$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, *Lusztig's $q$-analog of weight multiplicty* $K_{\lambda,\mu}(q)$ is a $q$-analog of the dimension $\dim V^{\lambda}_{\mu}$ of the $\mu$-weight space $V^{\lambda}_{\mu}$ of the highest weight $\mathfrak{g}$-irrep $V^{\lambda}$ with highest weight $\lambda$. There are a few ways to define it, but probably the simplest is to think of it as the Hilbert series associated to a filtration on $V^{\lambda}_{\mu}$ induced by the action of any regular nilpotent element of $\mathfrak{g}$. See for instance Lusztig's original paper "Singularities, character formulas, and a $q$-analog of weight multiplicities" and Joseph, Letzter and Zelikson - "On the Brylinski–Kostant filtration". In Type A, the $K_{\lambda,\mu}(q)$ are called *Kostka–Foulkes polynomials* and sometimes this name is used in other types too.

(Other ways to define $K_{\lambda,\mu}(q)$: via a $q$-analog of Kostant's partition function; in terms of intersection cohomology of Schubert varieties; as certain affine Kazhdan–Lusztig polynomials.)

On the other hand, let us define the *$q$-dimension* of $V^{\lambda}$ to be $\dim_q(V^{\lambda}) \mathrel{:=} \sum_{\mu} (\dim V^{\lambda}_{\mu}) q^{\langle \mu, \rho^{\vee}\rangle}$, where $\rho^{\vee}$ is the dual of the Weyl vector $\rho$, which is the sum of the fundamental weights. I believe the Weyl character formula tells us that $\dim_q(V^{\lambda}) = \prod_{\alpha \in \Phi^+} \frac{[\langle\lambda+\rho,\alpha\rangle]_q}{[\langle\lambda,\alpha\rangle]_q}$, where $\Phi^+$ are the positive roots of $\mathfrak{g}$ and $[m]_q \mathrel{:=} \frac{(q^{1/2}-q^{-1/2})^m}{(q^{1/2}-q^{-1/2})}$. Maybe I'm slightly off here, but something more-or-less like this should be true, and I think the notion of *$q$-Weyl dimension formula* is anyways an established thing.

**Question**: what is the relationship between the $K_{\lambda,\mu}(q)$ and $\dim_q(V^{\lambda})$? In particular, is there some way to write $\dim_q(V^{\lambda}) = \sum_{\mu} c_{\lambda,\mu}(q) K_{\lambda,\mu}(q)$ for some "simple" coefficients $c_{\lambda,\mu}(q)$?

Note that in Type A we have $s_{\lambda}(\mathbf{x}) = \sum_{\text{dominant $\mu$}} K_{\lambda,\mu}(q) P_{\lambda}(\mathbf{x})$, where $s_{\lambda}(\mathbf{x})$ is the Schur polynomial and $P_{\lambda}(\mathbf{x})$ is the Hall–Littlewood polynomial. And the $q$-Weyl dimension is essentially a principal specialization of $s_{\lambda}(\mathbf{x})$, and I believe the corresponding specialization of $P_{\lambda}(\mathbf{x})$ should give something like a $q$-binomial although I didn't fully work it out.

My main conceptual difficulty in linking $K_{\lambda,\mu}(q)$ and $\dim_q(V^{\lambda})$ is that, for $\dim_q(V^{\lambda})$, each weight space contributes a single power of $q$, while for the $K_{\lambda,\mu}(q)$ the weight spaces contribute many different powers of $q$.

But I still suspect something like what I'm asking for should be true and probably well-known, although I'm having trouble googling for the answer.

EDIT: I just noticed that in a comment to this MathOverflow answer of Jim Humphreys, Victor Protsak says "There is an interesting $q$-analogue of the character of a finite-dimensional module that involves the notion of $q$-multiplicity of weight due to Lusztig, and it specializes into a natural $q$-analogue of the Weyl dimension formula." An expansion of Victor's comment would likely answer my question.