The Canonical Component of the nilfibre for Parabolic adjoint action in type $A$

Abstract

This work is a continuation of [Fittouhi and Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type $A$]. Let $P$ be a parabolic subgroup of an irreducible simple algebraic group $G$, $P'$ its derived group and $\mathfrak m$ be the nilradical to its Lie algebra. A theorem of Richardson implies that the subalgebra $\mathbb C[\mathfrak m]^{P'}$, spanned by the $P$ semi-invariants in $\mathbb C[\mathfrak m]$, is polynomial. A linear subvariety $e+V$ of $\mathfrak m$ is is called a Weierstrass section for the action of $P'$ on $\mathfrak m$, if the restriction map induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$. Thus a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. The existence of a Weierstrass section $e+V$ in $\mathfrak m$ was established by a general combinatorial construction. Notably $e \in \mathscr N$ and is a sum of root vectors with linearly independent roots. The Weierstraass section $e+V$ looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the "canonical Weierstrass section". It was announced in [Fittouhi and Joseph, loc. cit.] that one may augment $e$ to an element $e_{VS}$ by adjoining root vectors. Then the linear span $E_{VS}$ of these root vectors lies in $\mathscr N^e$ and its closure is just $\mathscr N^e$. Yet this result shows that $\mathscr N^e$ need not admit a dense $P$ orbit. However this theorem was only verified in the special case needed to obtain the example showing that $\mathscr N^e$ may fail to admit a dense $P$ orbit. Here a general proof is given. Finally a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section.