On modules arising from quantum groups at $p^r$-th roots of unity

TL;DR

This paper explores the relationship between quantum group representations at roots of unity and algebraic group representations, establishing new connections between their extension groups and providing applications to Weyl modules.

Contribution

It introduces a method linking Ext groups of algebraic groups and quantum groups at $p^r$-th roots of unity, revealing new extension classes and homomorphisms.

Findings

Relation between Ext groups of G and quantum groups established

New proof of homomorphisms between Weyl modules provided

Differences between $p$-th and $p^r$-th roots of unity cases demonstrated

Abstract

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.