Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

We study rational actions of a linear algebraic groupGon an algebraV, and the induced actions on Rat(V), the spectrum of rational ideals ofV(a subset of Spec (V) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions onV. In particular, we relate properties of a rational idealJand itsorb, the ideal (J : G) = γ Gγ(J). The rational ideals ofVcontaining the orb ofJare precisely those in the Zariski-closureXof the orbit ofJin Rat(V). TheG-stratumofJconsists of all rational ideals inXwhose orbit is dense inX(i.e., whose orb is equal to the orb ofJ). We show that theG-stratum of a rational ideal consists of exactly oneG-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit ofJis open in its closure in Rat(V), provided thatJis locally closed. Among other results, we show that the semiprime ideal (J : G) is Goldie, and we relate the uniform and Gelfand–Kirillov dimensions ofV/JandV/(J : G).