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2] There is a natural graded Hopf algebra isomorphism q ∼ ϕ : H (Gr, C) −→ U(ge ). 3 Geometric Satake equivalence. Let Db (Gr) denote the bounded derived category of constructible complexes on Gr, to be understood as a direct limit of the corresponding bounded derived categories on finite dimensional projective subvarieties that exhaust Gr. One similarly defines Perv(Gr) ⊂ Db (Gr), the abelian category of perverse sheaves. 1. Let Perv G∨(O) (Gr) be the (full) abelian subcategory in Perv(Gr) formed by semisimple4 G∨ (O)-equivariant perverse sheaves on Gr.

2. 1): p+q = gr Extp+q +3 E∞ (k (λ), kA (µ)) DUb (A,Λ) A H p Ub , ExtqDY (Λ,Λ) (kA (λ), kA (µ)) = E2p,q Y F′∗ F∗   p+q +3 E∞ = gr Extp+q (k (lλ), kB (lµ)) . 3) 38 q The vertical arrow on the left of the diagram is induced by the map F′∗ : ExtDY (Λ,Λ) (kΛ , kΛ ) q → ExtDY (b,b) F′ (kΛ ) , F′ (kΛ ) . 1 implies that this map may be identified with the composite map q q deform ◦ evkb : ExtDY (Λ,Λ) (kΛ , kΛ ) ∼ = Sym(n∗ [−2]) −→ ExtDY (b,b) F′ (kΛ ) , F′ (kΛ ) . 1. 3). 3) is induced by the functor F.

4) using Frobenius reciprocity as follows: Extip IndBp (lλ) ,kp (lµ) = Extip (φIλ , kp (lµ)) ρ˜ U U → Extip (RIndB (φIλ ) , kp (lµ)) = Extip Indp (lλ) , kp (lµ) . Here the morphism ρ˜ is induced by the canonical B-module adjunction morphism ρ : U RIndB (φIλ ) −→ φIλ , restricted to p. , is an actual U-module and, moreover; (ii) the morphism ρ is a surjection, which is split as a morphism of p-modules. , Ri IndB (lλ) = 0, for all i = 0. 5) shows that U RIndB φIλ is an actual U-module, because λ − ν ∈ −Y++ , for ν large enough.

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