Setup
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements of the group ring $\mathbb Z[\Lambda]$ of $\Lambda$ as linear combinations of elements of the form $e^\lambda$, $\lambda \in \Lambda$. In particular, characters of finite-dimensional $B$-modules are elements of $\mathbb Z[\Lambda]$.
For dominant $\lambda \in \Lambda$ let $ V(\lambda) $ denote the Weyl module for $G$ with highest weight $\lambda$ and for any element $w$ in the Weyl group $W$ of $G$ let $V_w(\lambda)$ denote the Demazure submodule of $V(\lambda)$ associated to $w$; this is the $B$-submodule of $V(\lambda)$ generated by an extremal vector of weight $w\lambda$. (Remark in particular that $V_{w_0}(\lambda) = V(\lambda)$).
For any simple root $\alpha_i$ of $G$ with associated simple reflection $s_i \in W$ define the Demazure operator $D_{s_i} : \mathbb Z[\Lambda] \to \mathbb Z[\Lambda]$ by $$ D_{s_i}(e^\lambda) = \frac{ e^\lambda - e^{s_i \lambda - \alpha_i} }{ 1-e^{-\alpha_i} } . $$ It is easy to see that this is well-defined. For any word $\mathfrak w = (s_{i_1}, \ldots, s_{i_k})$ of simple reflections in $W$ we have a Demazure operator $D_{\mathfrak w}$ defined by the obvious composition.
We now have the following theorem: Choose $w \in W$ and let $ \mathfrak w $ be any (not necessarily reduced!) word of simple reflections representing $w$. Then the character of $V_w(\lambda)$ is $D_{\mathfrak w}(e^\lambda)$. [A reference for this is, say, section 3.3 of Brion-Kumar's Frobenius splitting book].
Question
Has anyone studied $q$-analogues of these Demazure operators? In light of recent work on $q$-character formulas and Kazhdan-Lusztig polynomials, this seems like a natural combinatorial thing to consider. For example, I would (perhaps naively) expect that an appropriate $q$-analogue of the Demazure operators computes, say, the $q$-analog of weight multiplicity considered by Kazhdan-Lusztig, R. Brylinski, Joseph, and others. (Also, I don't know much about the path model or crystal bases, but it seems as though there may be a connection to those as well).
