Actions of Algebraic Groups on the Spectrum of Rational Ideals Original Research Article

Letkbe an algebraically closed field andGa linear algebraic group overkacting rationally on ak-algebraV. Generalizing work of Moeglin and Rentschler in characteristic zero, we study the action ofGon the spectrum of rational ideals ofV. The main result is the following. Suppose thatVis semiprime left Goldie. LetLbe aG-stable commutative semisimple subalgebra of the total ring of fractionsQ(V) ofVsuch thatLG=k·1L. This occurs, for example, if the zero ideal ofVisG-rational andLis the center ofQ(V). Then there is, for some closed subgroupHofG, aG-equivariant embedding ν ofLintoQ(G/H) (the algebra of rational functions onG/H) such thatQ(G/H) is purely inseparable over ν(L). This has applications to the closure of the orbit of a rational ideal.