not scraped)