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

Abstract

AbstractThis 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 ′ be the derived group of P, and let $${\mathfrak {m}}$$ m be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra $${\mathbb {C}}[{\mathfrak {m}}]^{P'}$$ C [ m ] P ′ , spanned by the P semi-invariants in $${\mathbb {C}}[{\mathfrak {m}}]$$ C [ m ] , is polynomial. A linear subvariety $$e+V$$ e + V of $${\mathfrak {m}}$$ m is called a Weierstrass section for the action of $$P'$$ P ′ on $${\mathfrak {m}}$$ m , if the restriction map induces an isomorphism of $${\mathbb {C}}[{\mathfrak {m}}]^{P'}$$ C [ m ] P ′ onto $${\mathbb {C}}[e+V]$$ 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 $${\mathscr {N}}$$ N be zero locus of the augmentation $${\mathbb {C}}[{\mathfrak {m}}]^{P'}_+$$ C [ m ] + P ′ . It is called the nilfibre relative to this action. Suppose $$G=\textrm{SL}(n,{\mathbb {C}})$$ G = SL ( n , 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 $${\mathfrak {m}}$$ m was established by a general combinatorial construction. Notably, $$e \in {\mathscr {N}}$$ e ∈ 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 $${\mathscr {N}}^e$$ N e of $${\mathscr {N}}$$ 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 VS by adjoining root vectors. Then the linear span $$E_\textrm{VS}$$ E VS of these root vectors lies in $$\mathscr {N}^e$$ N e and its closure is just $${\mathscr {N}}^e$$ N e . Yet, this same result shows that $${\mathscr {N}}^e$$ 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 $${\mathscr {N}}^e$$ 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 $${\mathscr {N}}$$ 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.