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Current calculation.

By trdunlap2

Currently reading through Kazhdan-Lusztig and trying to make the following calculation.
Let V,V' be representations with the same central weight. K-L define a vector space \langle V,V' \rangle =V\otimes V' /\tilde{\mathfrak{g}}(V\otimes V' ) where \tilde{\mathfrak{g}} is \mathbb{C}[t,t^{-1}]\otimes\mathfrak{g}\oplus\mathbb{C}k whose action on V\otimes V' is not the usual action but a sort of anti-diagonal:
(t^nc)(x\otimes y)=((t^nc)x)\otimes y + x\otimes((-t)^{-n}c)(y)

This pairing has a second definition which generalizes: \langle V_1,V_2,\cdots,V_n\rangle. And satisfies the formula:
\langle V_1\dot\otimes V_2\dot\otimes\cdots\dot\otimes V_{r-1},V_r\rangle=\langle V_1,V_2,\cdots,V_{r-1},V_r\rangle

where V_1\dot\otimes V_2\dot\otimes\cdots\dot\otimes V_{r-1} is the tensor product that preserves central character.

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