Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 with itself and making the upper and lower halves adjoint.

The result I need is an extension of this to tensor products. Roughly, I would like a statement like:

There is a unique system of $U_q(\mathfrak{g})$-invariant Hermitian inner products on all tensor products $V_{\lambda_1}\otimes \cdots \otimes V_{\lambda_\ell}$ such that

- On $V_\lambda$, it is the Shapovalov form.
- The action of $E_i$ and $F_i$ are biadjoint (up to some powers of $q$).
- For any $j<\ell$, the natural map $V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_j}\hookrightarrow V_{\lambda_1}\otimes \cdots \otimes V_{\lambda_\ell}$ is an isometric embedding.

I said to myself, "Self, it would be silly to post this question on MathOverflow. You are in a math library, just feet away from Lusztig's book. Surely it is in there." However, I've had no luck finding it in Lusztig's book, which is sadly lacking in index. Is this actually written down anywhere?

**EDIT**: Jim asks for more motivation. I feel like this is the sort of question where motivation will not be very helpful in actually finding an answer, but there's no harm in saying a little (and it will allow me to put off real work).

One of the foundational principles of categorification is that things with nice categorifications have nice inner products (since Grothendieck groups have a nice inner product given by Euler characteristic of the Ext's between objects). I'm working right now on categorifying tensor products of representations, so it would be rather convenient for me to find some earlier references that used this form.

`$p$`

. Both of them found a miraculous determinant formula on each weight space. – Jim Humphreys Mar 22 '10 at 21:57SEMISIMPLELie algebra ..." At least, I can't imagine interpreting the rest of the post without this extra word. – Theo Johnson-Freyd Mar 23 '10 at 2:12