On large Dixmier algebras (Q1183301)

Let \(\mathfrak g\) be a complex semisimple Lie algebra. This paper studies Dixmier algebras (associative algebras \(A\) with the structure of Harish- Chandra bimodules). Call such an algebra \(A\) strongly prime if it has a (genuine) infinitesimal character and the product of any two of its nonzero subbimodules is nonzero. If \(A\) is strongly prime, then it is shown that \(A\) may be realized as a subalgebra and bimodule direct summand of \(L(M,M)\), the ring of \(\mathfrak g\)-finite endomorphisms of some module \(M\) in category \(\mathcal O\). If in addition \(A\) has maximum possible Gelfand-Kirillov dimension, then \(M\) may be taken to be a direct sum of submodules of Verma modules; moreover, \(A\) is then completely prime. Finally, if in addition to the above conditions all \(K\)-types of \(A\) lie in the root lattice, then \(A\) is the quotient of the enveloping algebra of \(\mathfrak g\) by a minimal primitive ideal.