The Canonical component of the nilfibre for parabolic adjoint action in type A

This work is a continuation of [Y. Fittouhi and A. 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. Let \(P'\) $P^{\prime }$ be the derived group of P, and let \({\mathfrak {m}}\) $\mathfrak{m}$ be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\) $\mathbb{C}{[\mathfrak{m}]}^{P^{\prime }}$ , spanned by the P semi-invariants in \({\mathbb {C}}[{\mathfrak {m}}]\) $\mathbb{C}[\mathfrak{m}]$ , is polynomial. A linear subvariety \(e+V\) $e+V$ of \({\mathfrak {m}}\) $\mathfrak{m}$ is called a Weierstrass section for the action of \(P'\) $P^{\prime }$ on \({\mathfrak {m}}\) $\mathfrak{m}$ , if the restriction map induces an isomorphism of \({\mathbb {C}}[{\mathfrak {m}}]^{P'}\) $\mathbb{C}{[\mathfrak{m}]}^{P^{\prime }}$ onto \({\mathbb {C}}[e+V]\) $\mathbb{C}[e+V]$ . Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let \({\mathscr {N}}\) $\mathcal{N}$ be zero locus of the augmentation \({\mathbb {C}}[{\mathfrak {m}}]^{P'}_+\) $\mathbb{C}{[\mathfrak{m}]}_{+}^{P^{\prime }}$ . It is called the nilfibre relative to this action. Suppose \(G=\textrm{SL}(n,{\mathbb {C}})\) $G=\text{SL}(n,\mathbb{C})$ , and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section \(e+V\) $e+V$ in \({\mathfrak {m}}\) $\mathfrak{m}$ was established by a general combinatorial construction. Notably, \(e \in {\mathscr {N}}\) $e\in \mathcal{N}$ and is a sum of root vectors with linearly independent roots. The Weierstrass section \(e+V\) $e+V$ 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”. Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a “canonical” component \({\mathscr {N}}^e\) ${\mathcal{N}}^e$ of \({\mathscr {N}}\) $\mathcal{N}$ containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element \(e_\textrm{VS}\) $e_{\text{VS}}$ by adjoining root vectors. Then the linear span \(E_\textrm{VS}\) $E_{\text{VS}}$ of these root vectors lies in \(\mathscr {N}^e\) ${\mathcal{N}}^e$ and its closure is just \({\mathscr {N}}^e\) ${\mathcal{N}}^e$ . Yet, this same result shows that \({\mathscr {N}}^e\) ${\mathcal{N}}^e$ need not admit a dense P orbit [Y. Fittouhi and A. Joseph, loc. cit., Lemma 6.10.7]. For the above [Y. Fittouhi and A. Joseph, loc. cit., Theorem 6.10.3] was needed. However, this theorem was only verified in the special case needed to obtain the example showing that \({\mathscr {N}}^e\) ${\mathcal{N}}^e$ may fail to admit a dense P orbit. Here a general proof is given (Theorem 4.4.5). 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. One may anticipate that the remaining components of \({\mathscr {N}}\) $\mathcal{N}$ can be similarly described. However, this is a long story and will be postponed for a subsequent paper. These results should form a template for general type.