On modules arising from quantum groups at $p^r$-th roots of unity
This paper studies the "reduction mod $p$" method, which constructs large classes of representations for a semisimple algebraic group $G$ from representations for the corresponding Lusztig quantum group $U_\zeta$ at a $p^r$-th root of unity. The $G$-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between $\operatorname{Ext}^n_G(V,W)$ and $\operatorname{Ext}^n_{U_\zeta}(V',W')$, when $V,W$ are obtained from $V',W'$ by reduction mod $p$. Since the dimensions of $\operatorname{Ext}^n$-spaces for $U_\zeta$-modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between certain rational $G$-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the $p$-th root of unity case and a general $p^r$-th root of unity case are essentially different.