rt.representation theory - Are there positive formulae for the inner product between elements of a Lie algebra representation in the Shapovalov form?

This may not be exactly what you want, but I'd suggest you look at Kashiwara's paper ``On crystal bass of the q-analogue of the universal enveloping algebra" (see MR1115118 ).

In section 2.5 Kashiwara discusses the quantum version of the Shapovalov form. More relevant to what I want to say is Proposition 3.4.4, which defines/proves existence of a modification of the Shapovalov form defined on $U_q^-$. Roughly what he is doing is the following: Consider the pairing of, for example, $F_1F_2F_1v_lambda$ and $F_1^2 F_2v_lambda$,
using the Shapovalov form on $M(lambda)$, and allow $lambda$ to vary. What you get is

$$P(lambda)/(q-q^{-1})^3$$ where $P$ is some Laurent polynomial in the $K_i$ and $q$, and evaluation at $lambda$ means
setting $K_i$ equal to $q^{(alpha_i, lambda)}.$ $P$ has a well defined highest order term in the $K_i$, and the coefficient of this term is a Laurent polynomial in $q$. Let $(F_1F_2F_1, F_1^2 F_2)$ be that leading coefficient. This will be Kashiwara's inner product on $U_q^-$, up to a power of $q$. In general, when pairing two monomials in the $F_i$ applied to $v_lambda$, the denominator in the above equation has a factor of $q_i-q_i^{-1}$ for each $F_i$ in the first monomial.

The way Kashiwara sets things up, it is clear that the inner product of two monomials in the $F_i$ is a Laurent polynomial in $q$ with positive coefficients. For monomials $m_1$ and $m_2$, the inner product will be zero unless $m_1$ and $m_2$ have the same weight (i.e. they are both the products of the same number of each $F_i$, but possibly in a different order). Furthermore, the sum of the coefficients is the number of ways of matching each $F_i$ in $m_1$ with an $F_i$ in $m_2$ for all $i$. This all follows from Equations (3.3.1) and (3.4.6) in Kashiwara's paper. I believe you can find the power of $q$ associated to a given matching by arranging the monomials correctly, drawing a line between each matching pair, and counting a contribution for each crossing in the resulting picture. So there should be a completely combinatorial formula.

As I said, this may well not be what you want. For instance, this construction does not depend on $lambda$. But maybe it is related.

In case you are wondering about the connection with crystal bases (i.e. the title of Kashiwara's paper), Kashiwara shows that the inner product of any two elements in the crystal lattice $L(infty)$ is regular at $q=0$, and a crystal basis is an orthonormal basis for the evaluation $(cdot, cdot)_0$ of the inner product at $q=0$. Of course for this to be true you need to get the powers of $q$ right, which I have not done here.

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