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5). We remark that the techniques of [Str03a] ensure that simple and indecomposable projective modules are gradable, both as objects of Oλ and of Op,q-pres (although the grading is different). We take their standard graded lifts λ to be determined by requiring that the simple head is concentrated in degree 0. 3. The parabolic category of q-presentable modules The properly stratified structure The results of [MS05, Section 2] extend to the categories Op,q-pres . Let us briefly sketch them. 6], one can define a simple-preserving duality on Op,q-pres .

Let P p (x · λ) be the projective cover of L(x · λ) in Op and let M p (λ) be the corresponding parabolic Verma module. The block decomposition of O induces a block decomposition Op = λ Opλ . ∼ mod−Aλ . Let e⊥ ∈ Aλ = End(P(λ)) be Let λ ∈ ✄+ with stabilizer Sλ , and recall that Oλ = p the idempotent projecting onto the direct sum of the projective modules P (x · λ) for x ∈ Sn p ∼ ⊥ such that xSλ ⊆ W p . Then End(P p (λ)) = Aλ /Aλ e⊥ p Aλ and O = mod− Aλ /Aλ ep Aλ . Since the idempotent e⊥ p is homogeneous, the latter quotient algebra inherits a graded structure.

5) Tλµ corresponds to • ⊗Aµ HomB λ VP(λ), B λ ⊗B µ VP(µ) . Hence these functors are gradable. 7) Tλµ = • ⊗Aµ HomB λ VP(λ), B λ ⊗B µ VP(µ) − (x0 ) , where x0 is the longest element in Sµ Sλ and Tλµ M (µ) = P (x0 · λ). short . 1. 8) Tλµ q (x0 ) Tµλ and Tµλ q− (x0 ) Tλµ . Chapter 4. Graded category O 45 Proof. 8) on the dominant Verma modules. 10) C = Hom(P (µ), M (µ)) = Hom(Tµλ M (λ), M (µ)) = Hom(M (λ), q − (x0 ) Tλµ M (µ)) = Hom(M (λ), q − (x0 ) P (x0 · λ)) = C. For the first calculation, we used the well-known fact that the composition factor L(x0 · λ) appears in M (λ) only once in degree (x0 ); for the second one, we used the also well-known fact that the shifted Verma module q (x0 ) M (λ) appears at the bottom of the projective module P (x0 · λ).

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