Hopf $2$-cocycles for certain affine algebraic reductive groups

Motivated by the open problem of classifying Hopf $2$-cocycles for affine algebraic reductive groups $G$ over $\mathbb{C}$, in particular by \cite[Question 7.2]{EG1} which concerns minimal Hopf $2$-cocycles, we classify (minimal) Hopf $2$-cocycles for affine algebraic reductive groups $G$ whose connected component of the identity is a torus $T$. To achieve this we utilize \cite[Proposition 5.4]{ENO} in our situation to classify finite indecomposable semisimple module categories over the infinite semisimple equivariantization tensor category $\Rep(T)^K\simeq \Rep(G)$, where $K:=G/T$. We then use it to classify the module categories over $\Rep(G)$ of rank $1$, and show that the corresponding fiber functors on $\Rep(G)$ are classical (that is, preserve dimensions), which implies that they are in bijection with Hopf $2$-cocycles for $G$.