$\DeclareMathOperator\dim{dim}$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.

  • $\begingroup$ For the coefficients $c_{\lambda,\mu}(q)$: I am happy to either sum over all $\mu$ or all dominant $\mu$. When $\mu$ is not dominant there are two different things $K_{\lambda,\mu}(q)$ could mean: one is defined in terms of the $q$-Kostant partition function and is "bad" because it can have negative coefficients; the one I want is the one coming from the Brylinski-Kostant filtration. It is known that this second one is equal to $q^{\textrm{some explicit power}} \cdot K_{\lambda,\nu}(q)$ where $\nu$ is the dominant representative of $W\mu$ (see the Joseph-Letzter-Zelikson paper). $\endgroup$ – Sam Hopkins Jan 10 '20 at 23:40

I think that the best reference for what I mention here is Stembridge's notes "Kostka-Foulkes Polynomials of General Type". 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" 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".

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", 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".

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    $\begingroup$ Thanks, especially for all the references; this is very illuminating. $\endgroup$ – Sam Hopkins Jan 21 '20 at 3:10
  • $\begingroup$ Is it possible that the principal specialization of the $P(\mu,q)$ gives exactly the sum of all the $q^{\textrm{some explicit power}}$ (from my comment under the question) for the Weyl orbit of $\mu$? So that one could simply sum all the $K_{\lambda,\mu}(q)$ over all including non-dominant $\mu$ to get the $q$-Weyl dimension? $\endgroup$ – Sam Hopkins Jan 21 '20 at 3:47
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    $\begingroup$ Disclosure of interest in this: Rhoades (arxiv.org/abs/1005.2568) proved cyclic sieving results for both the Kostka-Foulkes polynomials (see also Fontaine-Kamnitzer arxiv.org/abs/1212.1314), and the $q$-Weyl dimension polynomials. But his method of relating the two results is kind of subtle. I was wondering if the Kostak-Foulkes result formally implies the $q$-Weyl dimension result. $\endgroup$ – Sam Hopkins Jan 21 '20 at 4:58
  • $\begingroup$ @SamHopkins I haven't looked at Rhoades paper in a long time, but here are some superficial comments: In type A there are only so many natural candidates for q-analogs, so you will see $\frac{\text{q-thingy}}{\prod [h]_q}$ a lot. As Rhoades remarks, such expressions are $q$-dimensions but also Kostka-Foulkes polynomials. When you upgrade to the $q,t$-analogs you will see some expressions start to diverge, for example some $[h]_q$ will become $(1-q^at^{l+1})$ and some will be $(1-q^{a+1}t^l)$, similarly some expressions will start to diverge once you generalize outside of type A. [continued] $\endgroup$ – Gjergji Zaimi Jan 21 '20 at 5:19
  • $\begingroup$ If anything, the main idea in these papers is that the Kazhdan-Lusztig theory is quite crucial to get these cyclic sieving results. $\endgroup$ – Gjergji Zaimi Jan 21 '20 at 5:21

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