TITLE: Are there positive formulae for the inner product between elements of a Lie algebra representation in the Shapovalov form?
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Ben, my paper on the Shapovalov form does give a generating series for the entries of a Gram matrix in Corollary 3.4, and those entries are evidently positive. It is not very hard to  deduce a q-version of this generating series from the paper by Chari and Jing appearing in the references. This is not really what you want, though, since the calculations only apply to the basic representation.
In general, you are asking a hard question. The only place I can think to point you is Jantzen's thesis where he calculates the determinant of the Shapovalov form for irreducible representations of simple finite-dimensional Lie algebras. Of course, this isn't the easiest document to get and it is in German.
Otherwise, you might try to prove the Khovanov-Lauda conjecture on cyclotomic quotients of quiver Hecke algebras and see what that gets you. At any rate, there is a representation theoretic interpretation of the Shapovalov form on irreducible modules in type A in the "Graded Decomposition Numbers" paper of Brundan and Kleshchev.