Weierstrass sections for some truncated parabolic subalgebras

Abstract

In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ 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 $\mathfrak p$, 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 $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators. Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal $\mathfrak s\mathfrak l_2$-triple in the non reductive case.