The Canonical component of the nilfibre for parabolic adjoint action in type A
Authors: Fittouhi Yasmine , Joseph Anthony 
Published: February 2024

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.