On the imprimitivity theorem for algebraic groups (Q1073910)

Let G be an affine algebraic group, defined over an algebraically closed field k. Let H be an affine algebraic subgroup of G such that the homogeneous space G/H is affine. For a rational H-module X, define a right H-module structure on \(k[G]\otimes X\) by \((f\otimes v)h=f\cdot h\otimes h^{-1}v.\) Then \((k[G]\otimes X)^ H\) is a rational G-module called the G-module induced from H and denoted \(X|^ G_ H.\) Let G be as above and A be an affine k-algebra which is a rational G- module with G acting as algebra automorphisms. An \(A\cdot G\)-module M is an A-module which is also a rational G-module such that \(g(am)=g(a)g(m),\) \(g\in G\), \(a\in A\) and \(m\in M.\) Parshall and Scott have shown that the induction functor \(1/2-|^ G_ H\) is exact and using this, they have shown that the category, \(Mod(k[G/H]\cdot G)\), of \(k[G/H]\cdot G\)-modules is equivalent to the category, Mod(H), of rational H-modules. In this paper, the author proves a more general result and obtains the results of Parshall-Scott as consequences. Let G and A be as above. Let \(\alpha: A\to k\) be a k-algebra homomorphism and let B denote the Hopf algebra \(k[G]\otimes_ Ak\). The author shows that if A has no nontrivial G-stable ideals, then the functor \(1/2-\otimes_ Ak\) is an equivalence between the category of \(A\cdot G\)-modules and the category of B-modules. As another consequence of the main result, the author proves the following: Assume \(char(k)=p>0\). Let \(G_ n\) be as above. Let \(G_ n\) be the finite group scheme which is the kernel of the n-th power of the Frobenius on G. Let \(A_ n\) be the sub-algebra of k[G] consisting of all \(p^ n\)-powers. Then \(1/2-\otimes_{A_ n}k\) is an equivalence between the category of \(A_ n\cdot G\)-modules and the category of rational \(G_ n\)-modules.