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Abstract

Let 𝑮 be an algebraic group defined over an algebraically closed field of arbitrary characteristic, and let 𝑯 be a closed subgroup of 𝑮 with finite index. Let Varʜ and Varɢ denote the categories of 𝑯-varieties and 𝑮-varieties with equivariant morphisms, respectively. We show that the restriction functor Resᴳʜ: Varɢ —> Varʜ has both left and right adjoint functors -- we call them Indᴳʜ and CIndᴳʜ, respectively. We show that formulae similar to Mackey's decomposition in representation theory hold for these two functors. For the functor CIndᴳʜ, we study whether properties of 𝑯-varieties lift to their corresponding 𝑮-varieties. When 𝑮 is affine, we use the functor CIndᴳʜ to obtain an upper bound on the essential dimension of 𝑮 based on that of 𝑯. In addition, we show that the functor CIndᴳʜ can be used to construct equivariant compactifications of 𝑮-varieties from those of 𝑯-varieties. When combined with the functor Indᴳʜ and the results on regular compactification of (connected) reductive groups due to Bifet, DeConcini and Procesi, we obtain an application in constructing group compactifications of a disconnected affine algebraic group whose connected component is reductive.