On the orbit method for the Lie algebra of vector fields on a curve
Let \(G\) be the Lie algebra of vector fields on an affine smooth curve. With any element \(\lambda\in G^*\) having finite rank, one associates a canonical polarization \(P\). The main result of the paper presents a criterion for the induced module \(\text{Ind}_P^G(\lambda)\) to be simple. Moreover, it is proved that for any \(\lambda\) as above, the annihilator of \(\text{Ind}_P^G(\lambda)\) in \(U(G)\) is a primitive ideal. This theorem suggests the idea of a Dixmier map for \(G\). Finally, the author calculates the Gelfand-Kirillov dimension of \(\text{Ind}_P^G(\lambda)\) and conjectures that the annihilator of \(\text{Ind}_P^G(\lambda)\) in \(U(G)\) has finite Gelfand-Kirillov codimension.
scientific article; zbMATH DE number 1200014
scientific article