I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about the induction functor that seems contradictory to me.
Let $U_q$ be Lusztig version of the usual Drinfeld-Jimbo quantum group associated to a semisimple complex Lie algebra $\mathfrak g$, defined over $\mathcal A = \mathbb Z[v]_{(v-1,p)}$ (where $p \neq 2$ is a prime). We take $\mathfrak g = \mathfrak{sl}_2$. Let $U^0$ be the subalgebra generated by the $K$ and $U^{b}$ be the subalgebra generated by $U^0$ and the divided powers $F^{(r)}$.
Claim 1 (formula 2.11) : We have an isomorphism of $U^b$-modules, given by $H^0(U^b/U^0,M) \cong \bigoplus_{\lambda \in X} M_{\lambda} \otimes \mathcal A[U^b]_{-\lambda}$. Here subscripts mean weight spaces, and $\mathcal A[U^b]$ is the restricted dual, i.e $H^0(U^b/\mathcal A, \mathcal A)$.
Claim 2 (proof of proposition 4.2) : If $M = \lambda_m$ is a the one-dimensional module corresponding to the character $\lambda_m : U_0 \to \mathcal A, K \mapsto v^m$, then $H^0(U^b/U^0, \lambda_m)$ has weight $\{\lambda_{m+2i} : i \geq 0 \}$, each occuring with multiplicity one.
However, it seems that applying claim 1, since we know that $(\lambda_m)$ is 1-dimensional of weight $m$, hence $H^0(U^b/U^0, \lambda) = \mathcal A[U]_{-\lambda_m}$ which is absurd.
