# Why are the quantum Fock spaces in FLOTW the same as Uglov's?

Theorem 2.5 in the well-known FLOTW paper [1] and Theorem 2.1 in Uglov's paper [2] both refer to the original JMMO paper [3] to define quantum Fock spaces, i.e. Fock spaces for $$U_q(\widehat{\mathfrak{sl}}_n)$$, the affine quantum group. While Uglov even refers to the FLOTW paper as his source, I do not quite see how their definitions are equivalent: Let $$\lambda$$ be a multipartition and $$v_\lambda$$ the corresponding basis vector of a Fock space for a charge $$\mathbf{s}=(s_1,\dots, s_r)\in\mathbb{Z}^r$$.
In FLOTW, the element $$d$$ of $$U_q(\widehat{\mathfrak{sl}}_n)$$ acts as $$q^{-N_0(\lambda)}$$ on $$v_\lambda$$, where $$N_0(\lambda)$$ is the number of nodes of $$\lambda$$ which have residue $$0$$.
In Uglov's paper, it seems that this element acts as the scalar $$-(\Delta(\mathbf{s})+N_0(\lambda))$$.
The summand $$\Delta(\mathbf{s})$$ is some integer depending on $$\mathbf{s}$$. This integer is fine, as FLOTW only consider more restrictive $$\mathbf{s}$$, in which case $$\Delta(\mathbf{s})=0$$.
What does cause my problem is that in FLOTW the element $$d$$ acts as a power of $$q$$ and in Uglov's paper it does not. Why is that?

On a related note: In my understanding, a weight vector of weight $$\Lambda$$ would in particular be a vector on which $$d$$ acts as $$q^{\Lambda(d)}$$. However, this definition is not compatible with the action given in Uglov and the fact that all basis vectors $$v_\lambda$$ are weight vectors.

References
[1] FLOTW: https://arxiv.org/pdf/q-alg/9710007.pdf
[2] Uglov: https://arxiv.org/pdf/math/9905196.pdf