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Verma-flag multiplicities are standard-filtration ranks under V

Statement

Assume the Axiom of Choice. For a projective $P$ in a regular integral type-A block, choose its polynomial Soergel lift $B (P)$ determined by the preceding projective dictionary: if $P = ⨁_{w} P (w \cdot λ)^{m_{w}}$ , take $B (P) = ⨁_{w} B_{w^{- 1}}^{m_{w}}$ , so that its right reduction is $V (P)$ after forgetting auxiliary grading. Write $r_{x, d} (B (P))$ for the graded multiplicity of the polynomial standard graph module supported at $x$ in its standard support filtration, with the standard-generator normalization of the polynomial Soergel category. Then $(P : Δ (y \cdot λ)) = \sum_{d} r_{y^{- 1}, d} (B (P)) .$ Equivalently the polynomial Soergel character of this lift, evaluated at $v = 1$ and with inverse labels, recovers the ordinary Verma-flag character of $P$ . These are ranks before reduction; no separated Bruhat-support filtration intrinsic to the finite-dimensional module $V (P)$ is asserted, since the graph supports coalesce at the origin. The same ungraded multiplicity comparison holds with the corresponding Weyl-group polynomial/deformed lifts in other complex semisimple types.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), the regular integral block with $V = Hom (P (w_{0} \cdot λ), -)$ , the deformed family over $R = S (h)_{(0)}$ , and the type-A polynomial graph conventions.

[F1]

The projective/Soergel-module dictionary: $V$ restricts to an equivalence $Proj (O_{λ}) \tilde{\to} {ungraded Soergel modules}$ with $P (w \cdot λ) \mapsto {\overset{ˉ}{B}}_{w^{- 1}}$ , and the normalized polynomial lifts are the self-dual ones (Indecomposable projectives correspond to indecomposable Soergel modules).

[F2]

Deformed data: each lifted projective $P_{R}$ has a finite deformed Verma flag with the same multiplicities as its closed fiber, ${Hom}_{O_{R}} (P_{R}, N)$ is finite free for Verma-filtered $N$ with arbitrary base change, and a map that is an isomorphism on the residue fiber is an isomorphism; the central action on a deformed standard constituent of highest weight $x \cdot λ + τ$ is by ${HC}_{ρ} (z) (x (λ + ρ) + τ) = σ_{λ} ({HC}_{ρ} (z)) (x^{- 1} τ)$ , and $V θ_{s} ≅ C \otimes_{C^{s}} V$ (Lifting of indecomposable projectives to the deformed family, The central action on the deformed antidominant projective, Wall-crossing under the Soergel functor is tensoring with the coinvariant bimodule).

[F3]

The standard graph bimodule $S_{x}$ has ordinary left action and right action $r \cdot g = r x (g)$ , where $x (g) (τ) = g (x^{- 1} τ)$ . Untwisting its right action by $r \mapsto x^{- 1} (r)$ makes the right parameter ordinary and the left action $f \mapsto x^{- 1} (f)$ , whose value at $τ$ is $f (x τ)$ . Thus the central branch $f \mapsto f (y^{- 1} τ)$ has graph label $x = y^{- 1}$ . Support-layer ranks are preserved on passing to the fraction field, while origin reduction coalesces all supports (Standard graph bimodules, support filtrations and characters, The reduced Soergel module and the dominant Lefschetz operator, Indecomposable labels, characters and filtration ranks agree after real scalar extension).

[F4]

The antidominant projective has a Verma flag with the factors $Δ (w \cdot λ)$ , $w \in W$ , once each; by BGG reciprocity $(P (a) : Δ (w \cdot λ)) = [Δ (w \cdot λ) : L (a)] = 1$ for every $w$ , so $V Δ (w \cdot λ) ≅ k$ for every $w$ and $V_{R} Δ_{R} (w \cdot λ)$ is free of rank one over $R$ , its closed fiber being one-dimensional (Projectives in category O have finite Verma flags, BGG reciprocity, Finite Verma flags and their multiplicities, The endomorphisms of the antidominant projective form the coinvariant algebra, Soergel's Struktursatz: the functor V is fully faithful on projectives).

Proof

technique · direct

1.1F2F4

The functor $V_{R}$ is exact and sends each deformed standard $Δ_{R} (y \cdot λ)$ to a finite free $R$ -module of rank one: by [F4] its closed fiber is one-dimensional, and by [F2] the Hom module into the Verma-filtered standard is finite free with base change, so the rank is one.

1.2F2F3

On that rank-one free module the center acts on the generic constituent of highest weight $y \cdot λ + τ$ by the scalar ${HC}_{ρ} (z) (y (λ + ρ) + τ) = σ_{λ} ({HC}_{ρ} (z)) (y^{- 1} τ)$ , which in the polynomial graph convention [F3] is the branch of the graph label $x = y^{- 1}$ ; hence the graph label of the standard constituent is the inverse of its highest-weight label.

1.3F1F2F3algebra

The preceding dictionary proves the deformed, action-compatible wall isomorphism $V_{R} θ_{s} ≅ R \otimes_{R^{s}} V_{R}$ by adjunction, the translated central inclusion and the coefficient-of- $α_{s}$ Frobenius pairing; it also proves deformed full faithfulness and the natural reduction-of-Hom map. Iteration identifies the localized polynomial word with the $V_{R}$ -image of its lifted projective word. The unique top-label polynomial summand and the top projective summand match under this full faithfulness, by the dictionary's length induction. Restriction to their matching idempotents therefore gives an actual $A$ -module isomorphism $B (P) \otimes_{S} R ≅ V_{R} P_{R}$ , preserving every generic branch. It is not inferred from a mere origin isomorphism.

2.1F2F3step 1.1step 1.2step 1.3

Passing to the fraction field, the graph-support ranks of $B (P)$ at $x$ are therefore the multiplicities of the deformed Verma constituents of $P_{R}$ at the inverse label $y = x^{- 1}$ , and flat localization preserves these ranks; the deformed Verma flag of $P_{R}$ specializes exactly to the Verma flag of $P$ by [F2], so summing the graded multiplicities gives $(P : Δ (y \cdot λ)) = \sum_{d} r_{y^{- 1}, d} (B (P))$ , which is the stated comparison at $v = 1$ .

3.1F1F2F3step 1.1step 1.2step 2.1algebra∎

These are ranks before origin reduction, so no intrinsic separated support filtration on $V (P)$ is asserted. For another complex semisimple type use the deformed lift $V_{R} P_{R}$ and its generic branch ranks, constructed in the preceding dictionary without type-A diagrammatic classification. Steps 1.1--1.2 identify each standard with its rank-one inverse branch in any reflection realization, so the same equality holds for these Weyl-group deformed lifts; whenever a corresponding polynomial lift is chosen, its localization has those same ranks. The type-A polynomial support-filtration formula is the particular case of step 2.1, with the fixed average coordinate split off.

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Sources

Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46 full text) (standard reference, not scraped)
Arthur Garnier, The classical theory of Soergel bimodules and the link with Lusztig's conjecture, Oberwolfach Arbeitsgemeinschaft note (2022) (standard reference, not scraped)