Harish-Chandra homomorphism for generalized Takiff algebras
Let \({\mathfrak g}\) be a semisimple Lie algebra over an algebraically closed field \(k\) of characteristic zero. The generalized Takiff algebra \({\mathfrak g}_\infty\) of \({\mathfrak g}\) is the algebra obtained by extending scalars to the polynomial ring \(k[ T]\) in one variable over \(k\) and then restricting scalars back to \(k\). It is the inverse limit of its quotients \({\mathfrak g}_m:= {\mathfrak g}/ {\mathfrak p}_m\) by the ideals \({\mathfrak p}_m\) generated by \(T^m \otimes {\mathfrak g}\); the first such quotient \({\mathfrak g}_1\) is the (ordinary) Takiff algebra. In previous work the author has shown that the enveloping algebra of \({\mathfrak g}_1\) is a free module over its center (using methods that should extend to all \(m\)) [C. R. Acad. Sci., Paris, Sér. I 319, 11-14 (1994; Zbl 0812.17010)]. In this paper he studies the center itself, showing that it is isomorphic to the ring of invariant functions in the symmetric algebra by symmetrization, while the latter ring is embedded in the ring of Weyl group invariants on the symmetric algebra of a Cartan subalgebra via a map which becomes an isomorphism on passing to the quotient fields.
scientific article; zbMATH DE number 735310