On a simple representation of a simple Lie algebra, there is a unique bilinear form called the Shapovalov form for which the actions of $E_i$ and $F_i$ are biadjoint, and some fixed highest weight vector has $\langle v_h,v_h\rangle=1$.

The representation has a distinguished collection of vectors $F_{i_1}\cdots F_{i_n}v_h$ for all sequences $\mathbf{i}$. One can calculate any inner product $\langle F_{i_1}\cdots F_{i_n}v_h, F_{j_1}\cdots F_{j_n}v_h\rangle$, by simply moving the $F_j$'s to become $E_j$'s on the other side, and commuting them past the $F_i$'s. This is not hard to do computationally, but the formulas one gets are not positive, which is annoying for my purposes.

Does anyone know of positive formulae for these inner products? What about their $q$-analogues for quantum groups?

EDIT: I should note, following Allen's comment: I'm pretty sure that I know a vector space that has the dimension which is this inner product. There's also a positivity proof using the canonical basis (all the elements I'm interested in are positive linear combinations of canonical basis elements).

I'm trying to show that a surjective map to this vector space is an isomorphism, and do so by finding a spanning set of the domain that has the right cardinality.