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By Rau J., Mueller B.

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Zǫ ⊗ Zǫ ⊗ Zǫ −→ Zǫ ⊗ Zǫ −→ Zǫ → kZ . Now, given a ∈ Zǫ , let a ¯ ∈ Zǫ /Zǫ2 denote its image. g. [Lo]) that the assignment q q a1 ⊗ · · · ⊗ an −→ a ¯1 ∧ a ¯2 ∧ . . 5) 32 yields an isomorphism of cohomology. 5) is clearly Ub-equivariant, hence, we obtain a chain of Ub-equivariant quasi-isomorphisms q q L qis R ⊗Z R ∼ = Bar (Zǫ ) −→ ∧ (Zǫ /Zǫ2 ) = Λ . 7) as follows. Equip the vector space n[1] with the trivial Z-action (via the augmentation Z → k) and with the natural q adjoint Ub-action. Let P be an Ub ⋉ Z-module resolution of n[1] such that each term P i is free as a Z-module.

To prove this we observe that, for any finite dimensional G-module V (viewed as an Ugmodule), translation functors on block(U) commute with the functor M → M ⊗ φV . 2 we deduce dim ExtBi kB (lλ) , kB (lµ) ⊗ φV U B U i = dim Extblock(U) RIndB (lλ) , RIndB (lµ) ⊗ φV . 7) We put V = Vν , a simple module with highest weight ν. 5), in the same way as above. 6 is proved. 5. 1 Constructing an equivariant dg-resolution. 4. 1. There exists a (super)commutative dg-algebra R = Ub-action, and such that • • • i≤0 Ri , equipped with an The Ub-action on R preserves the grading, moreover, for each i, there is a direct sum decomposition Ri = ν∈Y Ri (ν) such that ur = ν(u) · r , ∀u ∈ Ut ⊂ Ub , r ∈ Ri (ν).

3 Construction of a bi-functor. 1 is the following bifunctor: DYU b Ub ⋉ (R ⊗Z R), Λ × DYB (B ⋉ Rb, b) −→ DYB (B ⋉ Rb, b), L M, N −→ M ⊗R N . 1) L In this formula, the tensor product M ⊗R N is taken with respect to the action on M of the second factor in the algebra R⊗Z R and with respect to the R-module structure on N obtained by restriction L to the subalgebra R ⊂ R ⊗Z B = Rb. The object M ⊗R N thus obtained has an additional R-action coming from the action of the first factor R ⊂ R ⊗Z R on M .

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