$\DeclareMathOperator\dim{dim}$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"](http://www.numdam.org/item/?id=AST_1983__101-102__208_0) and [Joseph, Letzter and Zelikson - "On the Brylinski–Kostant filtration"](https://www.jstor.org/stable/2646135). 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](https://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials). 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](https://mathoverflow.net/a/31685/25028), 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.



Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?
I think that the best reference for what I mention here is Stembridge's notes ["Kostka-Foulkes Polynomials of General Type"](https://www.aimath.org/WWN/kostka/stembridge.pdf). The picture you describe for type A actually generalizes nicely to all types.


Let's denote by $\Lambda$ the weight lattice, by $\Lambda^{+}$ the dominant weights, $W$ the Weyl group, and by $\chi\_{\lambda}=\sum\_{\mu} \dim(V^{\lambda}\_{\mu})e^{\mu}$ the Weyl characters that live in the ring $\mathbb Z[\Lambda]^W$. There are Hall-Littlewood polynomials $P(\lambda,q)$ defined in the notes above, that satisfy the important identity
$$\chi\_{\lambda}=\sum\_{\mu\in \Lambda^+}K\_{\lambda, \mu}(q)P(\mu,q) \tag{1}$$
In Lusztig's paper that you refer to, he proves this for one of the definitions of the $K\_{\lambda,\mu}$ (Kazhdan-Lusztig for affine Weyl group) and states that conjecturally this is the same if one defines them by taking the q-analog of the Kostant's partition function (conjecture 9.6). This conjecture was proved by Kato in ["Spherical Functions and a q-Analogue of Kostant's Weight Multiplicity Formula"](https://eudml.org/doc/142892) using methods from spherical analysis on p-adic groups. A more elementary proof was given by R. K. Brylinski in ["Characters and the q-analog of weight multiplicity"](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-36.1.68).


In your question you want to take the principal specialization of $(1)$, so that $\chi\_{\lambda}$ becomes $\operatorname{dim}\_q(V^{\lambda})$. As a result, $c\_{\lambda,\mu}(q)$ becomes the principal specialization of the Hall-Littlewood polynomial $P(\mu,q)$. Just like in type A, this is always a nice q-product. In fact, this was one of the motivating examples behind the evaluation conjecture for Macdonald Polynomials (See conjecture 12.10 in Macdonald's ["Orthogonal polynomials associated with root systems"](https://arxiv.org/abs/math/0011046), and the following discussion in (v) where it is explained that the principal specialization is known for the Hall-Littlewood case). The general principal specialization of Macdonald polynomials was proven later by Cherednik in ["Macdonald's Evaluation Conjectures and Difference Fourier Transform"](https://arxiv.org/abs/q-alg/9412016).


