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Wu, Xiaohan (Stella)

University of British Columbia

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 VarG and VarH denote the categories of H-varieties and G-varieties with equivariant morphisms, respectively. We show that the restriction functor ResᴳH: VarG —> VarH has both left and right adjoint functors -- we call them IndᴳH and CIndᴳH, respectively. We show that formulae similar to Mackey's decomposition in representation theory hold for these two functors. For the functor CIndᴳH, we study whether properties of H-varieties lift to their corresponding G-varieties. When G is affine, we use the functor CIndᴳH to obtain an upper bound on the essential dimension of G based on that of H. In addition, we show that the functor CIndᴳH can be used to construct equivariant compactifications of G-varieties from those of H-varieties. When combined with the functor IndᴳH 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.

Text

2025-04-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90686

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/