Induced ideals and Dixmier algebras
Let \({\mathfrak g}\) be a possibly infinite-dimensional Lie algebra over a field \(k\) and \(U({\mathfrak g})\) its enveloping algebra. Let \(u\to \check u\) be the principal antiautomorphism of \(U({\mathfrak g})\), acting by \(-1\) on \({\mathfrak g}\). Suppose that \({\mathfrak q}\) is a subalgebra of \({\mathfrak g}\) of finite codimension and that \(V\) is a representation of \({\mathfrak q}\). Then one can construct two ideals in \(U({\mathfrak g})\) induced from ideals in \(U({\mathfrak q})\), one by twisted induction from \(V\) (and taking the annihilator) and the other by twisted induction from \(V^*\); here the twist is by the modular character which is half the trace of the adjoint representation on \({\mathfrak g}/{\mathfrak q}\). \textit{M. Duflo} showed that these two ideals are related by the antiautomorphism \(u\to\check u\) if \({\mathfrak g}\) is finite-dimensional and conjectured that this still holds if \({\mathfrak g}\) is infinite-dimensional [Invent. Math. 67, 385-393 (1982; Zbl 0501.17006)]. Here the author gives a straightforward and purely algebraic proof which moreover generalizes to show that the existence of an antiautomorphism on a Dixmier algebra is preserved under induction of Dixmier algebras (as conjectured by Vogan). Duflo's result had earlier been proved by \textit{S. Chemla}, using the theory of \(D\)-modules [Ann. Inst. Fourier 44, 1067-1090 (1990; Zbl 0815.17022)].
scientific article; zbMATH DE number 954743