Weierstrass sections for some truncated parabolic subalgebras

Abstract

In this paper, using Bourbaki's convention, we consider a simple Lie algebra g⊂glm of type B, C or D and a parabolic subalgebra p of g associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with p, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space p⁎ of p and which allow to linearize semi-invariant generators. Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal sl2-triple in the non reductive case.