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The endomorphisms of the antidominant projective form the coinvariant algebra

Statement

Assume the Axiom of Choice. Let $g$ be finite-dimensional complex semisimple, $λ$ the dominant regular integral representative of a block and $O_{λ}$ its block with antidominant weight $w_{0} \cdot λ$ . Let $C$ be the coinvariant algebra of The coinvariant algebra of a finite Weyl group for the Cartan reflection representation, with its grading fixed there. Then:

(1) the natural maps $Z (U (g)) \to C$ (Harish-Chandra isomorphism, shifted by $λ + ρ$ in the published ${HC}_{ρ}$ convention, followed by the quotient map $ρ$ ) and $Z (U (g)) \to {End}_{g} P (w_{0} \cdot λ)$ (multiplication) are surjective with the same kernel;

(2) consequently the multiplication map induces a canonical algebra isomorphism ${End}_{g} P (w_{0} \cdot λ) ≅ C,$ identifying $C$ with the endomorphism ring of the antidominant projective, canonically and independently of the choice of projective cover inside its isomorphism class;

(3) the grading on the right-hand side is the internal grading of the coinvariant algebra; no graded lift of category $O$ is asserted or used.

This is Soergel's Endomorphismensatz [Soe90, Theorem 7], in the regular integral case; the companion general form for singular blocks replaces $C$ by the invariants $C^{W_{λ}}$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), the regular integral block of The regular integral block and its antidominant weight with $R = S (h)_{(0)}$ , the deformed antidominant lift $P_{R}$ of Lifting of indecomposable projectives to the deformed family, and its central-action algebra.

[F1]

The central-action lemma The central action on the deformed antidominant projective: for $R = S_{(0)}$ the map $a_{λ} ⁣ : Z \otimes R \to R \otimes_{R^{W}} R$ , $z \otimes r \mapsto σ_{λ} ({HC}_{ρ} (z)) \otimes r$ , and the multiplication map $Z \otimes R \to {End}_{g \otimes R} (P_{R})$ are surjective with the same kernel and induce an $R$ -algebra isomorphism ${End}_{g \otimes R} (P_{R}) ≅ R \otimes_{R^{W}} R$ ; on the closed right fiber the algebra becomes $C = S / (S S_{+}^{W})$ and, for dot-dominant integral $ν$ , $R^{W_{ν}} \otimes_{R^{W}} R$ specializes to $C^{W_{ν}}$ .

[F2]

Hom base change and residue fibers for the deformed lifts: for every map of deformation algebras $R \to R^{'}$ one has ${Hom}_{O_{R}} (P_{R}, N) \otimes_{R} R^{'} ≅ {Hom}_{O_{R^{'}}} (P_{R} \otimes_{R} R^{'}, N \otimes_{R} R^{'})$ , the residue fiber of $P_{R}$ at the closed point $0$ is $P (w_{0} \cdot λ)$ , and a $g \otimes R$ -linear map of deformed projectives that is an isomorphism on the residue fiber is itself an isomorphism (Lifting of indecomposable projectives to the deformed family).

[F3]

The shifted Harish-Chandra map ${HC}_{ρ} (z) (μ) = pr (z) (μ - ρ)$ is an algebra isomorphism $Z (U (g)) \to S (h)^{W}$ (Harish-Chandra isomorphism for the center); the invariant ring $S^{W}$ is polynomial and $S$ is free over it (Chevalley shephard todd for finite weyl groups, Weyl orbit sums form a basis of finite Weyl invariants); the coinvariant algebra $C = S / (S S_{+}^{W})$ has $C_{0} = C$ , its internal grading, one-dimensional socle in top degree and finite dimension $∣ W ∣$ (The coinvariant algebra of a finite Weyl group, The coinvariant algebra is a complete intersection with one-dimensional socle).

[F4]

The block conventions: the dominant regular integral representative $λ$ , the antidominant weight $w_{0} \cdot λ$ , and for a singular dominant integral weight the stabilizer $W_{λ}$ and the invariant algebra $C^{W_{λ}}$ (The regular integral block and its antidominant weight, The deformed category O family over the Cartan symmetric algebra).

Proof

technique · direct

1.1F1

By [F1] the multiplication map $Z \otimes R \to {End}_{g \otimes R} (P_{R})$ is surjective, has the same kernel as $a_{λ}$ , and makes the following diagram commute: $Z \otimes R \overset{a_{λ}}{\to} R \otimes_{R^{W}} R \tilde{\to} {End}_{g \otimes R} (P_{R})$ .

1.2F1F2F3

Specializing the right factor along $R \to C$ (the residue field at $0$ ) carries algebra isomorphisms to algebra isomorphisms, and [F2] identifies ${End}_{g \otimes R} (P_{R}) \otimes_{R} C ≅ {End}_{g} (P_{R} \otimes_{R} C) = {End}_{g} P (w_{0} \cdot λ)$ , while the target specializes to $(R \otimes_{R^{W}} R) \otimes_{R} C ≅ R / (S S_{+}^{W}) = C$ , and $a_{λ}$ specializes to the map sending $z$ to the class of ${HC}_{ρ} (z) (λ + ρ + τ)$ .

2.1F1F3step 1.2

Specializing the isomorphism of step 1.1 therefore gives the canonical algebra isomorphism ${End}_{g} P (w_{0} \cdot λ) ≅ C$ , induced by the multiplication action alone and hence independent of the chosen projective cover up to the canonical identification; the specialized multiplication map is the shifted Harish-Chandra composite $z \mapsto {HC}_{ρ} (z) (λ + ρ + τ)$ followed by the quotient map $R \to C$ , it is surjective, and the commutative diagram identifies its kernel with that of the multiplication map. Tensoring need not preserve kernels; no exactness of residue-field tensoring is used.

3.1F1F4step 2.1∎

The grading in (3) is the internal grading of $C$ transported along the isomorphism of step 2.1; no $Z$ -grading of category- $O$ objects or of the Hom spaces is used; for a singular dominant integral $λ$ the same specialization of the central-action lemma gives ${End}_{g} P ≅ C^{W_{λ}}$ , replacing the coinvariant algebra by its invariants.

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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)
Jonathan Brundan, Representations of the general linear Lie superalgebra in the BGG category O (survey; Section 4 with Soergel's Endomorphismensatz and Struktursatz) (standard reference, not scraped)