of the symmetric algebra 
, (iii) the structure of 
as 
-module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of 
1 Introduction
1.1
Let G be a simple algebraic group with
${\mathfrak g}={\mathrm {Lie\,}} G$
,
${\mathfrak g}={\mathfrak u}\oplus {\mathfrak t}\oplus {\mathfrak u}^-$
a fixed triangular decomposition, and
${\mathfrak b}={\mathfrak u}\oplus {\mathfrak t}$
the fixed Borel subalgebra. Then
$\Delta $
is the root system of
$({\mathfrak g},{\mathfrak t})$
,
$\Delta ^+$
is the set of positive roots corresponding to
${\mathfrak u}$
, and
$\theta $
is the highest root in
$\Delta ^+$
. Write
$U,T,B$
for the connected subgroups of G corresponding to
${\mathfrak u},{\mathfrak t},{\mathfrak b}$
.
Let
$P=L{\cdot }N$
be a parabolic subgroup of G, with the unipotent radical N and a Levi subgroup L. Then
${\mathfrak n}={\mathrm {Lie\,}} N$
is the nilradical of
${\mathfrak p}={\mathrm {Lie\,}} P$
. The unipotent radicals of the parabolic subgroups provide an important class of non-reductive groups. For instance, N is a Grosshans subgroup of G [Reference Grosshans11, Theorem 16.4]. Various results on the coadjoint representation of
${\mathfrak n}$
can be found in [Reference Elashvili9, Reference Elashvili and Ooms10, Reference Ooms19, Reference Panyushev22]. Our main goal is to elaborate on invariant-theoretic properties of the coadjoint representation
$(N:{\mathfrak n}^{*})$
, but we also consider actions of some larger unipotent groups on
${\mathfrak n}^{*}$
.
Without loss of generality, we may assume that
${\mathfrak p}$
is standard, i.e.,
${\mathfrak p}\supset {\mathfrak b}$
. Then
${\mathfrak n}\subset {\mathfrak u}$
is a sum of root spaces and
$\Delta ({\mathfrak n})$
denotes the corresponding set of positive roots. Unless otherwise stated, “a nilradical” (in
${\mathfrak g}$
) means “the nilradical of a standard parabolic subalgebra” of
${\mathfrak g}$
. Let 
be the Kostant cascade in
$\Delta ^+$
. It is a poset, and
$\beta _1=\theta $
is the unique maximal element of 
(see Section 2.2 for details). To each
${\mathfrak n}$
, we attach the subposet 
. Another ingredient is the optimization of
${\mathfrak n}$
. By definition, it is the maximal nilradical,
$\tilde {\mathfrak n}$
, such that 
. If
${\mathfrak n}=\tilde {\mathfrak n}$
, then
${\mathfrak n}$
is said to be optimal. An explicit description of
$\tilde {\mathfrak n}$
via 
is given in Section 2.3. Properties of the optimal nilradicals are better, and in order to approach arbitrary nilradicals, it is convenient to consider first the optimal ones. Roughly speaking, the output of this article is that to a great extent invariant-theoretic properties of
${\mathfrak n}$
are determined by 
and
$\tilde {\mathfrak n}$
.
1.2
Let
${\mathfrak q}={\mathrm {Lie\,}} Q$
be a Lie algebra. As usual,
$\xi \in {\mathfrak q}^{*}$
is said to be regular, if the stabilizer
${\mathfrak q}^{\xi }$
has minimal dimension. Then
${\mathfrak q}^{*}_{\mathsf {reg}}$
denotes the set of all regular points and
${{\mathrm {ind}}\,{\mathfrak q}:=\dim {\mathfrak q}^{\xi }}$
for any
$\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}$
. If
${\mathrm {ind}}\,{\mathfrak q}=0$
, then
${\mathfrak q}$
is called Frobenius. Set
$\boldsymbol {b}({\mathfrak q}):=(\dim {\mathfrak q}+{\mathrm {ind}}\,{\mathfrak q})/2$
. By definition, the Frobenius semiradical of
${\mathfrak q}$
is
${\mathcal F}({\mathfrak q})=\sum _{\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}}{\mathfrak q}^{\xi }$
. Hence,
${\mathcal F}({\mathfrak q})=0$
if and only if
${\mathfrak q}$
is Frobenius. Clearly,
${\mathcal F}({\mathfrak q})$
is a characteristic ideal of
${\mathfrak q}$
. This notion and basic results on it are due to Ooms [Reference Ooms18, Reference Ooms19].
The symmetric algebra of
${\mathfrak q}$
, 
, is a Poisson algebra equipped with the Lie–Poisson bracket
$\{\ ,\,\}$
. The algebra of symmetric invariants 
is the center of 
, i.e.,
If the group Q is connected, then 
, i.e., the Poisson center 
is also the algebra of Q-invariant polynomial functions on
${\mathfrak q}^{*}$
. If 
is a Poisson-commutative subalgebra, then
${\mathrm {trdeg\,}}\mathcal P\leqslant \boldsymbol {b}({\mathfrak q})$
[Reference Vinberg28, 0.2] and this upper bound is always attained [Reference Sadetov25]. Therefore, if
${\mathfrak r}\subset {\mathfrak q}$
is a Lie subalgebra, then
$\boldsymbol {b}({\mathfrak r})\leqslant \boldsymbol {b}({\mathfrak q})$
. The passage
${\mathfrak n}\leadsto \tilde {\mathfrak n}$
has the property that
$\boldsymbol {b}({\mathfrak n})=\boldsymbol {b}(\tilde {\mathfrak n})$
. This implies that 
(see [Reference Panyushev22, Proposition 5.5]).
An abelian subalgebra
${\mathfrak a}\subset {\mathfrak q}$
is called a commutative polarization (= CP), if
${\dim {\mathfrak a}=\boldsymbol {b}({\mathfrak q})}$
. Then
$\boldsymbol {b}({\mathfrak a})=\boldsymbol {b}({\mathfrak q})$
. A complete classification of the nilradicals with CP is obtained in [Reference Panyushev22]. In Section 3.4, we use Rosenlicht’s theorem to provide simple proofs of some basic properties of CP’s.
1.3
For
${\mathfrak n}={\mathfrak p}^{\mathsf {nil}}$
, let
${\mathfrak n}^-\subset {\mathfrak u}^-$
be the opposite nilradical, i.e.,
$\Delta ({\mathfrak n}^-)=-\Delta ({\mathfrak n})$
. Then
${\mathfrak g}={\mathfrak p}\oplus {\mathfrak n}^-$
. Let
${\mathfrak g}_{\gamma }$
denote the roots space of
$\gamma \in \Delta $
and
$e_{\gamma }\in {\mathfrak g}_{\gamma }$
a nonzero vector. Consider the space 
and 
. Using the vector space isomorphism
${\mathfrak n}^{*}={\mathfrak g}/{\mathfrak p}\simeq {\mathfrak n}^-$
, one can regard
${\mathfrak k}$
as a subspace of
${\mathfrak n}^{*}$
and
${\boldsymbol {\zeta }}$
as an element of
${\mathfrak n}^{*}$
. We say that
${\mathfrak k}\subset {\mathfrak n}^{*}$
is a cascade subspace and
${\boldsymbol {\zeta }}\in {\mathfrak k}$
is a cascade point. As usual,
$\xi \in {\mathfrak n}^{*}$
is called N-generic, if there is an open subset
$\Omega \in {\mathfrak n}^{*}$
such that
$\xi \in \Omega $
and the stabilizer
${\mathfrak n}^{\xi }$
is N-conjugate to
${\mathfrak n}^{\xi '}$
for any
$\xi '\in \Omega $
. Any stabilizer
${\mathfrak n}^{\nu }$
with
$\nu \in \Omega $
is said to be N-generic, too.
In Section 4, we prove that the action
$(N:{\mathfrak n}^{*})$
has N-generic stabilizers if and only if the stabilizer
${\mathfrak n}^{\boldsymbol {\zeta }}$
is generic (and the latter is not always the case!). Moreover, N-generic stabilizers always exist if
${\mathfrak n}$
is optimal. If
${\mathfrak n}$
is not optimal, then one can consider the linear action of the larger group
$\tilde N=\exp (\tilde {\mathfrak n})\supset N$
on
${\mathfrak n}^{*}$
. We prove that the action
$(\tilde N:{\mathfrak n}^{*})$
always has an
$\tilde N$
-generic stabilizer, and
$\tilde {\mathfrak n}^{\boldsymbol {\zeta }}$
is such a stabilizer. Actually, the equality
$\tilde {\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak n}^{\boldsymbol {\zeta }}$
holds here. For any
${\mathfrak n}$
, we give an explicit formula for
${\mathfrak n}^{\boldsymbol {\zeta }}$
via 
and
$\tilde {\mathfrak n}$
, which shows that
${\mathfrak n}^{\boldsymbol {\zeta }}$
is T-stable. Using that formula and the criterion for the coadjoint representations [Reference Tauvel and Yu27, Corollary 1.8(i)], one easily verifies whether the stabilizer
${\mathfrak n}^{\boldsymbol {\zeta }}$
is generic in each concrete example. For
${\mathsf {\mathbf {{A}}}}_{n}$
, the nilradicals having a generic stabilizer for
$(N:{\mathfrak n}^{*})$
are explicitly described, while for
${\mathsf {\mathbf {{C}}}}_{n}$
, all nilradicals have a generic stabilizer (see Section 5). A general construction of nilradicals without generic stabilizers is also provided.
We prove that
${\mathcal F}({\mathfrak n})$
is the
${\mathfrak b}$
-stable ideal of
${\mathfrak n}$
generated by
${\mathfrak n}^{\boldsymbol {\zeta }}$
(regardless of the presence of generic stabilizers). Then our formula for
${\mathfrak n}^{\boldsymbol {\zeta }}$
allows us to explicitly describe
${\mathcal F}({\mathfrak n})$
for
${\mathsf {\mathbf {{A}}}}_{n}$
and
${\mathsf {\mathbf {{C}}}}_{n}$
. For any
${\mathfrak g}$
, we provide a criterion for the equality
${\mathcal F}({\mathfrak n})={\mathfrak n}$
and give the complete list of nilradicals with this property. Another observation is that if
${\mathfrak n}\subset {\mathfrak n}'\subset \tilde {\mathfrak n}$
, then
${\mathcal F}({\mathfrak n}')\subset {\mathcal F}({\mathfrak n})$
.
1.4
Since
${\mathfrak n}$
is B-stable, one can consider the algebra of Q-invariants 
for any subgroup
$Q\subset B$
, and we are primarily interested in the unipotent subgroups U and
$\tilde N=\exp (\tilde {\mathfrak n})\subset U$
. In Section 6, we prove that
and this common algebra is polynomial, of Krull dimension 
. In particular, for any optimal nilradical
$\tilde {\mathfrak n}$
, the Poisson center of 
is a polynomial algebra. If
${\mathfrak n}\ne \tilde {\mathfrak n}$
, then 
does not occur in (1.1). This algebra is not always polynomial, and its Krull dimension equals 
. Nevertheless, 
shares many properties with algebras of invariants of reductive groups. For instance, 
is finitely generated [Reference Joseph12, Lemma 4.6], and we prove that the affine variety 
has rational singularities.
Using results on 
and
${\mathfrak n}^{\boldsymbol {\zeta }}$
, we describe the nilradicals having the property that
${\mathrm {ind}}\,{\mathfrak n}=\dim {\mathfrak z}({\mathfrak n})$
, where
${\mathfrak z}({\mathfrak n})$
is the center of
${\mathfrak n}$
. By [Reference Moore and Wolf15], this property is equivalent to that the Lie group N has a “square integrable representation.” Therefore, such nilpotent Lie algebras are sometimes called “square integrable” (see [Reference Elashvili9, Reference Elashvili and Ooms10]). From a modern point of view, the square integrable nilradicals are precisely the “quasi-reductive” ones (see [Reference Baur and Moreau1, Reference Duflo, Khalgui and Torasso7, Reference Moreau and Yakimova16]). Our description shows that all square integrable nilradicals are metabelian.
As 
is a polynomial algebra, we are interested in question whether 
is a free 
-module. Equivalently, when is the quotient map 
equidimensional? We prove several assertions for U or (what is the same)
$\tilde N$
.
• If
${\mathfrak n}$
has a CP, then 
is a free module over 
. This includes all nilradicals for
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
.
• In particular, if
${\mathfrak n}$
is the optimization of a nilradical with CP (any
${\mathfrak g}$
) or any optimal nilradical in
${\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
, then 
is a free module over its Poisson center 
. This also implies that in these cases, the enveloping algebra 
is a free module over its center.
1.5 Structure of the article
In Section 2, we recall basic facts on 
and (optimal) nilradicals. In Section 3, the necessary information is gathered on generic stabilizers, the Frobenius semiradical, and commutative polarizations. We also include invariant-theoretic proofs for some properties of commutative polarizations. Our results on generic stabilizers and
${\mathcal F}({\mathfrak n})$
for a nilradical
${\mathfrak n}\subset {\mathfrak g}$
are gathered in Section 4, whereas Section 5 contains explicit results for
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
. In Section 6, we study various algebras of invariants related to the coadjoint representation of
${\mathfrak n}$
, and in Section 7, we classify the square integrable nilradicals. Topics related to the equidimensionality of the quotient map
$\pi :{\mathfrak n}^{*}\to {\mathfrak n}^{*}/\!\!/ U$
are treated in Section 8. The lists of cascade roots for all
${\mathfrak g}$
and the Hasse diagrams of some posets 
are presented in Appendix A.
Main notation. Throughout,
${\mathfrak g}={\mathrm {Lie\,}}(G)$
is a simple Lie algebra. Then:
-
–
${\mathfrak b}$
is a fixed Borel subalgebra of
${\mathfrak g}$
with
${\mathfrak u}=[{\mathfrak b},{\mathfrak b}]$
. -
–
${\mathfrak t}$
is a fixed Cartan subalgebra in
${\mathfrak b}$
and
$\Delta $
is the root system of
${\mathfrak g}$
with respect to
${\mathfrak t}$
. -
–
$\Delta ^+$
is the set positive roots corresponding to
${\mathfrak u}$
, and
$\theta \in \Delta ^+$
is the highest root. -
–
$\Pi =\{\alpha _1,\dots ,\alpha _{{\mathsf {rk\,}}{\mathfrak g}}\}$
is the set of simple roots in
$\Delta ^+$
. -
–
${\mathfrak t}^{*}_{\mathbb Q}$
is the
${\mathbb Q}$
-vector subspace of
${\mathfrak t}^{*}$
spanned by
$\Delta $
, and
$(\ ,\, )$
is the positive-definite form on
${\mathfrak t}^{*}_{\mathbb Q}$
induced by the Killing form on
${\mathfrak g}$
. -
– If
$\gamma \in \Delta $
, then
${\mathfrak g}_{\gamma }$
is the root space in
${\mathfrak g}$
and
$e_{\gamma }\in {\mathfrak g}_{\gamma }$
is a nonzero vector. -
– If
${\mathfrak c}\subset {\mathfrak u}^{\pm }$
is a
${\mathfrak t}$
-stable subspace, then
$\Delta ({\mathfrak c})\subset \Delta ^{\pm }$
is the set of roots of
${\mathfrak c}$
. -
–
$\boldsymbol {b}({\mathfrak q})=(\dim {\mathfrak q}+{\mathrm {ind}}\,{\mathfrak q})/2$
for a Lie algebra
${\mathfrak q}$
. -
– In the explicit examples, the Vinberg–Onishchik numbering of simple roots of
${\mathfrak g}$
is used (see [Reference Onishchik and Vinberg17, Table 1]).
2 Generalities on the cascade and nilradicals
2.1 The root order in
$\Delta ^+$
and the Heisenberg subset
We identify
$\Pi $
with the vertices of the Dynkin diagram of
${\mathfrak g}$
. For any
$\gamma \in \Delta ^+$
, let
$[\gamma :\alpha ]$
be the coefficient of
$\alpha \in \Pi $
in the expression of
$\gamma $
via
$\Pi $
. The support of
$\gamma $
is
${\mathsf {supp}}(\gamma )=\{\alpha \in \Pi \mid [\gamma :\alpha ]\ne 0\}$
. As is well known,
${\mathsf {supp}}(\gamma )$
is a connected subset of the Dynkin diagram. For instance,
${\mathsf {supp}}(\theta )=\Pi $
and
${\mathsf {supp}}(\alpha )=\{\alpha \}$
. Let “
$\preccurlyeq $
” denote the root order in
$\Delta ^+$
, i.e., we set
$\gamma \preccurlyeq \gamma '$
if
$[\gamma :\alpha ]\leqslant [\gamma ':\alpha ]$
for all
$\alpha \in \Pi $
. Then
$(\Delta ^+,\preccurlyeq )$
is a graded poset, and we write
$\gamma \prec \gamma '$
if
$\gamma \preccurlyeq \gamma '$
and
$\gamma \ne \gamma '$
.
An upper ideal of
$(\Delta ^+,\preccurlyeq )$
is a subset I such that if
$\gamma \in I$
and
$\gamma \prec \gamma '$
, then
$\gamma '\in I$
. Therefore, I is an upper ideal of
$\Delta ^+$
if and only if
${\mathfrak r}=\bigoplus _{\gamma \in I} {\mathfrak g}_{\gamma }$
is a
${\mathfrak b}$
-stable ideal of
${\mathfrak u}$
, i.e.,
$[{\mathfrak b},{\mathfrak r}]\subset {\mathfrak r}$
.
For a dominant weight
$\lambda \in {\mathfrak t}^{*}_{\mathbb Q}$
, set
$\Delta _{\lambda }=\{\gamma \in \Delta \mid (\lambda ,\gamma )=0 \}$
and
$\Delta ^{\pm }_{\lambda }=\Delta _{\lambda }\cap \Delta ^{\pm }$
. Then
$\Delta _{\lambda }$
is the root system of a semisimple subalgebra
${\mathfrak g}^{\perp \lambda }\subset {\mathfrak g}$
and
$\Pi _{\lambda }=\Pi \cap \Delta ^+_{\lambda }$
is the set of simple roots in
$\Delta ^+_{\lambda }$
. Then:
-
•
${\mathfrak p}_{\lambda }={\mathfrak g}^{\perp \lambda }+{\mathfrak b}$
is a standard parabolic subalgebra of
${\mathfrak g}$
. -
• The set of roots of the nilradical
${\mathfrak n}_{\lambda }={\mathfrak p}_{\lambda }^{\mathsf {nil}}$
is
$\Delta ^+\setminus \Delta ^+_{\lambda }$
. It is also denoted by
$\Delta ({\mathfrak n}_{\lambda })$
.
If
$\lambda =\theta $
, then
${\mathfrak n}_{\theta }$
is a Heisenberg Lie algebra (Heisenberg nilradical) and 
is called the Heisenberg subset (of
$\Delta ^+$
).
2.2 The cascade poset
The recursive construction of the Kostant cascade in
$\Delta ^+$
begins with
$\beta _1=\theta $
. On the next step, we take the highest roots in the irreducible subsystems of
$\Delta _{\theta }$
. These roots are called the descendants of
$\beta _1$
. The same construction is then applied to every descendant of
$\beta _1$
, and so on. This procedure eventually terminates and yields a set 
, which is called the Kostant cascade. The roots in 
are strongly orthogonal, which means that
$\beta _i\pm \beta _j\not \in \Delta $
for all
$i,j$
. We make 
a poset by letting that
$\beta _i$
covers
$\beta _j$
if and only if
$\beta _j$
is a descendant of
$\beta _i$
. Then
$\beta _1$
is the unique maximal element of 
. We refer to [Reference Joseph12, Section 2], [Reference Kostant13], and [Reference Panyushev22, 2.2] for more details. Let us summarize the main features of 
.
-
•
is a maximal set of strongly orthogonal roots in
$\Delta ^+$
. -
• Each
$\beta _i$
is the highest root of the irreducible root system
$\Delta \langle i\rangle \subset \Delta $
with simple roots
${\mathsf {supp}}(\beta _i)$
. -
•
is also a subposet of
$(\Delta ^+,\preccurlyeq )$
, which provides the same poset structure as above. -
• One has
$\beta _j\prec \beta _i$
if and only if
${\mathsf {supp}}(\beta _j)\varsubsetneq {\mathsf {supp}}(\beta _i)$
. -
•
$\beta _j$
and
$\beta _i$
are incomparable in if and only if
${\mathsf {supp}}(\beta _j)\cap {\mathsf {supp}}(\beta _i)=\varnothing $
. -
• The numbering of
is not canonical. It is only required to be a linear extension of , i.e., if
$\beta _j\prec \beta _i$
, then
$j>i$
. In specific examples considered below, we use the numbering of cascade roots given in Appendix A.
is a maximal set of strongly orthogonal roots in 
is also a subposet of 
if and only if 
is not canonical. It is only required to be a linear extension of 
, i.e., if 
Using the decomposition 
and induction on
${\mathsf {rk\,}}{\mathfrak g}$
, one obtains the disjoint union parametrized by 
:
where 
is the Heisenberg subset of
$\Delta \langle i\rangle ^+$
and 
. For
$1\leqslant i\leqslant m$
, let
${\mathfrak g}\langle i\rangle \subset {\mathfrak g}$
be the simple Lie algebra with root system
$\Delta \langle i\rangle $
. The geometric counterpart of (2.1) is the vector space sum
${\mathfrak u}=\bigoplus _{i=1}^m {\mathfrak h}_i$
, where
${\mathfrak h}_i$
is the Heisenberg Lie algebra in
${\mathfrak g}\langle i\rangle $
and 
. In particular,
${\mathfrak h}_1={\mathfrak n}_{\theta }$
. For each 
, we set 
. It then follows from (2.1) that 
. Note that
$\#\Phi (\beta _i)\leqslant 2$
and
$\#\Phi (\beta _i)= 2$
if and only if the algebra
${\mathfrak g}\langle i\rangle $
is of type
${\mathsf {\mathbf {{A}}}}_{n}$
with
$n\geqslant 2$
. Our definition of the subsets
$\Phi (\beta _i)$
yields the well-defined map 
, where
$\Phi ^{-1}(\alpha )=\beta _i$
if
$\alpha \in \Phi (\beta _i)$
. Note that
$\alpha \in {\mathsf {supp}}(\Phi ^{-1}(\alpha ))$
and
$\alpha \in \Phi (\Phi ^{-1}(\alpha ))$
. We think of the cascade poset as a triple 
. The corresponding Hasse diagrams, with subsets
$\Phi (\beta _i)$
attached to every node, are depicted in [Reference Panyushev22, Section 6]. Some of them are included in Appendix A.
Obviously, 
, and 
if and only if each
$\beta _i$
is a multiple of a fundamental weight for
${\mathfrak g}\langle i\rangle $
. Recall that
$\theta $
is a multiple of a fundamental weight of
${\mathfrak g}$
if and only if
${\mathfrak g}$
is not of type
${\mathsf {\mathbf {{A}}}}_{n}$
,
$n\geqslant 2$
. It is well known that 
if and only if
${\mathrm {ind}}\,{\mathfrak b}=0$
. This happens exactly if
${\mathfrak g}\not \in \{{\mathsf {\mathbf {{A}}}}_{n}, {\mathsf {\mathbf {{D}}}}_{2n+1}, {\mathsf {\mathbf {{E}}}}_{6}\}$
and then
$\Phi ^{-1}$
yields a bijection between 
and
$\Pi $
.
2.3 Nilradicals and optimal nilradicals
Let
${\mathfrak p}\supset {\mathfrak b}$
be a standard parabolic subalgebra of
${\mathfrak g}$
, with nilradical
${\mathfrak n}={\mathfrak p}^{\mathsf {nil}}$
. If 
, then we write 
and 
. Here, 
is the set of minimal elements of the poset
$(\Delta ({\mathfrak n}), \preccurlyeq )$
and 
is the set of simple roots for the standard Levi subalgebra 
. Clearly, 
if and only if 
.
The integer 
is the depth of 
. Letting
one obtains the partition 
and the canonical
${\mathbb Z}$
-grading
The following is well known and easy.
Lemma 2.1 If 
denotes the lower central series of 
, then 
. The center of 
is 
. Hence, 
is abelian if and only if 
, i.e., 
and
$[\theta :\alpha ]=1$
.
Set 
. Then 
for any nonzero nilradical 
.
, one has 
is an upper ideal of 
.
and 
. A standard parabolic subalgebra 
is said to be optimal if
This goes back to [Reference Joseph12, 4.10], and we also apply this term to 
. Then 
is optimal if and only if 
. For a nonempty 
, set 
and consider the nilradical 
. Then 
and 
. Hence, 
is optimal, it is the minimal optimal nilradical containing 
, and it is the maximal element of the set of nilradicals 
.
Definition 1 The nilradical 
is called the optimization of 
.
If 
is not specified for a given nilradical
${\mathfrak n}$
, then 
and we write
$\tilde {\mathfrak n}$
for the optimization of
${\mathfrak n}$
.
Proposition 2.3 (cf. [Reference Joseph12, 2.4] and [Reference Panyushev22, 2.3])
Let
$\tilde {\mathfrak n}$
be the optimization of a nilradical
${\mathfrak n}$
. Then:
-
•
. -
•
and
$\boldsymbol {b}({\mathfrak n})=\boldsymbol {b}(\tilde {\mathfrak n})$
.
.
and 
Remark 2.4 The merit of optimization is that the passage from
${\mathfrak n}$
to
$\tilde {\mathfrak n}$
does not change 
and
$\boldsymbol {b}({\mathfrak n})$
. More generally, if two nilradicals
${\mathfrak n}'$
and
${\mathfrak n}$
have the same optimization, then 
and
$\boldsymbol {b}({\mathfrak n}')=\boldsymbol {b}({\mathfrak n})$
. For instance, this happens if
${\mathfrak n}\subset {\mathfrak n}'\subset \tilde {\mathfrak n}$
.
Example 2.5 (1) If
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
, then
${\mathfrak u}$
is the set of strictly upper-triangular matrices,
$\alpha _i=\varepsilon _i-\varepsilon _{i+1}$
(
$1\leqslant i\leqslant n$
), and 
, where
$t=[(n+1)/2]$
and
$\beta _i=\alpha _i+\dots +\alpha _{n+1-i}=\varepsilon _i-\varepsilon _{n+2-i}$
. Here, 
is a chain and
$\Phi (\beta _i)=\{\alpha _i,\alpha _{n+1-i}\}$
.
(2) Take 
and the nilradical 
. Then 
and therefore 
(cf. the matrices below).
The cells with ball represent the cascade, and the thick lines depict the Heisenberg subset attached to an element of the cascade. By Proposition 2.3, we have
${\mathrm {ind}}\,\tilde {\mathfrak n}=2$
,
${\mathrm {ind}}\,{\mathfrak n}=6$
, and
$\boldsymbol {b}({\mathfrak n})=\boldsymbol {b}(\tilde {\mathfrak n})=10$
.
Example 2.6 For a square matrix A, let
$\hat A$
denote its transpose with respect to the antidiagonal. Choose the skew-symmetric form defining
${\mathfrak g}={\mathfrak {sp}}_{2n}\subset {\mathfrak {sl}}_{2n}$
such that
$$\begin{align*}{\mathfrak {sp}}_{2n}=\left\{ \begin{pmatrix} A & M \\ M' & -\hat A\end{pmatrix}\mid M=\hat M \ \& \ M'=\hat M'\right\} , \end{align*}$$
where
$A,M,M'$
are
$n\times n$
matrices. Then
${\mathfrak u}$
(resp.
${\mathfrak t}$
) is the set of symplectic strictly upper triangular (resp. diagonal) matrices. Hence,
${\mathfrak t}=\{\mathsf {diag}(\varepsilon _1,\dots ,\varepsilon _n,-\varepsilon _n,\dots ,-\varepsilon _1)\mid \varepsilon _i\in {\mathbb C} \}$
. Recall that
$\alpha _i=\varepsilon _i-\varepsilon _{i+1}$
(
$i<n$
) and
$\alpha _n=2\varepsilon _n$
. Then 
is a chain, where
$\beta _i=2\varepsilon _i$
and
$\Phi (\beta _i)=\{\alpha _i\}$
for all i. Here,
${\mathfrak n}_{\{\alpha _{n}\}}=\Big\{\Big(\begin {matrix} 0 & M \\ 0 & 0\end {matrix}\Big)\mid M=\hat M \Big\}$
is the nilradical of the maximal parabolic subalgebra with 
. It is the only (standard) abelian nilradical in
${\mathfrak {sp}}_{2n}$
and 
corresponds to the antidiagonal entries of M.
3 Generic stabilizers, the Frobenius semiradical, and commutative polarizations
Let Q be a connected algebraic group with
${\mathfrak q}={\mathrm {Lie\,}} Q$
. If
$\rho :Q\to GL({\mathbb {V}})$
is a representation of Q, then the corresponding Q-action on
${\mathbb {V}}$
is denoted by
$(Q:{\mathbb {V}})$
. For
$q\in Q$
and
$v\in {\mathbb {V}}$
, we write
$q{\cdot }v$
in place of
$\rho (q)v$
. Likewise,
$({\mathfrak q}:{\mathbb {V}})$
corresponds to
$d\rho :{\mathfrak q}\to {\mathfrak {gl}}({\mathbb {V}})$
.
3.1 Generic stabilizers
Let
$(Q:{\mathbb {V}})$
be a linear action. We say that
$v\in {\mathbb {V}}$
is Q-generic, if there is a dense open subset
$\Omega \subset {\mathbb {V}}$
such that
$v\in \Omega $
and the stabilizer
${\mathfrak q}^x$
is Q-conjugate to
${\mathfrak q}^v$
for any
$x\in \Omega $
. Then any
${\mathfrak q}^x$
(
$x\in \Omega )$
is called a Q-generic stabilizer for the representation
$({\mathfrak q}:{\mathbb {V}})$
, and we say that
$({\mathfrak q}: {\mathbb {V}})$
has a Q-generic stabilizer. (One can consider similar notions for non-connected groups, for arbitrary actions of Q, and for stationary subgroups
$Q^x\subset Q$
, but we do need it now.) By semi-continuity of orbit dimensions, the set Q-generic points is contained in the set of Q-regular points
$$ \begin{align} {\mathbb{V}}_{\mathsf{reg}}=\{v\in{\mathbb{V}}\mid \dim Q{\cdot}v \ \text{is maximal}\}, \end{align} $$
but usually, this inclusion is proper. By a result of Richardson [Reference Richardson24], if Q is reductive, then Q-generic stabilizers exist for any action of Q on a smooth affine variety. But this is no longer true for non-reductive groups, and one of our goals is to study (the presence of) generic stabilizers for the coadjoint representation of a nilradical in
${\mathfrak g}$
.
A practical method for proving the existence of Q-generic points and finding Q-generic stabilizers is given by Elashvili [Reference Elashvili8, Lemma 1]. Let
$\mathsf {T}_v(Q{\cdot }v)={\mathfrak q}{\cdot }v$
be the tangent space of the orbit
$Q{\cdot }v$
at v and
${\mathbb {V}}^{{\mathfrak q}_v}$
the fixed point subspace of
${\mathfrak q}^v$
in
${\mathbb {V}}$
. Then
$$ \begin{align} v\in {\mathbb{V}} \text{ is } Q\text{-generic if and only if} \ {\mathbb{V}}={\mathfrak q}{\cdot}v+{\mathbb{V}}^{{\mathfrak q}_v}. \end{align} $$
The main case of interest for us is the coadjoint representation of Q, when
${\mathbb {V}}={\mathfrak q}^{*}$
. For the coadjoint representation, we usually skip “Q” from notation and refer to “generic” and “regular” points (in
${\mathfrak q}^{*}$
) and “generic” stabilizers (in
${\mathfrak q}$
). Translating Elashvili’s criterion (3.2) into the setting of coadjoint representations and taking annihilators, one obtains the following nice formula (see [Reference Tauvel and Yu27, Corollary 1.8(i)]). Given
$\xi \in {\mathfrak q}^{*}$
, the stabilizer
${\mathfrak q}^{\xi }$
is generic (i.e.,
$\xi $
is a Q-generic point) if and only if
$$ \begin{align} [{\mathfrak q},{\mathfrak q}^{\xi}]\cap {\mathfrak q}^{\xi} =\{0\}. \end{align} $$
The reason is that
$({\mathfrak q}{\cdot }\xi )^{\perp }={\mathfrak q}^{\xi }$
and
$(({\mathfrak q}^{*})^{{\mathfrak q}_{\xi }})^{\perp }=[{\mathfrak q},{\mathfrak q}^{\xi }]$
, where
$(\cdot )^{\perp }$
stands for the annihilator in the dual space.
3.2 The Frobenius semiradical
For the Q-module
${\mathbb {V}}={\mathfrak q}^{*}$
, the set of Q-regular (or just “regular”) points
${\mathfrak q}^{*}_{\mathsf {reg}}$
consists of all
$\xi \in {\mathfrak q}^{*}$
such that the stabilizer
${\mathfrak q}^{\xi }$
has the minimal possible dimension. If
$\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}$
, then
${\mathrm {ind}}\,{\mathfrak q}:=\dim {\mathfrak q}^{\xi }$
is the index of (a Lie algebra)
${\mathfrak q}$
. The Frobenius semiradical
${\mathcal F}({\mathfrak q})$
of a Lie algebra
${\mathfrak q}$
is introduced by Ooms (see [Reference Ooms18, Reference Ooms19]). By definition,
$$\begin{align*}{\mathcal F}({\mathfrak q})=\sum_{\xi\in {\mathfrak q}^{*}_{\mathsf{reg}}}{\mathfrak q}^{\xi}. \end{align*}$$
Obviously,
${\mathcal F}({\mathfrak q})$
is a characteristic ideal of
${\mathfrak q}$
, and
${\mathcal F}({\mathfrak q})=0$
if and only
${\mathrm {ind}}\,{\mathfrak q}=0$
(i.e.,
${\mathfrak q}$
is a Frobenius Lie algebra). Note that if
$({\mathfrak q}:{\mathfrak q}^{*})$
has a generic stabilizer, then any Q-generic point in the sense of Section 3.1 is regular, but not vice versa.
Lemma 3.1 (cf. [Reference Ooms19, Proposition 1.7])
If
$({\mathfrak q}:{\mathfrak q}^{*})$
has a generic stabilizer and
$\xi \in {\mathfrak q}^{*}$
is any generic point, then
${\mathcal F}({\mathfrak q})$
is the
${\mathfrak q}$
-ideal generated by the sole stabilizer
${\mathfrak q}^{\xi }$
.
Proof By [Reference Ooms19, Lemma 1.2], if
$\Psi $
is open and dense in
${\mathfrak q}^{*}_{\mathsf {reg}}$
, then
${\mathcal F}({\mathfrak q})=\sum _{\eta \in \Psi }{\mathfrak q}^{\eta }$
. Applying this to the set of generic points
$\Omega \subset {\mathfrak q}^{*}_{\mathsf {reg}}$
, we obtain
${\mathcal F}({\mathfrak q})=\sum _{\eta \in \Omega }{\mathfrak q}^{\eta }=\sum _{g\in Q}{\mathfrak q}^{g{\cdot }\xi }$
. Clearly, the last sum yields the ideal of
${\mathfrak q}$
generated by
${\mathfrak q}^{\xi }$
.
If
${\mathfrak q}$
is quadratic, i.e.,
${\mathfrak q}\simeq {\mathfrak q}^{*}$
as Q-module, then
${\mathcal F}({\mathfrak q})=\sum _{x\in {\mathfrak q}_{\mathsf {reg}}}{\mathfrak q}^x$
. Since
$x\in {\mathfrak q}^x$
for any
$x\in {\mathfrak q}$
, we see that here
${\mathcal F}({\mathfrak q})={\mathfrak q}$
(cf. [Reference Elashvili and Ooms10, Theorem 3.2]). In particular, this is the case if
${\mathfrak q}$
is reductive. Following Ooms,
${\mathfrak q}$
is said to be quasi-quadratic if
${\mathcal F}({\mathfrak q})={\mathfrak q}$
. Another interesting property of the functor
${\mathcal F}(\cdot )$
is that
${\mathrm {ind}}\,{\mathfrak q}\leqslant {\mathrm {ind}}\,{\mathcal F}({\mathfrak q})$
and
${\mathcal F}({\mathcal F}({\mathfrak q}))={\mathcal F}({\mathfrak q})$
[Reference Ooms19].
3.3 Commutative polarizations
If
${\mathfrak a}$
is an abelian subalgebra of
${\mathfrak q}$
, then
$\dim {\mathfrak a}\leqslant \boldsymbol {b}({\mathfrak q})$
(see [Reference Vinberg28, 0.2] or [Reference Ooms18, Theorem 14]). If
$\dim {\mathfrak a}= \boldsymbol {b}({\mathfrak q})$
, then
${\mathfrak a}$
is called a commutative polarization (=CP) of
${\mathfrak q}$
, and we say that
${\mathfrak q}$
has a CP. If
${\mathfrak a}$
is a CP and also an ideal of
${\mathfrak q}$
, then it is called a CP-ideal. If
${\mathfrak q}$
is solvable and has a CP, then it also has a CP-ideal (see [Reference Elashvili and Ooms10, Theorem 4.1]). More generally, a similar argument shows that if
${\mathfrak q}$
is an ideal of a solvable Lie algebra
${\mathfrak r}$
and
${\mathfrak q}$
has a CP, then
${\mathfrak q}$
has a CP-ideal that is
${\mathfrak r}$
-stable. A standard nilradical
${\mathfrak n}$
is an ideal of
${\mathfrak b}$
. Therefore, if
${\mathfrak n}$
has a CP, then it also has a CP-ideal that is
${\mathfrak b}$
-stable. Henceforth, “a CP-ideal of
${\mathfrak n}$
” means “a
${\mathfrak b}$
-stable CP-ideal of
${\mathfrak n}$
.”
Basic results on commutative polarizations are presented in [Reference Elashvili and Ooms10]. It is also shown therein that if
${\mathfrak g}$
is of type
${\mathsf {\mathbf {{A}}}}_{n}$
or
${\mathsf {\mathbf {{C}}}}_{n}$
, then every nilradical in
${\mathfrak g}$
has a CP. A complete classification of the nilradicals having a CP is obtained in [Reference Panyushev22]. By Lemma 2.1, 
is abelian if and only if 
and
$[\theta :\alpha ]=1$
. The abelian nilradical
${\mathfrak n}_{\{\alpha \}}$
play a key role in our theory. By Theorems 3.10 and 4.1 in [Reference Panyushev22], a nilradical
${\mathfrak n}$
has a CP if and only if at least one of the following two conditions is satisfied:
-
(1)
${\mathfrak n}={\mathfrak n}_{\theta }={\mathfrak h}_1$
is the Heisenberg nilradical. In this case, if
${\mathfrak a}$
is any maximal abelian ideal of
${\mathfrak b}$
, then
${\mathfrak a}\cap {\mathfrak n}$
is a CP-ideal of
${\mathfrak n}$
, and vice versa. -
(2) There is an abelian nilradical
${\mathfrak n}_{\{\alpha \}}$
such that
${\mathfrak n}$
is contained in
$\widetilde {{\mathfrak n}_{\{\alpha \}}}$
, the optimization of
${\mathfrak n}_{\{\alpha \}}$
. (There can be several abelian nilradicals with this property, and, for a “right” choice of such
$\check \alpha \in \Pi $
,
${\mathfrak n}\cap {\mathfrak n}_{\{\check \alpha \}}$
is a CP-ideal of
${\mathfrak n}$
(cf. also Section 8)).
If
${\mathfrak g}$
has no parabolic subalgebras with abelian nilradicals, then the Heisenberg nilradical
${\mathfrak n}_{\theta }={\mathfrak h}_1$
is the only nilradical with CP. This happens precisely if
${\mathfrak g}$
is of type
${\mathsf {\mathbf {{G}}}}_{2}$
,
${\mathsf {\mathbf {{F}}}}_{4}$
,
${\mathsf {\mathbf {{E}}}}_{8}$
. Another result of [Reference Panyushev22] is that
${\mathfrak n}$
has a CP if and only if
$\tilde {\mathfrak n}$
has.
3.4 The role of commutative polarizations
If
${\mathfrak a}$
is a CP of a Lie algebra
${\mathfrak q}$
, then:
-
(1)
${\mathcal F}({\mathfrak q})\subset {\mathfrak a}$
[Reference Ooms18, Proposition 20] and thereby
${\mathcal F}({\mathfrak q})$
is an abelian ideal. (However, it can happen that
${\mathcal F}({\mathfrak q})$
is abelian, whereas
${\mathfrak q}$
has no CP (see Example 7.3).) -
(2) Since
${\mathfrak a}$
is abelian,
$\boldsymbol {b}({\mathfrak a})=\dim {\mathfrak a}=\boldsymbol {b}({\mathfrak q})$
. Therefore, the Poisson center is contained in [Reference Panyushev22, Proposition 5.5]. Hence, if
${\mathfrak a}$
is a CP-ideal of
${\mathfrak q}$
, then . -
(3) Thus, if
${\mathfrak a}_1,\dots ,{\mathfrak a}_s$
are different CP in
${\mathfrak q}$
, then
${\mathcal F}({\mathfrak q})\subset \bigcap _{i=1}^s {\mathfrak a}_i$
and .
is contained in 
[
.
. In many cases, the presence of a CP in
${\mathfrak n}$
allows us to quickly describe the Frobenius semiradical for
${\mathfrak n}$
(see Sections 5.1 and 5.2). For the reader’s convenience, we provide an invariant-theoretic proof for two basic results on abelian subalgebras mentioned above.
Proposition 3.2 Let
${\mathfrak a}$
be an abelian subalgebra of
${\mathfrak q}$
. Then:
-
(1)
$\dim {\mathfrak a}\leqslant \boldsymbol {b}({\mathfrak q})$
. -
(2) If
$\dim {\mathfrak a}=\boldsymbol {b}({\mathfrak q})$
, then
${\mathfrak q}^{\xi }\subset {\mathfrak a}$
for any
$\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}$
, i.e.,
${\mathcal F}({\mathfrak q})\subset {\mathfrak a}$
.
Proof We show that both assertions are immediate consequences of Rosenlicht’s theorem [Reference Brion4, Chapter 1.6]. Let
$A\subset Q$
be the connected (abelian) subgroup with
${{\mathrm {Lie\,}} A={\mathfrak a}}$
. For the A-action on
${\mathfrak q}^{*}$
, let
${\mathbb C}({\mathfrak q}^{*})^A$
denote the field of A-invariant rational functions on
${\mathfrak q}^{*}$
.
(1) Since
${\mathfrak a}$
is abelian, we have 
; hence,
${\mathrm {trdeg\,}}{\mathbb C}({\mathfrak q}^{*})^A\geqslant \dim {\mathfrak a}$
. By Rosenlicht’s theorem,
$$\begin{align*}{\mathrm{trdeg\,}}{\mathbb C}({\mathfrak q}^{*})^A=\dim{\mathfrak q}-\max_{\xi\in{\mathfrak q}^{*}}A{\cdot}\xi= \dim{\mathfrak q}-\dim{\mathfrak a}+\min_{\xi\in{\mathfrak q}^{*}}\dim{\mathfrak a}^{\xi}. \end{align*}$$
Therefore,
$$\begin{align*}2\dim{\mathfrak a}\leqslant \dim{\mathfrak q}+\min_{\xi\in{\mathfrak q}^{*}}\dim{\mathfrak a}^{\xi}\leqslant \dim{\mathfrak q}+{\mathrm{ind}}\,{\mathfrak q}=2\boldsymbol{b}({\mathfrak q}). \end{align*}$$
(2) If
$\dim {\mathfrak a}=\boldsymbol {b}({\mathfrak q})$
, then
$\min _{\xi \in {\mathfrak q}^{*}}\dim {\mathfrak a}^{\xi }={\mathrm {ind}}\,{\mathfrak q}$
. For
$\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}$
, one has
$\dim {\mathfrak q}^{\xi }={\mathrm {ind}}\,{\mathfrak q}$
. That is,
${\mathfrak a}^{\xi }={\mathfrak q}^{\xi }\subset {\mathfrak a}$
for any
$\xi \in {\mathfrak q}^{*}_{\mathsf {reg}}$
.
The proof above also implies that if
$\dim {\mathfrak a}=\boldsymbol {b}({\mathfrak q})$
, then
${\mathrm {trdeg\,}}{\mathbb C}({\mathfrak q}^{*})^A=\dim {\mathfrak a}$
. Therefore, 
and
${\mathbb C}({\mathfrak q}^{*})^A$
is the fraction field of 
.
4 Generic stabilizers and the Frobenius semiradical for the nilradicals
In this section, we study the Frobenius semiradical,
${\mathcal F}({\mathfrak n})$
, of a nilradical
${\mathfrak n}\subset {\mathfrak g}$
and the existence of generic stabilizers for the coadjoint representation of
$N=\exp ({\mathfrak n})$
. We also say that “
${\mathfrak n}$
has a generic stabilizer,” if the coadjoint representation
$(N:{\mathfrak n}^{*})$
has.
Let
${\mathfrak n}={\mathfrak p}^{\mathsf {nil}}$
be a standard nilradical and
${\mathfrak n}^-=({\mathfrak p}^-)^{\mathsf {nil}}$
the opposite nilradical, i.e.,
$\Delta ({\mathfrak n}^-)=-\Delta ({\mathfrak n})$
. Using the vector space sum
${\mathfrak g}={\mathfrak p}\oplus {\mathfrak n}^-$
and the P-module isomorphism
${\mathfrak n}^{*}\simeq {\mathfrak g}/{\mathfrak p}$
, we identify
${\mathfrak n}^{*}$
with
${\mathfrak n}^-$
and thereby regard
${\mathfrak n}^-$
as P-module. In terms of
${\mathfrak n}^-$
, the
${\mathfrak p}$
-action on
${\mathfrak n}^{*}$
is given by the Lie bracket in
${\mathfrak g}$
, with the subsequent projection to
${\mathfrak n}^-$
. In particular, the coadjoint representation
$ {\mathrm {ad}}^{*} _{\mathfrak n}$
of
${\mathfrak n}$
has the following interpretation. If
$x\in {\mathfrak n}$
and
$\xi \in {\mathfrak n}^-$
, then
$( {\mathrm {ad}}^{*} _{\mathfrak n} x){\cdot }\xi =\mathsf {pr}_{{\mathfrak n}^-} ([x,\xi ])$
, where
$\mathsf {pr}_{{\mathfrak n}^-}: {\mathfrak g}\to {\mathfrak n}^-$
is the projection with kernel
${\mathfrak p}$
.
Recall that 
. Set 
and 
. We say that
${\mathfrak k}$
is a cascade subspace of
${\mathfrak n}^{*}$
and
${\boldsymbol {\zeta }}$
is a cascade point of
${\mathfrak n}^{*}$
. Clearly,
${\boldsymbol {\zeta }}$
depends on the choice of nonzero root vectors
$e_{-\beta }\in {\mathfrak g}_{-\beta }$
, but all such points
${\boldsymbol {\zeta }}$
form a sole dense T-orbit in
${\mathfrak k}$
, which is denoted by
${\mathfrak k}_0$
. That is, 
. It was proved by Joseph [Reference Joseph12, 2.4] that upon the identification of
${\mathfrak n}^-$
and
${\mathfrak n}^{*}$
as P-modules and hence B-modules,
$B{\cdot }{\boldsymbol {\zeta }}$
is dense in
${\mathfrak n}^{*}$
and
$B{\cdot }{\boldsymbol {\zeta }}\subset {\mathfrak n}^{*}_{\mathsf {reg}}$
. In particular,
${\mathfrak k}_0\subset {\mathfrak n}^{*}_{\mathsf {reg}}$
.
Proposition 4.1 Let
${\mathfrak n}$
be an arbitrary nilradical and
${\boldsymbol {\zeta }}\in {\mathfrak n}^{*}$
a cascade point. Then:
-
(i)
${\mathcal F}({\mathfrak n})$
is the
${\mathfrak b}$
-stable ideal of
${\mathfrak n}$
generated by
${\mathfrak n}^{\boldsymbol {\zeta }}$
. -
(ii)
$({\mathfrak n}:{\mathfrak n}^{*})$
has a generic stabilizer if and only if the stabilizer
${\mathfrak n}^{\boldsymbol {\zeta }}$
is generic.
Proof
(i) Since
$B{\cdot }{\boldsymbol {\zeta }}$
is dense in
${\mathfrak n}^{*}_{\mathsf {reg}}$
, a result of Ooms [Reference Ooms19, Lemma 1.2] implies that
${\mathcal F}({\mathfrak n})=\sum _{b\in B}{\mathfrak n}^{b{\cdot }{\boldsymbol {\zeta }}}$
. Clearly, the last sum is the smallest
${\mathfrak b}$
-stable ideal containing
${\mathfrak n}^{\boldsymbol {\zeta }}$
.
(ii) If
${\boldsymbol {\zeta }}$
is not generic, then
$[{\mathfrak n},{\mathfrak n}^{{\boldsymbol {\zeta }}}]\cap {\mathfrak n}^{{\boldsymbol {\zeta }}} \ne \{0\}$
and hence
$[{\mathfrak n},{\mathfrak n}^{b{\cdot }{\boldsymbol {\zeta }}}]\cap {\mathfrak n}^{b{\cdot }{\boldsymbol {\zeta }}} \ne \{0\}$
for any
$b\in B$
. Therefore, neither of the points
${b{\cdot }{\boldsymbol {\zeta }}}$
(
$b\in B$
) can be generic. Since
$B{\cdot }{\boldsymbol {\zeta }}$
is dense in
${\mathfrak n}^{*}$
, this means that generic points cannot exist in such a case.
The point of Proposition 4.1
(i) is that the existence of a generic stabilizer for
${\mathfrak n}$
is not required (cf. Lemma 3.1). We use instead the action of B on
${\mathfrak n}^{*}$
.
Proposition 4.2 If
${\mathfrak n}$
is an optimal nilradical, then:
-
(i)
${\mathfrak n}^{\boldsymbol {\zeta }}$
is a generic stabilizer for
$({\mathfrak n}:{\mathfrak n}^{*})$
. -
(ii) The N-saturation of
${\mathfrak k}$
is dense in
${\mathfrak n}^{*}$
, i.e.,
$\overline {N{\cdot }{\mathfrak k}}={\mathfrak n}^{*}$
.
Proof
(i) For an optimal nilradical
${\mathfrak n}$
, one has 
[Reference Joseph12, 2.4]. Since the roots in 
are strongly orthogonal, condition (3.3) is satisfied for
$\xi ={\boldsymbol {\zeta }}$
.
(ii) Recall that 
and
${\mathrm {codim\,}}_{{\mathfrak n}^{*}}({\mathfrak n}{\cdot }\xi )={\mathrm {ind}}\,{\mathfrak n}$
for any
$\xi \in {\mathfrak k}_0$
. Furthermore,
${\mathfrak n}{\cdot }\xi \cap {\mathfrak k}=\{0\}$
for any
$\xi \in {\mathfrak k}$
. (Use again the strong orthogonality.) Hence,
${\mathfrak n}{\cdot }\xi \oplus {\mathfrak k}={\mathfrak n}^{*}$
for any
$\xi \in {\mathfrak k}_0$
, which implies that
$N{\cdot }{\mathfrak k}_0$
is open and dense in
${\mathfrak n}^{*}$
(cf. [Reference Elashvili8, Lemma 1.1]).
Recall that
$\tilde {\mathfrak n}$
denotes the optimization of
${\mathfrak n}$
. Then 
, 
, and
The cascade subspaces and cascade points associated with
${\mathfrak n}$
or
$\tilde {\mathfrak n}$
are the same, if regarded as objects in
${\mathfrak u}^-$
. But one has to distinguish them as objects in
${\mathfrak n}^{*}$
or
$\tilde {\mathfrak n}^{*}$
. For this reason, working simultaneously with
${\mathfrak n}$
and
$\tilde {\mathfrak n}$
, we write
$\tilde {\boldsymbol {\zeta }}$
for a cascade point in the cascade subspace
$\tilde {\mathfrak k}\subset \tilde {\mathfrak n}^{*}$
. Note that, for the natural projection
$\tau : \tilde {\mathfrak n}^{*}\to {\mathfrak n}^{*}$
, one has
$\tau (\tilde {\mathfrak k})={\mathfrak k}$
and
$\tau (\tilde {\boldsymbol {\zeta }})={\boldsymbol {\zeta }}$
. Since
$\tau $
is B-equivariant, this allows us to transfer some good properties from the coadjoint action
$(\tilde N:\tilde {\mathfrak n}^{*})$
to the coadjoint action
$(N: {\mathfrak n}^{*})$
.
If
${\mathfrak n}$
is not optimal, then it may or may not have a generic stabilizer. To see this, we provide an explicit description of
${\mathfrak n}^{\boldsymbol {\zeta }}$
for an arbitrary nilradical
${\mathfrak n}$
, which generalizes Joseph’s description for the optimal nilradicals.
Because
${\boldsymbol {\zeta }}\in {\mathfrak n}^{*}_{\mathsf {reg}}$
and
$\tilde {\boldsymbol {\zeta }}\in \tilde {\mathfrak n}^{*}_{\mathsf {reg}}$
, it follows from (4.1) that
$\dim {\mathfrak n}^{\boldsymbol {\zeta }}=\dim (\tilde {\mathfrak n}/{\mathfrak n})+\dim \tilde {\mathfrak n}^{\tilde {\boldsymbol {\zeta }}}$
. Since
$\tau $
is B-equivariant, we have 
. If 
, then
${\mathfrak n}\subset \tilde {\mathfrak n}=\bigoplus _{j=1}^k{\mathfrak h}_j$
and
${\mathfrak n}=\bigoplus _{j=1}^k ({\mathfrak h}_j\cap {\mathfrak n})$
(see Section 2.3). For any
$\gamma \in \Delta ({\mathfrak h}_j)\setminus \{\beta _j\}=:\overline {\Delta ({\mathfrak h}_j)}$
, we have
$\beta _j-\gamma \in \Delta ^+$
.
Lemma 4.3 Under the previous notation, if
${\mathfrak n}\ne \tilde {\mathfrak n}$
and 
, then the root space
${\mathfrak g}_{\beta _j-\gamma }$
belongs to
${\mathfrak n}^{\boldsymbol {\zeta }}$
.
Proof Assume that
$\beta _j-\gamma \not \in \Delta ({\mathfrak n})$
. Then
$[\gamma :\alpha ]=[\beta _j-\gamma :\alpha ]=0$
for all 
. Hence,
$[\beta _j:\alpha ]=0$
. However, since
$\beta _j\in \Delta ({\mathfrak n})$
, there is an 
such that
$\beta _j\succ \alpha '$
, i.e.,
$[\beta _j:\alpha ']>0$
. This contradiction shows that
$\gamma ':=\beta _j-\gamma \in \Delta ({\mathfrak n})$
.
Let us prove that
${\mathfrak g}_{\gamma '}\subset {\mathfrak n}^{\boldsymbol {\zeta }}$
. Since
$\gamma '\in \Delta ({\mathfrak h}_j)$
and hence
${\mathsf {supp}}(\gamma ')\subset {\mathsf {supp}}(\beta _j)$
, properties of 
imply that
$\beta _j$
is the only element
$\beta _i$
of 
such that
$\beta _i-\gamma '$
is a root. Indeed,
-
• if
$\beta _i$
and
$\beta _j$
are incomparable, then
${\mathsf {supp}}(\beta _i)\cap {\mathsf {supp}}(\beta _j)=\varnothing $
; -
• if
$\beta _i\prec \beta _j$
, then
${\mathsf {supp}}(\beta _i)\subset {\mathsf {supp}}(\beta _j)\setminus \Phi (\beta _j)$
, while
${\mathsf {supp}}(\gamma ')\cap \Phi (\beta _j)\ne \varnothing $
; -
• if
$\beta _i\succ \beta _j$
, then
$(\beta _i,\gamma ')=0$
and
$\gamma '$
does not belong to the Heisenberg subset of
$\Delta \langle i\rangle $
.
Therefore,
$\gamma '-\beta _j=-\gamma $
is the only root that might occur in
$[e_{\gamma '},{\boldsymbol {\zeta }}]$
. But
$\gamma \not \in \Delta ({\mathfrak n})$
, i.e.,
$-\gamma \not \in \Delta ({\mathfrak n}^{*})$
and therefore
$ {\mathrm {ad}}^{*} _{\mathfrak n}(e_{\gamma '})({\boldsymbol {\zeta }})=0$
.
Theorem 4.4 Let
${\mathfrak n}$
be an arbitrary nilradical and
${\boldsymbol {\zeta }}\in {\mathfrak k}_0\subset {\mathfrak n}^{*}$
a cascade point. Then:
-
(i)
${\mathfrak n}^{\boldsymbol {\zeta }}$
is T-stable and -
(ii)
. -
(iii) For any
$\xi \in {\mathfrak k}_0$
, we have
${\mathfrak n}^{\xi }={\mathfrak n}^{\boldsymbol {\zeta }}$
.
.
Proof The number of roots in 
equals
$\dim (\tilde {\mathfrak n}/{\mathfrak n})=\dim {\mathfrak n}^{\boldsymbol {\zeta }}-\dim \tilde {\mathfrak n}^{\tilde {\boldsymbol {\zeta }}}$
, and each such root yields a root subspace in
${\mathfrak n}^{\boldsymbol {\zeta }}$
(Lemma 4.3), hence (i). Clearly, (ii) is just a reformulation of (i). The last assertion follows from the fact that
${\mathfrak n}^{\boldsymbol {\zeta }}$
is T-stable and
$T{\cdot }{{\boldsymbol {\zeta }}}={\mathfrak k}_0$
.
The advantage of
${\boldsymbol {\zeta }}\in {\mathfrak k}_0$
is that
${\mathfrak n}^{\boldsymbol {\zeta }}$
is a sum of root spaces. Therefore, Eq. (3.3) is easily verified in practice. It is convenient to restate it as follows.
Condition 4.5 The stabilizer
${\mathfrak n}^{\boldsymbol {\zeta }}$
of
${\boldsymbol {\zeta }}\in {\mathfrak k}_0\subset {\mathfrak n}^{*}$
is generic (equivalently,
$({\mathfrak n}:{\mathfrak n}^{*})$
has a generic stabilizer) if and only if the difference of any two roots in
$\Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
does not belong to
$\Delta ({\mathfrak n})$
.
Example 4.6 (1) If
${\mathfrak g}=\mathfrak {sl}_7$
and 
, then 
is not optimal and 
, where 
(see Example 2.5). Here, 
and the matrices therein show that
$\dim (\tilde {\mathfrak n}/{\mathfrak n})=4$
. More precisely,
Therefore,
$\Delta ({\mathfrak n}^{\boldsymbol {\zeta }})=\{ \alpha _2, \alpha _2+\alpha _3, [2,4], [2,5], [2,6], [1,6]\}$
. Since
$[2,6]-\alpha _2=[3,6]\in \Delta ({\mathfrak n})$
, Condition 4.5 is not satisfied for
${\boldsymbol {\zeta }}$
, and
$({\mathfrak n}:{\mathfrak n}^{*})$
does not have a generic stabilizer.
(2) If
${\mathfrak g}=\mathfrak {sl}_7$
and 
, then
${\mathfrak n}_{T_2}$
has the same optimization
$\tilde {\mathfrak n}$
as 
, but now 
and 
. Here, Condition 4.5 is satisfied and 
is a generic stabilizer.
Thus, the action
$(N:{\mathfrak n}^{*})$
does not always have N-generic stabilizers. A remedy is to consider the action of the larger group
$\tilde N$
on
${\mathfrak n}^{*}$
.
Theorem 4.7 For any nilradical
${\mathfrak n}$
and the cascade subspace
${\mathfrak k}\subset {\mathfrak n}^{*}$
, we have:
-
(i)
$\dim N{\cdot }{\mathfrak k}=2\dim {\mathfrak n}-\dim \tilde {\mathfrak n}$
, i.e.,
${\mathrm {codim\,}}_{{\mathfrak n}^{*}}N{\cdot }{\mathfrak k}=\dim (\tilde {\mathfrak n}/{\mathfrak n})$
; -
(ii) the
$\tilde N$
-saturation of
${\mathfrak k}$
is dense in
${\mathfrak n}^{*}$
, i.e.,
$\overline {\tilde N{\cdot }{\mathfrak k}}={\mathfrak n}^{*}$
; -
(iii)
$\tilde {\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak n}^{\boldsymbol {\zeta }}$
is a generic stabilizer for the linear action
$(\tilde N:{\mathfrak n}^{*})$
.
Proof Recall that
$\dim {\mathfrak k}={\mathrm {ind}}\,\tilde {\mathfrak n}$
and
$\max _{\xi \in {\mathfrak n}^{*}}\dim N{\cdot }\xi =\dim N{\cdot }{\boldsymbol {\zeta }}=\dim {\mathfrak n}-{\mathrm {ind}}\,{\mathfrak n}$
.
(i) Since
$\mathsf T_{\boldsymbol {\zeta }}(N{\cdot }{\boldsymbol {\zeta }})={\mathfrak n}{\cdot }{\boldsymbol {\zeta }}=({\mathfrak n}^{\boldsymbol {\zeta }})^{\perp }$
, it follows from Theorem 4.4
(iii) that all tangent spaces
${\mathfrak n}{\cdot }\xi $
,
$\xi \in {\mathfrak k}_0$
, are the same. Therefore, using the differential of the map
$$\begin{align*}\kappa: N\times{\mathfrak k}\to \overline{N{\cdot}{\mathfrak k}}\subset{\mathfrak n}^{*} \end{align*}$$
at
$(1,{\boldsymbol {\zeta }})$
, we obtain
$\dim \overline {N{\cdot }{\mathfrak k}}=\dim ({\mathsf {Im\,}} \textit {d}\kappa _{(1,{\boldsymbol {\zeta }})})= \dim ({\mathfrak n}{\cdot }{\boldsymbol {\zeta }}+{\mathfrak k})$
. Because the roots in 
are strongly orthogonal, we have
${\mathfrak n}{\cdot }{\boldsymbol {\zeta }}\cap {\mathfrak k}=\{0\}$
and hence
$$\begin{align*}\dim({\mathfrak n}{\cdot}{\boldsymbol{\zeta}}+{\mathfrak k})=\dim{\mathfrak n}-{\mathrm{ind}}\,{\mathfrak n}+{\mathrm{ind}}\,\tilde{\mathfrak n}=2\dim{\mathfrak n}-\dim\tilde{\mathfrak n}. \end{align*}$$
(ii) Since
$\tilde N{\cdot }\tilde {\mathfrak k}$
is dense in
$\tilde {\mathfrak n}^{*}$
(Proposition 4.2) and
$\tau $
is B-equivariant, we conclude that
$\tau (\tilde N{\cdot }\tilde {\mathfrak k})=\tilde N{\cdot }{\mathfrak k}$
is dense in
$\tau (\tilde {\mathfrak n}^{*})={\mathfrak n}^{*}$
.
(iii) Let us first prove that
$$ \begin{align} \max_{\xi\in{\mathfrak n}^{*}}\dim \tilde N{\cdot}\xi=\max_{\xi\in{\mathfrak n}^{*}}\dim N{\cdot}\xi+\dim(\tilde{\mathfrak n}/{\mathfrak n}). \end{align} $$
Clearly, inequality “
$\leqslant $
” holds. By Eq. (4.1), the RHS equals
$\dim \tilde {\mathfrak n}-{\mathrm {ind}}\,{\mathfrak n}=\dim {\mathfrak n}-\dim {\mathfrak k}$
. On the other hand, part (ii) implies that
$\dim {\mathfrak n}=\dim \tilde N{\cdot }{\mathfrak k}\leqslant \max _{\xi \in {\mathfrak k}}\dim \tilde N{\cdot }\xi +\dim {\mathfrak k}$
. This proves Eq. (4.2) and also shows that almost all
$\xi \in {\mathfrak k}$
satisfy this relation. Moreover, since
${\mathfrak k}$
is T-stable and
${\mathfrak k}_0=T{\cdot }{\boldsymbol {\zeta }}$
is dense in
${\mathfrak k}$
, we obtain that
$$\begin{align*}\dim \tilde N{\cdot}\xi=\dim N{\cdot}\xi + \dim(\tilde{\mathfrak n}/{\mathfrak n}) \end{align*}$$
for every
$\xi \in {\mathfrak k}_0$
. Hence,
$\tilde {\mathfrak n}^{\xi }={\mathfrak n}^{\xi }$
for every
$\xi \in {\mathfrak k}_0$
. Combining this with Theorem 4.4
(iii) and part (ii), we conclude that
$\tilde {\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak n}^{\boldsymbol {\zeta }}$
is a generic stabilizer for the action
$(\tilde N:{\mathfrak n}^{*})$
.
Using Theorem 4.4, we describe certain nilradicals that do not have a generic stabilizer.
Proposition 4.8 Let
$\beta _j$
be a descendant of 
and
$\alpha \in \Phi (\beta _i)$
. Suppose that 
and
$[\beta _i:\nu ]>[\beta _j:\nu ]>0$
for some 
. Then 
does not have a generic stabilizer.
Proof Recall that
$\beta _i$
is the highest root in
$\Delta \langle i\rangle ^+$
and
$\beta _j$
is a maximal root in
$\Delta \langle i\rangle ^+\setminus \Delta ({\mathfrak h}_i)$
. If
$\Phi (\beta _i)=\{\alpha \}$
, then
$\alpha +\beta _j\in \Delta ({\mathfrak h}_i)\subset \Delta \langle i\rangle ^+$
, because
$\beta _j$
is not the highest root of
$\Delta \langle i\rangle ^+$
. If
$\Phi (\beta _i)=\{\alpha ,\alpha '\}$
, then
$\Delta \langle i\rangle $
is of type
${\mathsf {\mathbf {{A}}}}_{p}$
and hence both
$\beta _j+\alpha $
and
$\beta _j+\alpha '$
belong to
$\Delta ({\mathfrak h}_i)$
. In any case, if
$\alpha \in \Phi (\beta _i)$
, then
$\beta _i-\alpha -\beta _j\in \Delta ^+$
.
Since 
and
$[\beta _j:\nu ]>0$
, we have
$\beta _j\in \Delta ({\mathfrak n})$
and also
$\beta _i\in \Delta ({\mathfrak n})$
. That is, 
. Because
$\alpha \in \Delta ({\mathfrak h}_i)\setminus \Delta ({\mathfrak n})$
, we have
$\beta _i-\alpha \in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
(see Theorem 4.4
(ii)). The assumption on
$\nu $
implies that
$[\beta _i-\alpha -\beta _j:\nu ]>0$
. Thus, we have
$\beta _i-\alpha ,\beta _j\in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
and
$\beta _i-\alpha -\beta _j\in \Delta ({\mathfrak n})$
. By Condition 4.5, this means that
${\mathfrak n}$
has no generic stabilizers.
Example 4.9 For applications, it suffices to consider the case in which
$\beta _i=\theta $
, i.e.,
$i=1$
.
(1) If
${\mathfrak g}$
is exceptional, then
$\#\Phi (\beta _1)=1$
and
$\beta _1$
has the unique descendant
$\beta _2$
. Although
$\beta _1-\beta _2$
is not a root, its support is defined as in Section 2.1. Then Proposition 4.8 applies to any
$\nu \in {\mathsf {supp}}(\beta _1-\beta _2)\setminus \Phi (\beta _1)$
and 
.
For instance, let
${\mathfrak g}$
be of type
${\mathsf {\mathbf {{E}}}}_{6}$
. Then
$\Phi (\beta _1)=\{\alpha _6\}$
,
${\mathsf {supp}}(\beta _1-\beta _2)=\{\alpha _2,\alpha _3,\alpha _4,\alpha _6\}$
, and
${\mathsf {supp}}(\beta _2)\supset \{\alpha _2,\alpha _3,\alpha _4\}$
(see formulae for 
in Appendix A). Hence, one can take
$\nu =\alpha _i$
with
$i\in \{2,3,4\}$
. More generally, if 
and 
, then 
has no generic stabilizers.
(2) This method also works for
${\mathfrak g}={\mathfrak {so}}_{n}$
with
$n\geqslant 7$
, where
$\Phi (\beta _1)=\{\alpha _2\}$
.
– If
$n\ne 8$
, then
$\beta _1=\varepsilon _1+\varepsilon _2$
has two descendants,
$\beta _2=\varepsilon _1-\varepsilon _2$
, and
$\beta _3=\varepsilon _3+\varepsilon _4$
(
$\varepsilon _4=0$
, if
$n=7$
) (see Appendix A). For
$(\beta _1,\beta _3)$
, Proposition 4.8 applies, if one takes 
. While for
$(\beta _1,\beta _2)$
, we can take 
such that 
and 
with
$j\geqslant 3$
.
– If
$n=8$
, then
$\beta _1$
has three descendants, and we can take 
with
$i,j\ne 2$
.
Proposition 4.10 If
${\mathfrak n}\subset {\mathfrak n}'$
and
$\tilde {\mathfrak n}=\tilde {{\mathfrak n}'}$
, then
${\mathcal F}({\mathfrak n}')\subset {\mathcal F}({\mathfrak n})\subset {\mathfrak n}$
.
Proof Let
${\mathfrak k}$
(resp.
${\mathfrak k}'$
) be the cascade subspace of
${\mathfrak n}^{*}$
(resp.
$({\mathfrak n}')^{*}$
). Take
${\boldsymbol {\zeta }}\in {\mathfrak k}_0$
and
${\boldsymbol {\zeta }}'\in {\mathfrak k}^{\prime }_0$
. Since
$\Delta ({\mathfrak n})\subset \Delta ({\mathfrak n}')$
and 
, we have 
for all 
. Then Theorem 4.4 shows that
${\mathfrak n}^{\boldsymbol {\zeta }}\supset ({\mathfrak n}')^{{\boldsymbol {\zeta }}'}$
. By Proposition 4.1, this yields the required embedding.
Our description of
$\Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
yields a criterion for
${\mathfrak n}$
to be quasi-quadratic.
Theorem 4.11 For a nilradical 
, the following assertions are equivalent:
-
•
${\mathcal F}({\mathfrak n})={\mathfrak n}$
, i.e.,
${\mathfrak n}$
is quasi-quadratic. -
• For each
, one of the two conditions is satisfied:-
– either
; -
– or
$\Phi (\Phi ^{-1}(\alpha ))=\{\alpha ,\alpha '\}$
, and if C is the chain in the Dynkin diagram that connects
$\alpha $
and
$\alpha '$
, then (in particular, ).
-
, one of the two conditions is satisfied:
;
(in particular, 
).Proof Since
${\mathcal F}({\mathfrak n})$
is the
${\mathfrak b}$
-ideal generated by
${\mathfrak n}^{\boldsymbol {\zeta }}$
(Proposition 4.1), it is clear that
${\mathcal F}({\mathfrak n})={\mathfrak n}$
if and only if
$\alpha \in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
for each 
.
(1) If 
, then
$\Phi (\alpha )=\alpha $
and
$\alpha \in \Delta (\tilde {\mathfrak n}^{\boldsymbol {\zeta }})\subset \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
.
(2) If 
and 
, then
$\alpha \ne \beta _j$
and there are two possibilities.
–
$\Phi (\beta _j)=\alpha $
. Then the whole Heisenberg algebra
${\mathfrak h}_{j}$
belongs to
${\mathfrak n}$
and hence 
. Then it follows from Theorem 4.4
(ii) that
$\Delta ({\mathfrak h}_j\cap {\mathfrak n}^{\boldsymbol {\zeta }})=\{\beta _j\}$
, i.e.,
$\alpha \not \in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
.
–
$\Phi (\beta _j)=\{\alpha , \alpha '\}$
. Here,
$\beta _j$
is the highest root in a root system of type
${\mathsf {\mathbf {{A}}}}_{p}$
(
$p\geqslant 2$
). Therefore, if
$C=\{\alpha =\alpha _1,\alpha _2,\dots ,\alpha _p=\alpha '\}$
is the chain connecting
$\alpha $
and
$\alpha '$
, then
$\beta _j=\alpha _1+\alpha _2+\dots +\alpha _p$
. Then
$\alpha \in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
if and only if
$\beta _j-\alpha =\alpha _2+\dots +\alpha _p\not \in \Delta ({\mathfrak n})$
, and this is only possible if 
.
Using Theorem 4.11, one readily obtains the list of all quasi-quadratic nilradicals.
Proposition 4.12 The quasi-quadratic nilradicals are as follows.
-
(1) If
${\mathfrak g}$
is not of type
${\mathsf {\mathbf {{A}}}}_{n},{\mathsf {\mathbf {{D}}}}_{2n+1},\text { and } {\mathsf {\mathbf {{E}}}}_{6}$
, then if and only if . -
(2) If
${\mathfrak g}$
is of type
${\mathsf {\mathbf {{A}}}}_{n}$
, then if and only if ,
$i=1,2,\dots ,n$
. -
(3) If
${\mathfrak g}$
is of type
${\mathsf {\mathbf {{D}}}}_{2n+1}$
, then if and only if and . -
(4) If
${\mathfrak g}$
is of type
${\mathsf {\mathbf {{E}}}}_{6}$
, then if and only if ,
$i\ne 6$
.
if and only if 
.
if and only if 
, 
if and only if 
and 
.
if and only if 
, 
Proof (1) In this case,
$\Phi ^{-1}$
is a bijection; hence,
$\Phi (\Phi ^{-1}(\alpha ))=\{\alpha \}$
for any
$\alpha \in \Pi $
.
(2) Here, each
${\mathfrak n}$
has a CP [Reference Elashvili and Ooms10]; hence,
${\mathcal F}({\mathfrak n})$
is abelian. That is,
${\mathcal F}({\mathfrak n})={\mathfrak n}$
must be an abelian nilradical.
(3) In this case, 
and there is a unique 
such that
$\#\Phi (\beta )=2$
. Namely,
$\Phi (\beta _{2n-1})=\{\alpha _{2n},\alpha _{2n+1}\}$
(see Appendix A). The chain connecting
$\alpha _{2n}$
and
$\alpha _{2n+1}$
in the Dynkin diagram is
$C=\{\alpha _{2n},\alpha _{2n-1}, \alpha _{2n+1}\}$
. Hence the answer.
(4) Here, 
,
$\Phi (\beta _3)=\{\alpha _2,\alpha _4\}$
,
$\Phi (\beta _2)=\{\alpha _1,\alpha _5\}$
(see Appendix A). Since
$\alpha _1,\alpha _2,\alpha _3,\alpha _4,\alpha _5$
form a chain in the Dynkin diagram, the result follows.
5 Generic stabilizers and the Frobenius semiradical in case of
${\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
First, we explicitly describe the nilradicals in
${\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
having a generic stabilizer. To state the result for
${\mathfrak {sl}}_{n+1}$
, we need some notation. Recall that, for
${\mathfrak {sl}}_{n+1}$
, the poset 
is a chain
$\beta _1\succ \beta _2\succ \dots \succ \beta _t$
, where
$t=[(n+1)/2]$
and
$\Phi (\beta _i)=\{\alpha _i,\alpha _{n+1-i}\}$
. Let 
be a nilradical. Then 
for some
$k\leqslant t$
(cf. Lemma 2.2). Therefore, 
and 
. Set 
and 
.
Theorem 5.1 Suppose that
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
and 
. Then
(That is,
$\sigma (\alpha _j)=\alpha _{n+1-j}$
.) In particular, if 
, then 
has a generic stabilizer.
Proof
1
$^o$
. For
$k=1$
, we have 
and
${\mathfrak n}\subset {\mathfrak h}_1$
. If 
, then
${\mathfrak n}={\mathfrak h}_1$
is optimal. If 
, then
${\mathfrak n}$
is abelian. In both cases, there is a generic stabilizer for
$({\mathfrak n}:{\mathfrak n}^{*})$
, as required.
2
$^o$
. Suppose that
$k\geqslant 2$
and the symmetry of 
and 
fails for some
$j\leqslant k-1$
. W.l.o.g, we may assume that 
, whereas 
. Then
$\beta _j-\alpha _{n+1-j}=\beta _{j+1}+\alpha _j\in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
. Since 
and
$\alpha _j\in \Delta ({\mathfrak n})$
, Condition 4.5 is not satisfied.
3
$^o$
. If 
, then one can directly describe
${\mathfrak n}^{\boldsymbol {\zeta }}$
and see that Condition 4.5 is satisfied. It is necessary to distinguish two cases: (a) 
and (b) only 
.
(a) Here, 
and
$\Phi (\beta _k)\subset \Delta ({\mathfrak n})$
. Hence,
${\mathfrak h}_k\subset {\mathfrak n}$
and 
. Theorem 4.4
(ii) shows that
${\mathfrak n}^{\boldsymbol {\zeta }}\subset {\mathfrak n}_{\{\alpha _k\}}\cap {\mathfrak n}_{\{\alpha _{n+1-k}\}}$
, the last intersection being the north-east
$k\times k$
square of
$(n+1)\times (n+1)$
matrices in
${\mathfrak {sl}}_{n+1}$
. More precisely, if 
, then
$\Delta ({\mathfrak n}^{\boldsymbol {\zeta }})=\Gamma _1\cup \dots \cup \Gamma _j$
, where
$$ \begin{gather*} \Gamma_1=\{\varepsilon_i-\varepsilon_j \mid 1\leqslant i\leqslant i_1, \ n+2-i_1\leqslant j\leqslant n+1\} , \\ \Gamma_2=\{\varepsilon_i-\varepsilon_j \mid i_1+1\leqslant i\leqslant i_2, \ n+2-i_2\leqslant j\leqslant n+1-i_1\}, \text{etc.} \end{gather*} $$
Here,
$\{\Gamma _s\}_{s=1}^j$
is a string of square blocks located along the antidiagonal in the
$k\times k$
square (see Figure 1a). The size of the sth square is
$i_s-i_{s-1}$
, where
$i_0=0$
. It is readily seen that if
$\gamma $
and
$\gamma '$
belong to different blocks, then
$\gamma -\gamma '$
is not a root, while if
$\gamma $
and
$\gamma '$
belong to the same block, then
$\gamma -\gamma '$
is either not a root or a root of the standard Levi subalgebra corresponding to
${\mathfrak n}$
. Thus,
$\gamma -\gamma '\not \in \Delta ({\mathfrak n})$
and Condition 4.5 is satisfied.
Figure 1: The generic stabilizer
${\mathfrak n}^{\boldsymbol {\zeta }}$
: cases (a) and (b).
(b) Here,
${\mathfrak n}$
is smaller than in part (a), but 
remains the same. Now only “half” of
${\mathfrak h}_k$
belongs to
${\mathfrak n}$
. Therefore, 
, and the subsets 
with
$i_{j-1}< s< k$
become larger than in (a). Hence,
$\Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
becomes larger and, along with
$\Gamma _1\cup \dots \cup \Gamma _j$
, it also contains the strip of roots
$$ \begin{align*} \tilde\Gamma=\{\varepsilon_i-\varepsilon_j\mid i_{j-1}+1\leqslant i\leqslant i_j=k, \ k+1\leqslant j\leqslant n+1-k\} \end{align*} $$
attached to
$\Gamma _j$
(see Figure 1b). The new feature is that the difference of roots from
$\Gamma _j$
and
$\tilde \Gamma $
can be a root in
$\bigcup _{s=i_{j-1}}^k \Delta ({\mathfrak h}_s)$
. But such a difference belongs to the parts of sets
$\Delta ({\mathfrak h}_s)$
that are missing in
$\Delta ({\mathfrak n})$
. Thus, Condition 4.5 is still satisfied here.
The symmetry condition of Theorem 5.1 means that the matrix shape of
${\mathfrak n}\subset {\mathfrak {sl}}_{n+1}$
must be “almost” symmetric w.r.t. the antidiagonal. That is, the symmetry may only fail in case (b) for roots in
$\Delta ({\mathfrak h}_s)$
with
$i_{j-1}< s \leqslant i_j=k$
(if 
, but 
).
Remark 5.2 Using Theorem 5.1, one easily computes the number of nontrivial (standard) nilradicals with generic stabilizers. For
${\mathsf {\mathbf {{A}}}}_{2n-1}$
, it is
$2^{n+1}-3$
; for
${\mathsf {\mathbf {{A}}}}_{2n}$
, it is
$3(2^{n}-1)$
. Hence, the ratio
$\#\{\text {nilradicals with generic stabilizer}\}/\#\{\text {all nilradicals}\}$
exponentially decreases.
Theorem 5.3 If
${\mathfrak g}={\mathfrak {sp}}_{2n}$
, then a generic stabilizer exists for every nilradical
${\mathfrak n}$
.
Proof For an appropriate choice of a skew-symmetric bilinear form defining
${\mathfrak {sp}}_{2n}\subset {\mathfrak {sl}}_{2n}$
, a Borel subalgebra of
${\mathfrak {sp}}_{2n}$
,
${\mathfrak b}({\mathfrak {sp}}_{2n})$
, is the set of symplectic upper-triangular matrices (see Example 2.6). Then the matrix shape of any (standard) nilradical in
${\mathfrak {sp}}_{2n}$
is symmetric w.r.t. the antidiagonal, and the approach in the proof of 3
$^o$
(a) in Theorem 5.1 applies to any nilradical in
${\mathfrak {sp}}_{2n}$
. If 
and 
, where
$i_1<i_2<\dots < i_j=k$
, then
${\mathfrak n}^{\boldsymbol {\zeta }}$
belongs to the north-east
$k\times k$
square in the abelian nilradical
${\mathfrak n}_{\{\alpha _{n}\}}$
. Here, the blocks
$\Gamma _j$
inside this square represent matrices that are symmetric w.r.t. the antidiagonal (cf. Figure 1a).
For these two series, we explicitly describe
${\mathcal F}({\mathfrak n})$
for any
${\mathfrak n}$
. This also demonstrates the role of commutative polarizations.
5.1 The Frobenius semiradical
${\mathcal F}({\mathfrak n})$
for
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
Let 
be a nilradical such that 
. Then 
, 
, and 
. We may assume that 
and then, as in the proof of Theorem 5.1, there are two possibilities.
(a) 
. Then
${\mathfrak n}_{\{\alpha _k\}}$
and
${\mathfrak n}_{\{\alpha _{n+1-k}\}}$
are CP-ideals of
${\mathfrak n}$
[Reference Panyushev22, Example 3.8], and hence
${\mathcal F}({\mathfrak n})\subset {\mathfrak n}_{\{\alpha _k\}}\cap {\mathfrak n}_{\{\alpha _{n+1-k}\}}$
(cf. Section 3.4(3)). Here,
${\mathfrak s}:={\mathfrak n}_{\{\alpha _k\}}\cap {\mathfrak n}_{\{\alpha _{n+1-k}\}}$
is the north-east square of size k in
${\mathfrak {sl}}_{n+1}$
. On the other hand,
${\mathfrak n}^{\boldsymbol {\zeta }}\subset {\mathfrak s}$
(see Theorem 5.1) and
${\mathfrak b}{\cdot }{\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak s}$
. Hence,
${\mathcal F}({\mathfrak n})={\mathfrak s}$
and
$\dim {\mathcal F}({\mathfrak n})=k^2$
(see Figure 2). For the special case of
${\mathfrak n}={\mathfrak u}$
, the description of
${\mathcal F}({\mathfrak u})$
is obtained in [Reference Ooms19, Theorem 4.1].
Figure 2:
${\mathcal F}({\mathfrak n})={\mathfrak n}_{\{\alpha _k\}}\cap {\mathfrak n}_{\{\alpha _{n+1-k}\}}$
,
${\mathfrak n}\supset {\mathfrak n}_{\{\alpha _k\}}\cup {\mathfrak n}_{\{\alpha _{n+1-k}\}}$
.
Actually, here,
${\mathfrak a}_j:={\mathfrak n}_{\{\alpha _j\}}\cap {\mathfrak n}$
is a CP-ideal of
${\mathfrak n}$
for any j such that
$k\leqslant j\leqslant n+1-k$
(see Example 4.8 in [Reference Panyushev22]).
(b) 
. Then only
${\mathfrak n}_{\{\alpha _k\}}$
is a CP-ideal and
${\mathcal F}({\mathfrak n})\subset {\mathfrak n}_{\{\alpha _k\}}$
. On the other hand,
${\mathsf {supp}}(\beta _k-\alpha _k)=\{\alpha _{k+1},\dots ,\alpha _{n+1-k}\}$
. Hence,
$\beta _k-\alpha _k\not \in \Delta ({\mathfrak n})$
and
$\alpha _k\in \Delta ({\mathfrak n}^{\boldsymbol {\zeta }})$
(see Theorem 4.4
(ii)). By Proposition 4.1
(i), this implies that
${\mathfrak n}_{\{\alpha _k\}}\subset {\mathcal F}({\mathfrak n})$
. Thus, here,
${\mathcal F}({\mathfrak n})={\mathfrak n}_{\{\alpha _k\}}$
and
$\dim {\mathcal F}({\mathfrak n})=k(n+1-k)$
. Since
${\mathcal F}({\mathfrak n})$
appears to be a CP, we conclude that
${\mathfrak n}_{\{\alpha _k\}}$
is the only CP for
${\mathfrak n}$
.
It follows from the descriptions above that for
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
,
${\mathcal F}({\mathfrak n})$
is always equal to the intersection of all CP-ideals of
${\mathfrak n}$
. But this is not true for other simple Lie algebras.
5.2 The Frobenius semiradical
${\mathcal F}({\mathfrak n})$
for
${\mathfrak g}={\mathfrak {sp}}_{2n}$
Let 
be a nilradical such that 
. Then 
, 
and 
. The abelian nilradical
${\mathfrak n}_{\{\alpha _{n}\}}$
is identified with the space of
$n\times n$
matrices that are symmetric w.r.t. the antidiagonal (see Example 2.6). Then
${\mathfrak n}^{\boldsymbol {\zeta }}$
belongs to the north-east
$k\times k$
square in
${\mathfrak n}_{\{\alpha _{n}\}}$
, and the south-west corner of this square,
${\mathfrak g}_{\beta _k}$
, lies in
${\mathfrak n}^{\boldsymbol {\zeta }}$
. Hence,
${\mathcal F}({\mathfrak n})={\mathfrak b}{\cdot }{\mathfrak n}^{\boldsymbol {\zeta }}$
is equal to this k-square and
$\dim {\mathcal F}({\mathfrak n})=k(k+1)/2$
. In this case,
${\mathfrak a}:={\mathfrak n}_{\{\alpha _n\}}\cap {\mathfrak n}$
is the only CP-ideal of
${\mathfrak n}$
and the inclusion
${\mathcal F}({\mathfrak n})\subset {\mathfrak a}$
is proper unless
$k=n$
(see the following picture).
Remark 5.4 The series
${\mathsf {\mathbf {{A}}}}_{n}$
and
${\mathsf {\mathbf {{C}}}}_{n}$
are easily handled, because 
is a chain for them and any nilradical has a CP. Therefore,
${\mathcal F}({\mathfrak n})$
is an abelian ideal of
${\mathfrak n}$
, and there is a natural upper bound on
${\mathcal F}({\mathfrak n})$
. However, it can happen that
${\mathfrak n}$
does not have a CP, but
${\mathcal F}({\mathfrak n})$
is abelian (see Example 7.3).
6 The Poisson center and U-invariants
Recall that
$P=L{\cdot }N$
is a standard parabolic subgroup of G and
${\mathfrak p}={\mathfrak l}\oplus {\mathfrak n}$
. Since N is connected, one has 
, and this algebra is the center of the Poisson algebra 
. Because
${\mathfrak n}$
is a P-module, one can consider algebras of invariants in 
for any subgroup
$Q\subset P$
. Specifically, we are interested in the groups U and
$\tilde N$
, where
${\mathrm {Lie\,}} \tilde N=\tilde {\mathfrak n}$
is the optimization of
${\mathfrak n}$
. We also wish to compare the algebras of Q-invariants in 
and 
. The algebras of interest for us are organized in the following diagram:
If an algebra 
is finitely generated, then we also consider the associated quotient morphism
$\pi _Q: {\mathfrak n}^{*}\to {\mathfrak n}^{*}/\!\!/ Q:={\mathsf {Spec\,}}({\mathbb C}[{\mathfrak n}^{*}]^Q)$
.
For any nilradical
${\mathfrak n}$
, one can form the solvable Lie algebra
$$\begin{align*}{\mathfrak f}_{\mathfrak n}={\mathfrak f}_{\tilde{\mathfrak n}}={\mathfrak t}_{\mathfrak n}\oplus\tilde{\mathfrak n} \subset{\mathfrak b} , \end{align*}$$
where 
. By [Reference Panyushev22, Proposition 5.1], the corresponding connected group
$F_{\mathfrak n}\subset B$
has an open orbit in
${\mathfrak f}_{\mathfrak n}^{*}$
, i.e.,
${\mathfrak f}_n$
is a Frobenius Lie algebra. For this reason,
${\mathfrak f}_{\mathfrak n}$
is called the Frobenius envelope of
${\mathfrak n}$
. Note that the unipotent radical of
$F_{\mathfrak n}$
is
$\tilde N$
.
Lemma 6.1 The four algebras of invariants forming the square in (6.1) are polynomial. In particular, for any optimal nilradical
$\tilde {\mathfrak n}$
, the Poisson center 
is polynomial.
Proof
1
$^o$
. For any nilradical
${\mathfrak n}$
, the action
$(B:{\mathfrak n}^{*})$
is locally transitive and
$B{\cdot }{\boldsymbol {\zeta }}$
is the dense orbit in
${\mathfrak n}^{*}$
[Reference Joseph12, 2.4]. Therefore, 
is a polynomial algebra and 
equals the number of prime divisors in
${\mathfrak n}^{*}\setminus B{\cdot }{\boldsymbol {\zeta }}$
(see [Reference Panyushev21, Theorem 4.4]). Hence, the algebras in the first column of (6.1) are polynomial.
2
$^o$
. More generally, the
$F_{\mathfrak n}$
-equivariant embeddings
${\mathfrak n}\hookrightarrow \tilde {\mathfrak n}\hookrightarrow {\mathfrak f}_{\mathfrak n}$
yield the
$F_{\mathfrak n}$
-equivariant projections
${\mathfrak f}_{\mathfrak n}^{*}\to \tilde {\mathfrak n}^{*}\to {\mathfrak n}^{*}$
, which implies that
$F_{\mathfrak n}$
has a dense orbit in
${\mathfrak n}^{*}$
, too. Therefore, arguing as in [Reference Panyushev21, Section 4], one proves that 
is also a polynomial algebra. Thus, the algebras in the second column of (6.1) are polynomial, too.
If
${\mathfrak n}\ne \tilde {\mathfrak n}$
, then the Poisson center 
is not always polynomial (see examples below).
Lemma 6.2 For any nilradical
${\mathfrak n}$
, we have 
.
Proof It is clear that 
. On the other hand, by [Reference Panyushev22, Proposition 5.5], the equality
$\boldsymbol {b}({\mathfrak n})=\boldsymbol {b}(\tilde {\mathfrak n})$
(see Proposition 2.3) implies that 
. Hence, 
.
Lemma 6.3 For any nilradical
${\mathfrak n}$
, we have 
.
Proof
1
$^o$
. Let us first prove that 
. Since
$\tilde N$
is unipotent, the field of invariants
${\mathbb C}({\mathfrak n}^{*})^{\tilde N}$
is the quotient field of the algebra of invariants 
[Reference Brion4, Chapter 1.4]. Furthermore, by Rosenlicht’s theorem [Reference Brion4, Chapter 1.6], one has
$$\begin{align*}{\mathrm{trdeg\,}}{\mathbb C}({\mathfrak n}^{*})^{\tilde N}+\max_{\xi\in{\mathfrak n}^{*}}\dim \tilde N{\cdot}\xi=\dim{\mathfrak n} , \end{align*}$$
and the same holds for U in place of
$\tilde N$
. Therefore, since
$$\begin{align*}\max_{\xi\in{\mathfrak n}^{*}}\dim U{\cdot}\xi\geqslant \max_{\xi\in{\mathfrak n}^{*}}\dim \tilde N{\cdot}\xi, \end{align*}$$
it suffices to prove that one has equality here. By Theorem 4.7,
$\tilde N{\cdot }{\mathfrak k}$
is dense in
${\mathfrak n}^{*}$
. Hence,
$U{\cdot }{\mathfrak k}$
is dense in
${\mathfrak n}^{*}$
, too.
Now,
${\mathfrak u}=({\mathfrak u}\cap \tilde {\mathfrak l})\oplus \tilde {\mathfrak n}$
, where
$\tilde {\mathfrak l}$
is the standard Levi subalgebra of
$\tilde {\mathfrak p}:=\mathsf {norm}_{\mathfrak g}(\tilde {\mathfrak n})$
. Since
$\tilde {\mathfrak n}$
is optimal,
${\mathfrak u}\cap \tilde {\mathfrak l}$
stabilizes any
$\xi \in {\mathfrak k}_0\subset {\mathfrak k}$
. Hence,
${\mathfrak u}^{\xi }=({\mathfrak u}\cap \tilde {\mathfrak l})\oplus \tilde {\mathfrak n}^{\xi }$
and
$\dim U{\cdot }\xi =\dim \tilde N{\cdot }\xi $
. Therefore,
$\max _{\xi \in {\mathfrak n}^{*}}\dim U{\cdot }\xi = \max _{\xi \in {\mathfrak n}^{*}}\dim \tilde N{\cdot }\xi $
, and we are done.
2
$^o$
. By the first part, the extension 
is algebraic. But the algebra of invariants of a connected algebraic group Q acting on an affine variety X is algebraically closed in
${\mathbb C}[X]$
(see [Reference Kraft14, p. 100]).
Combining previous lemmas yields our main result on diagram (6.1).
Theorem 6.4 The four algebras of invariants that form the square in (6.1) are equal, i.e.,
These algebras are polynomial, and their common transcendence degree is 
. If
${\mathfrak n}\ne \tilde {\mathfrak n}$
, then 
. Thus, there are at most two different algebras of invariants in (6.1).
Proof (1) By Lemma 6.1, these four algebras are polynomial.
(2) By Lemmas 6.2 and 6.3, these four algebras are equal. (Note that Lemma 6.3 applies also to
$\tilde {\mathfrak n}$
in place of
${\mathfrak n}$
.)
(3) Since both
$\tilde N$
and N are unipotent, it follows from the Rosenlicht theorem that 
and 
. Hence, the last relation is just Eq. (4.1).
Remark 6.5 The algebra of U-invariants in 
is polynomial for an arbitrary
${\mathfrak b}$
-stable ideal
${\mathfrak r}$
of
${\mathfrak u}$
. That is, if
${\mathfrak r}\subset {\mathfrak u}$
and
$[{\mathfrak b},{\mathfrak r}]\subset {\mathfrak r}$
, then 
is a polynomial algebra. The reason is that B has an open orbit in
${\mathfrak r}^{*}$
(see [Reference Panyushev21, Section 4] for details).
Corollary 6.6 For any nilradical
${\mathfrak n}$
and a cascade point
${\boldsymbol {\zeta }}\in {\mathfrak k}_0\subset {\mathfrak n}^{*}$
, we have:
-
(1)
; -
(2)
$B{\cdot }{\boldsymbol {\zeta }}=F_{\mathfrak n}{\cdot }{\boldsymbol {\zeta }}$
.
;
Proof (1) It is known that:
-
•
; -
•
[Reference Panyushev21, Theorem 4.4]; -
•
.
;
[
.The last equality relies on the facts that the orbit
$F_{\mathfrak n}{\cdot }{\boldsymbol {\zeta }}$
is open in
${\mathfrak n}^{*}$
and
$\tilde N$
is the unipotent radical of
$F_{\mathfrak n}$
. Hence, [Reference Panyushev21, Theorem 4.4] applies also in this case.
(2) Since B and
$F_{\mathfrak n}$
are solvable, the orbits
$B{\cdot }{\boldsymbol {\zeta }}$
and
$F_{\mathfrak n}{\cdot }{\boldsymbol {\zeta }}$
are affine. Therefore, both
${\mathfrak n}^{*}\setminus B{\cdot }{\boldsymbol {\zeta }}$
and
${\mathfrak n}^{*}\setminus F_{\mathfrak n}{\cdot }{\boldsymbol {\zeta }}$
are the union of divisors and the assertion follows from (1).
If
${\mathfrak n}$
is not optimal, then 
is not always polynomial (see Example 6.10). Actually, there are Lie algebras
${\mathfrak q}$
such that 
is not finitely generated! A construction of such
${\mathfrak q}$
utilizing the Nagata counterexample to Hilbert’s 14th problem is given by Dixmier in [Reference Dixmier5, 4.9.20(c)]. Nevertheless, we demonstrate below that, for the nilradicals in
${\mathfrak g}$
, the algebra 
is as good as the algebras of invariants of linear actions of reductive groups. Recall that a commutative associative
${\mathbb C}$
-algebra
${\mathcal A}$
equipped with action of an algebraic group Q is called a rational Q-algebra if, for any
$a\in {\mathcal A}$
, the linear span in
${\mathcal A}$
of the orbit
$Q{\cdot }a$
is finite-dimensional and the representation of Q on
$\langle Q{\cdot }a\rangle $
is rational (see, e.g., [Reference Grosshans11, p.1]).
Theorem 6.7 Let
${\mathfrak n}$
be an arbitrary nilradical in
${\mathfrak g}$
. Then:
-
(i) the algebra
is finitely generated; -
(ii) the quotient variety
${\mathfrak n}^{*}/\!\!/ N$
has rational singularities.
is finitely generated;
Proof
(i) Let
${\mathfrak p}$
be the parabolic subalgebra with
${\mathfrak n}={\mathfrak p}^{\mathsf {nil}}$
. Let
${\mathfrak l}$
be the standard Levi subalgebra of
${\mathfrak p}$
and L the corresponding Levi subgroup of
$P=\mathsf {Norm}_G({\mathfrak n})\subset G$
. Since
$[{\mathfrak l},{\mathfrak n}]\subset {\mathfrak n}$
, the algebra 
is a rational L-algebra. Set
$U(L)=U\cap L$
. It is a maximal unipotent subgroup of L, and U is a semi-direct product of
$U(L)$
and N. By Theorem 6.4, 
is a polynomial algebra. Now, we can apply Corollary 4 from [Reference Popov23, Section 3], which asserts that if
${\mathcal A}$
is a rational L-algebra, then
${\mathcal A}$
is finitely generated if and only if
${\mathcal A}^{U(L)}$
is. In our case, 
and
${\mathcal A}^{U(L)}$
is a polynomial algebra, with 
generators. Hence, 
is finitely generated.
(ii) Let
(P)
be a so-called “stable property” of local rings of algebraic varieties (see [Reference Popov23, Section 6] for the details). By [Reference Popov23, Theorem 6],
${\mathcal A}$
has property
(P)
if and only if
${\mathcal A}^{U(L)}$
has. Since rationality of singularities is such a property, we obtain the second assertion.
Remark 6.8 (1) Part (i) of Theorem 6.7 appears in [Reference Joseph12, Lemma 4.6(ii)], with another proof. Namely, in place of reference to [Reference Popov23], one can provide the following simple argument for the implication:
$$ \begin{align*} {\mathcal A}^{U(L)} \textit{ is finitely generated } \Longrightarrow {\mathcal A} \textit{ is finitely generated}. \end{align*} $$
Let
$f_1,\dots ,f_l$
be a set of generators for
${\mathcal A}^{U(L)}$
. We may also assume that each
$f_i$
is a T-eigenvector. Then the
${\mathbb C}$
-linear span of
$L{\cdot }f_i\subset {\mathcal A}$
is a simple finite-dimensional L-module, say
$\mathsf {V}_i$
. It is easily seen that the
${\mathbb C}$
-algebra generated by
$\sum _{i=1}^l \mathsf {V}_i$
is L-stable and contains all simple L-modules from
${\mathcal A}$
. Hence, the finite-dimensional space
$\sum _{i=1}^l \mathsf {V}_i$
generates
${\mathcal A}$
.
(2) There is also an alternate approach to part (ii). By a result of Kostant, the algebra
${\mathbb C}[{\mathfrak n}^{*}]^N$
is a multiplicity free L-module (see [Reference Joseph12, 4.5]). Being the algebra of invariants of a linear action,
${\mathbb C}[{\mathfrak n}^{*}]^N$
is also integrally closed. Therefore,
${\mathfrak n}^{*}/\!\!/ N$
is a spherical L-variety. By [Reference Popov23, Theorem 10], this implies that
${\mathfrak n}^{*}/\!\!/ N$
has rational singularities.
Remark 6.9
(1) The general treatment of “stable properties” is due to Popov [Reference Popov23], but the assertion that relates rationality of singularities for
${\mathcal A}$
and
${\mathcal A}^{U(L)}$
goes back to Kraft and Luna (see exposition in Brion’s thesis [Reference Brion2, Chapter I(c)]). It is worth noting that, for a rational L-algebra
${\mathcal A}$
, the equivalence
$$ \begin{align*} {\mathcal A} \textit{ is finitely generated } \Longleftrightarrow {\mathcal A}^{U(L)} \textit{ is finitely generated} \end{align*} $$
holds over an algebraically closed field of an arbitrary characteristic (see [Reference Grosshans11, Theorem 16.2]).
(2) A list of “stable properties” is found in [Reference Popov23, Section 6].
Example 6.10
(1) Take
${\mathfrak g}=\mathfrak {sl}_5$
and 
with 
. Then
$\dim {\mathfrak n}=7$
, 
, and
$\tilde {\mathfrak n}={\mathfrak u}$
. Hence,
${\mathrm {ind}}\,{\mathfrak n}=5$
. Let
$\{e_{ij}\mid 1\leqslant i<j\leqslant 5, j=4,5\}$
be the matrix units corresponding to
${\mathfrak n}$
(i.e., a basis for
${\mathfrak n}$
). We regard them as (linear) functions on
${\mathfrak n}^-\simeq {\mathfrak n}^{*}$
. Here,
$e_{15}\in {\mathfrak g}_{\beta _1}$
and
$e_{24}\in {\mathfrak g}_{\beta _2}$
. Obviously,
$e_{15}$
and
$f_{12}=\begin {vmatrix} e_{14} & e_{15} \\ e_{24} & e_{25} \end {vmatrix}$
belong to 
. For the standard Levi subalgebra
${\mathfrak l}$
of
${\mathfrak p}=\mathsf {norm}_{\mathfrak g}({\mathfrak n})$
, one has
$[{\mathfrak l},{\mathfrak l}]={\mathfrak {sl}}_3$
; hence, 
is an
${\mathfrak {sl}}_3$
-module. Taking the
${\mathfrak {sl}}_3$
-modules generated by
$e_{15}$
and
$f_{12}$
, one obtains six functions that generate 
:
$$\begin{align*}e_{15},e_{25},e_{35},f_{12}=\begin{vmatrix} e_{14} & e_{15} \\ e_{24} & e_{25} \end{vmatrix}, \ f_{13}= \begin{vmatrix} e_{14} & e_{15} \\ e_{34} & e_{35} \end{vmatrix}, \ f_{23}=\begin{vmatrix} e_{24} & e_{25} \\ e_{34} & e_{35} \end{vmatrix}. \end{align*}$$
Since
$f_{12}\not \in {\mathbb C}[e_{15},e_{25},e_{35}]$
and a minimal generating system can be chosen to be
$SL_3$
-invariant, the set of generators above is minimal. These generators satisfy the relation
$e_{15}f_{23}-e_{25}f_{13}+e_{35}f_{12}=0$
; hence,
${\mathfrak n}^{*}/\!\!/ N$
is a hypersurface. Under passage to
$\tilde N$
or U, the situation improves, because 
.
(2) More generally, for
${\mathfrak g}=\mathfrak {sl}_N$
and 
with
$k\leqslant N/2$
, one has
Here,
$[{\mathfrak l},{\mathfrak l}]=\mathfrak {sl}_{N-k}$
and 
, where
$F_i$
is the “anti-principal” minor of degree i, i.e.,
$$\begin{align*}F_1=e_{1,N}, \ F_2=\begin{vmatrix} e_{1,N-1} & e_{1,N} \\ e_{2,N-1} & e_{2,N} \end{vmatrix},\dots, F_k=\begin{vmatrix} e_{1,N-k+1} & \cdots & e_{1,N} \\ \cdots & \cdots & \cdots \\ e_{k,N-k+1} & \cdots & e_{k,N} \end{vmatrix}. \end{align*}$$
The
$SL_{N{-}k}$
-module generated by
$F_i$
,
$\langle SL_{N-k}{\cdot }F_i\rangle $
, is isomorphic to
$\wedge ^i \mathbb V$
, where
$\mathbb V$
is the standard
$SL_{N{-}k}$
-module. It is also easily seen that
$F_j$
does not belong to the
${\mathbb C}$
-algebra generated by
$\sum _{i=1}^{j-1}\langle SL_{N-k}{\cdot }F_i\rangle $
. Therefore, the minimal number of generators of 
is
If
$k=1$
, then 
is abelian and 
. If
$k\geqslant 2$
, then 
and the equality holds only for
$k=2, N=4$
.
(3) For
${\mathsf {\mathbf {{E}}}}_{6}$
and 
, one has 
, 
, and 
. Here,
$[{\mathfrak l},{\mathfrak l}]= {\mathfrak {sl}}_2\oplus \mathfrak {sl}_{5}$
and 
, where
$\deg F_i=1,2,4$
for
$i=1,2,3$
, respectively. Using the
${\mathfrak l}$
-modules generated by
$F_1$
and
$F_2$
, one can prove that 
.
7 On the square integrable nilradicals
Recall that any nilradical 
is equipped with the canonical
${\mathbb Z}$
-grading 
, where 
and 
is the center of
${\mathfrak n}$
. Accordingly, one obtains the partition 
(see Section 2.3).
Clearly,
${\mathrm {ind}}\,{\mathfrak n}\geqslant \dim {\mathfrak z}({\mathfrak n})$
and 
. Following [Reference Elashvili9, Reference Elashvili and Ooms10], we say that
${\mathfrak n}$
is square integrable, if
${\mathrm {ind}}\,{\mathfrak n}=\dim {\mathfrak z}({\mathfrak n})$
. Then
${\mathfrak n}^{\xi }={\mathfrak z}({\mathfrak n})$
for any
$\xi \in {\mathfrak n}^{*}_{\mathsf {reg}}$
and hence
${\mathcal F}({\mathfrak n})={\mathfrak z}({\mathfrak n})$
is abelian. Since 
, the extension 
is algebraic. This implies that 
and hence 
is a free 
-module. It is easily seen that a Heisenberg algebra
${\mathfrak h}$
is square integrable, with
${\mathrm {ind}}\,{\mathfrak h}=1$
. Using 
and our description of
${\mathfrak n}^{\boldsymbol {\zeta }}$
, we classify all square integrable nilradicals. Recall that 
is a poset.
Lemma 7.1 For any 
, we have 
.
Proof Take 
and 
. Then there are the following possibilities:
-
•
$\beta =\alpha $
. Then . -
•
$\beta \ne \alpha $
and
$\Phi (\beta )=\{ \alpha \}$
. Recall that
$\beta $
is the highest root in the root system with basis
${\mathsf {supp}}(\beta )$
. Since
$\beta $
is a minimal element of , we have . Then
$[\beta :\alpha ]=2$
and . -
•
$\Phi (\beta )=\{\alpha ,\alpha '\}$
. Here,
${\mathsf {supp}}(\beta )$
is a basis for the root system
${\mathsf {\mathbf {{A}}}}_{p}$
(
$p\geqslant 2$
) and . Then either and , or and .
.
, we have 
. Then 
.
. Then either 
and 
, or 
and 
.Theorem 7.2 Let 
be an arbitrary nilradical. Then
Proof
(1) Let
${\boldsymbol {\zeta }}\in {\mathfrak k}_0$
be a cascade point. Then 
. If
${\mathfrak n}$
is square integrable, then 
. Hence, 
. By Lemma 7.1, this is only possible, if 
. Thus, the implication “
$\Rightarrow $
” is proved.
(2) If 
, then the assumption on 
is vacuous and
${\mathfrak n}$
is abelian, hence square integrable. Therefore, to prove the implication “
$\Leftarrow $
,” we may assume that 
. Then
${\mathfrak z}({\mathfrak n})={\mathfrak n}(2)=[{\mathfrak n},{\mathfrak n}]$
and 
. Now, there are two possibilities for 
.
• Suppose that 
and
$[\theta :\alpha ]=2$
.
Set
$\beta _i=\Phi ^{-1}(\alpha )$
. Since 
, we have
$[\beta _i:\alpha ]=2$
. Let us prove that
$\beta _i$
is the unique minimal element of 
. Assume that 
, i.e., there is 
such that
$\beta '\prec \beta $
. Then
${\mathsf {supp}}(\beta ')\subset {\mathsf {supp}}(\beta _i)\setminus \{\alpha \}$
and 
. A contradiction! Hence,
$\beta _i$
is minimal in 
. Assume that there is another minimal element of 
, say
$\tilde \beta $
. Then
${\mathsf {supp}}(\beta _i)$
and
${\mathsf {supp}}(\tilde \beta )$
are disjoint, and we must have 
, which is impossible. This contradiction shows that 
.
Since
$B{\cdot }{\boldsymbol {\zeta }}$
is dense in
${\mathfrak n}^{*}$
,
$B{\cdot }{\boldsymbol {\zeta }}\subset {\mathfrak n}^{*}_{\mathsf {reg}}$
, and
${\mathfrak n}(2)$
is B-stable, it suffices to prove that
${\mathfrak n}^{\boldsymbol {\zeta }}\subset {\mathfrak n}(2)$
(and then, actually,
${\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak n}(2)$
). Recall that 
. For the unique minimal element 
, we have
${\mathfrak h}_i\subset {\mathfrak n}$
and hence
${\mathfrak h}_i\cap {\mathfrak n}^{\boldsymbol {\zeta }}={\mathfrak g}_{\beta _i}$
. If
$\beta _i\prec \beta _j$
, then
$[\beta _j:\alpha ]=2$
and the contribution from
${\mathfrak h}_j\cap {\mathfrak n}$
to
${\mathfrak n}^{\boldsymbol {\zeta }}$
is
(see Theorem 4.4). Since 
, we have
${\mathfrak g}_{\beta _j}\subset {\mathfrak n}(2)$
. The very definition of 
says that 
. Hence,
$[\gamma :\alpha ]=0$
and
$[\beta _j-\gamma :\alpha ]=2$
. Thus, 
and
${\mathfrak h}_j\cap {\mathfrak n}^{\boldsymbol {\zeta }}\subset {\mathfrak n}(2)$
.
• Suppose that 
and
$[\theta :\alpha ]=[\theta :\alpha ']=1$
.
The argument here is similar to that in the previous part. Set
$\beta =\Phi ^{-1}(\alpha )$
and
${\beta '=\Phi ^{-1}(\alpha ')}$
. Using the hypothesis that 
, one first proves that
$\beta =\beta '$
(hence
$\Phi (\beta )=\{\alpha ,\alpha '\}$
) and that
$\beta $
is the unique minimal element of 
. Then the use of Theorem 4.4 allows us to check that
${\mathfrak n}^{\boldsymbol {\zeta }}\subset {\mathfrak n}(2)$
.
Using Theorem 7.2, it is not hard to get the list of square integrable nilradicals in all simple Lie algebras. For 
, 
is abelian and these cases are well known. If 
, then 
. The condition that 
belongs to the highest graded component of 
is quite strong. Therefore, not all nilradicals with 
are square integrable.
Example 7.3 (1) The list of square integrable nilradicals with 
is given below.
(2) The list contains all the cases in which
${\mathfrak n}={\mathfrak h}_1$
is the Heisenberg nilradicals. For the exceptional algebras, this corresponds to 
indicated first, whereas for the classical series, this corresponds to the cases with
$k=1$
.
(3) Using [Reference Panyushev22], one verifies that 
has no CP for
${\mathsf {\mathbf {{B}}}}_{n}$
with
$k\geqslant 2$
,
$({\mathsf {\mathbf {{E}}}}_{8}, \alpha _7)$
, and
$({\mathsf {\mathbf {{F}}}}_{4}, \alpha _1)$
.
(4) For
${\mathsf {\mathbf {{D}}}}_{2n}$
and 
, 
is not square integrable. Indeed, here, 
, 
, and 
contains two elements,
$\alpha _{2n}=\beta _{2n-1}$
and
$\alpha _{2n-1}=\beta _{2n}$
. (We use the notation of Appendix A.) Hence, 
, while the other elements of 
belong to 
. Here, 
and 
. (Of course, there are other nilradicals with 
that are not square integrable.)
Remark 7.4 A real nilpotent Lie group Q has a unitary square integrable representation if and only if
${\mathrm {ind}}\,{\mathfrak q}=\dim {\mathfrak z}({\mathfrak q})$
[Reference Moore and Wolf15, Theorem 1]. For this reason, Elashvili applied the term “square integrable” to the nilpotent Lie algebras
${\mathfrak q}$
satisfying that equality, over an algebraically closed field of characteristic zero [Reference Elashvili9] (cf. also [Reference Elashvili and Ooms10]). Afterward, the theory of square integrable representations was extended to the setting of arbitrary Lie groups [Reference Duflo6]. The relevant notions are those of a quasi-reductive Lie group and a coadjoint orbit of reductive type (see [Reference Duflo, Khalgui and Torasso7]). Therefore, Theorem 7.2 provides a classification of the quasi-reductive nilradicals in
${\mathfrak g}$
. Various results on quasi-reductive seaweed (= biparabolic) subalgebras of
${\mathfrak g}$
are obtained in [Reference Baur and Moreau1, Reference Moreau and Yakimova16].
8 When is the quotient morphism equidimensional?
It is well known that if a graded polynomial algebra
${\mathcal A}$
is a free module over a graded subalgebra
${\mathcal B}$
, then
${\mathcal B}$
is necessarily polynomial. Furthermore, if
${\mathcal B}$
is a graded polynomial subalgebra of
${\mathcal A}$
, then
${\mathcal A}$
is a free
${\mathcal B}$
-module if and only if the induced morphism
$\pi : {\mathsf {Spec\,}}({\mathcal A})\to {\mathsf {Spec\,}}({\mathcal B})$
is equidimensional, i.e.,
$\dim \pi ^{-1}(\pi (x))=\dim {\mathcal A}-\dim {\mathcal B}$
for any
$x\in {\mathsf {Spec\,}}({\mathcal A})$
[Reference Schwarz26, Proposition 17.29]. By a theorem of Chevalley, dominant equidimensional morphisms are open. Therefore, in the graded situation,
$\pi $
is also onto (see [Reference Vinberg and Gindikin29, 2.4]).
In this section, we point out certain nilradicals
${\mathfrak n}$
in
${\mathfrak g}$
such that 
is a free module over 
. The last algebra is always polynomial (Lemma 6.1); hence, our task is to guarantee that the quotient morphism
$$\begin{align*}\pi: {\mathsf{Spec\,}}({\mathcal A})={\mathfrak n}^{*}\to {\mathfrak n}^{*}/\!\!/ U={\mathsf{Spec\,}}({\mathcal B}) \end{align*}$$
is equidimensional. Since 
(Lemma 6.3), the optimal nilradicals of such type provide examples, where 
is a free module over its Poisson center 
. We begin with a simple observation.
Lemma 8.1 If 
, then 
is a free 
-module.
Proof By Theorem 6.4, we have 
. Since 
is an element of degree
$1$
, the hyperplane
${\mathfrak n}^{*}_0=\{\xi \in {\mathfrak n}^{*}\mid \xi (e_{\theta })=0\}$
is U-stable and
${\mathfrak n}^{*}_0/\!\!/ U\simeq \mathbb A^s$
, where 
. By a result of Brion [Reference Brion3] on invariants of unipotent groups,
$\pi _0:{\mathfrak n}^{*}_0\to {\mathfrak n}^{*}_0/\!\!/ U$
is equidimensional, and this implies that
$\pi $
is equidimensional, too.
Theorem 8.2 Let
${\mathfrak n}_{\{\alpha \}}$
be an abelian nilradical, i.e.,
$[\theta :\alpha ]=1$
. Suppose that a nilradical
${\mathfrak n}$
is contained between
${\mathfrak n}_{\{\alpha \}}$
and its optimization
$\widetilde {{\mathfrak n}_{\{\alpha \}}}$
. Then:
-
(1)
${\mathfrak n}_{\{\alpha \}}$
is a CP-ideal of
${\mathfrak n}$
; -
(2)
is a free module over .
is a free module over 
.Proof (1) By Remark 2.4, we have
$\dim {\mathfrak n}_{\{\alpha \}}=\boldsymbol {b}({\mathfrak n}_{\{\alpha \}})=\boldsymbol {b}({\mathfrak n})$
.
(2) Consider the commutative diagram 
corresponding to the U-equivariant inclusion
${\mathfrak n}_{\{\alpha \}}\hookrightarrow {\mathfrak n}$
. Since
${\mathfrak n}$
and
${\mathfrak n}_{\{\alpha \}}$
have the same optimization, Theorem 6.4 implies that 
, i.e.,
$\varphi /\!\!/ U$
is an isomorphism. By a general result on U-invariants for the abelian nilradicals [Reference Panyushev20, Theorem 4.6], 
is a free module over 
. Hence, the morphism
$\pi _{\{\alpha \}}$
is equidimensional and onto. Because the projection
$\varphi $
is also onto and equidimensional,
$\pi $
must be equidimensional, too. Since
${\mathfrak n}^{*}/\!\!/ U$
is an affine space, the morphism
$\pi $
is flat and
${\mathbb C}[{\mathfrak n}^{*}]$
is a free
${\mathbb C}[{\mathfrak n}^{*}]^U$
-module.
Corollary 8.3 For the optimal nilradical
$\tilde {\mathfrak n}=\widetilde {\mathfrak n}_{\{\alpha \}}$
, the algebra 
is a free module over 
.
Theorem 8.2 implies interesting consequences for some types of simple Lie algebras.
Proposition 8.4 For every nilradical
${\mathfrak n}$
in
${\mathfrak {sl}}_{n+1}$
, there is an
$\alpha \in \Pi $
such that
${\mathfrak n}_{\{\alpha \}}\subset {\mathfrak n}\subset \widetilde {{\mathfrak n}_{\{\alpha \}}}$
. Therefore, 
is a free module over 
for any
${\mathfrak n}$
.
Proof We use the notation of Example 2.5(1). Recall that
$[\theta :\alpha ]=1$
for each
$\alpha \in \Pi $
. If 
and 
(
$k\leqslant [(n+1)]/2$
), then 
. Then any
$\alpha $
in this intersection will do. (The construction of such a simple root
$\alpha $
is essentially contained in the proof of Theorem 6.2 in [Reference Elashvili and Ooms10]. For, then
${\mathfrak n}_{\{\alpha \}}$
is a CP-ideal of
${\mathfrak n}$
. But our approach that refers to 
is shorter.)
Of course, the first assertion of Proposition 8.4 appears to be true because all simple roots of
${\mathfrak {sl}}_{n+1}$
provide abelian nilradicals. Nevertheless, the second assertion can be proved for the nilradicals in
${\mathfrak {sp}}_{2n}$
, although the proof becomes more involved.
Proposition 8.5 For any nilradical
${\mathfrak n}$
in
${\mathfrak {sp}}_{2n}$
, 
is a free module over 
.
Proof Here,
${\mathfrak n}_{\{\alpha _{n}\}}$
is the only abelian nilradical and 
. Hence,
$\widetilde {{\mathfrak n}_{\{\alpha _{n}\}}}={\mathfrak u}$
.
(1) If 
and 
, then
${\mathfrak n}_{\{\alpha _{n}\}}\subset {\mathfrak n}$
and Theorem 8.2 applies. In particular, this shows that here 
is a free module over 
.
(2) Suppose that 
. Then 
for some
$k<n$
, 
, and
${\mathfrak a}:={\mathfrak n}\cap {\mathfrak n}_{\{\alpha _{n}\}}$
is a CP-ideal of
${\mathfrak n}$
. If
${\mathfrak n}_{\{\alpha _{n}\}}$
is identified with the space
$\widehat {\mathsf {Sym}}_n$
of
$n\times n$
matrices that are symmetric w.r.t. the antidiagonal (see Example 2.6), then
Since
${\mathfrak a}$
is a
${\mathfrak b}$
-stable ideal of
${\mathfrak u}$
, 
is a polynomial algebra (see Remark 6.5). Let us prove that 
is a free 
-module, i.e., that
$\bar \pi :{\mathfrak a}^{*}\to {\mathfrak a}^{*}/\!\!/ U$
is equidimensional. Consider the commutative diagram
corresponding to the U-equivariant inclusion
${\mathfrak a}\hookrightarrow {\mathfrak n}_{\{\alpha _{n}\}}$
. Here, 
(cf. Section 3.4(2)), i.e., 
. Upon the identification of
${\mathfrak n}_{\{\alpha _{n}\}}^{*}$
with
${\mathfrak n}_{\{\alpha _{n}\}}^-=\left \{\begin {pmatrix} 0 & 0 \\ C & 0\end {pmatrix}\mid C=\hat C \right \}$
and thereby with the dual space
$\widehat {\mathsf {Sym}}^{*}_n$
, the algebra
${\mathbb C}[{\mathfrak n}_{\{\alpha _{n}\}}^{*}]^U$
is freely generated by the principal minors of C. Let
$f_i$
be the principal minor of degree i; see the figure 
. Then 
and
$\pi _{\alpha _n}(M)=(f_1(M),\dots ,f_n(M))$
. On the other hand, 
for
$i\leqslant k$
and hence 
. Furthermore, if
$\bar M=\psi (M)\in {\mathfrak a}^{*}$
, then
$f_i(\bar M)=f_i(M)$
for
$i\leqslant k$
. Then
$\psi /\!\!/ U$
takes
$(f_1(M),\dots ,f_n(M))$
to
$(f_1(M),\dots ,f_k(M))$
and thereby
$\psi /\!\!/ U$
is surjective and equidimensional. Since we have already proved that
$\pi _{\alpha _n}$
is onto and equidimensional, this yields the same conclusion for
$\bar \pi $
.
Once it is proved that
$\bar \pi $
is onto and equidimensional, an argument similar to that in Theorem 8.2 can be applied to the embedding
${\mathfrak a}\hookrightarrow {\mathfrak n}$
. Consider the diagram
Since
${\mathfrak a}$
is a CP-ideal of
${\mathfrak n}$
, we have 
[Reference Panyushev22, Proposition 5.5]. Hence, 
and thereby 
, i.e.,
$\phi /\!\!/ U$
is an isomorphism. This implies that
$\pi $
is equidimensional, and we are done.
The proofs of Propositions 8.4 and 8.5 exploit the fact that every nilradical in
${\mathfrak {sl}}_{n+1}$
or
${\mathfrak {sp}}_{2n}$
has a CP (and hence a CP-ideal). Using Theorem 6.4, we get a joint corollary to these propositions:
For
${\mathfrak n}\ne \tilde {\mathfrak n}$
, one cannot even guarantee that 
is a polynomial algebra (see Example 6.10). It might be interesting to characterize non-optimal nilradicals having a polynomial algebra 
.
By [Reference Panyushev22], if
${\mathfrak n}\subset \widetilde {\mathfrak n}_{\{\alpha \}}$
with
$[\theta :\alpha ]=1$
, then
${\mathfrak n}$
has a CP. However, to point out a CP-ideal of
${\mathfrak n}$
, one needs some precautions:
(1) If
$\Phi (\Phi ^{-1}(\alpha ))=\{\alpha \}$
, then
${\mathfrak n}\cap {\mathfrak n}_{\{\alpha \}}$
is a CP-ideal of
${\mathfrak n}$
.
(2) If
$\Phi (\Phi ^{-1}(\alpha ))=\{\alpha ,\alpha '\}$
and
${\mathfrak g}\ne {\mathfrak {sl}}_{n+1}$
, then at least one of
${\mathfrak n}\cap {\mathfrak n}_{\{\alpha \}}$
and
${\mathfrak n}\cap {\mathfrak n}_{\alpha '}$
is a CP-ideal of
${\mathfrak n}$
.
(3) If
${\mathfrak g}={\mathfrak {sl}}_{n+1}$
, then one should choose the minimal
$\widetilde {\mathfrak n}_{\{\alpha \}}$
containing
${\mathfrak n}$
. More precisely, since
${\mathfrak n}\subset \widetilde {\mathfrak n}_{\{\alpha \}}$
, we have 
. Here, one has to take
$\alpha $
such that 
. Then item (2) applies.
In any case,
${\mathfrak n}\cap {\mathfrak n}_{\{\alpha \}}$
is a CP-ideal of
${\mathfrak n}$
for a “right” choice of
$\alpha \in \Pi $
, and the hypothesis that
${\mathfrak n}_{\{\alpha \}}\subset {\mathfrak n}$
is not required for the presence of CP. That is, Theorem 8.2 does not apply to all nilradicals with CP. Nevertheless, using a case-by-case argument, we can prove the following.
Theorem 8.6 If 
has a CP, then 
is a free 
-module.
Proof By the preceding discussion, we may assume that
${\mathfrak n}\subset \widetilde {\mathfrak n}_{\{\alpha \}}$
and
${\mathfrak a}:={\mathfrak n}\cap {\mathfrak n}_{\{\alpha \}}$
is a CP-ideal of
${\mathfrak n}$
. As in the proof of Proposition 8.5, one has the commutative diagram (8.1), where
$\phi /\!\!/ U$
is an isomorphism. Therefore, it suffices to prove that
$\bar {\pi }:{\mathfrak a}^{*}\to {\mathfrak a}^{*}/\!\!/ U$
is equidimensional. Consider all simple Lie algebras having abelian nilradicals.
(1) The algebras
${\mathfrak {sl}}_{n+1}$
and
${\mathfrak {sp}}_{2n}$
have been considered above.
(2) For
${\mathfrak {so}}_{2n+1}$
, the only abelian nilradical corresponds to
$\alpha _1=\varepsilon _1-\varepsilon _2$
and 
. Therefore, 
and Lemma 8.1 applies.
(3) For
${\mathfrak {so}}_{2n}$
, the abelian nilradicals correspond to
$\alpha _1,\alpha _{n-1}$
, and
$\alpha _n=\varepsilon _{n-1}+\varepsilon _n$
.
• For
$\alpha _1=\varepsilon _1-\varepsilon _2$
, the situation is the same as for
${\mathfrak g}={\mathfrak {so}}_{2n+1}$
.
• Since
${\mathfrak n}_{\{\alpha _{n-1}\}}\simeq {\mathfrak n}_{\{\alpha _{n}\}}$
, we consider only the last possibility. The case of
$({\mathfrak {so}}_{2n}, \alpha _n)$
is similar to
$({\mathfrak {sp}}_{2n}, \alpha _n)$
, which is considered in Proposition 8.5. The distinction is that now
${\mathfrak n}_{\{\alpha _{n}\}}$
consists of
$n\times n$
matrices that are skew-symmetric w.r.t. the antidiagonal, and the basic U-invariants are pfaffians of the principal minors of even order. The embedding
${\mathfrak a}\subset {\mathfrak n}_{\{\alpha _{n}\}}$
can be described by the figure in the proof of Proposition 8.5, only now k must be even and the matrices should be skew-symmetric w.r.t. the antidiagonal.
More precisely, if
$n=2p+1$
, then 
and
$\Phi ^{-1}(\alpha _n)=\beta _{2p-1}$
. If
$n=2p$
, then
$\alpha _{n}=\beta _{2p-1}$
. In both cases, 
. If
${\mathfrak n}\subset \widetilde {\mathfrak n}_{\{\alpha _{n}\}}$
and
${\mathfrak n}_{\{\alpha \}}\not \subset {\mathfrak n}$
, then 
, where
$s<p=[n/2]$
. Then 
and 
is generated by the pfaffians of order
$2,4,\dots ,2s$
. The fact that these pfaffians form a regular sequence in 
can be proved as in Proposition 8.5 for
${\mathfrak {sp}}_{2n}$
.
(4) For
${\mathsf {\mathbf {{E}}}}_{6}$
, the abelian nilradicals correspond to
$\alpha _1$
and
$\alpha _5$
. In both cases, we have 
and hence Lemma 8.1 applies to any
${\mathfrak n}\subset \widetilde {{\mathfrak n}_{\{\alpha _i\}}}$
,
$i=1,5$
.
(5) For
${\mathsf {\mathbf {{E}}}}_{7}$
, the only abelian nilradical corresponds to
$\alpha _1$
. Here, 
(see Appendix A). Therefore, Lemma 8.1 applies here.
Our computations suggest that Theorem 8.6 holds in the general case.
Conjecture 8.7 For any nilradical
${\mathfrak n}$
in a simple Lie algebra
${\mathfrak g}$
, 
is a free 
-module. In particular, for any optimal nilradical
$\tilde {\mathfrak n}$
, 
is a free 
-module.
So far, this conjecture is proved for (i) the square integrable nilradicals (Section 7), (ii) the nilradicals with CP, (iii) the series
${\mathsf {\mathbf {{A}}}}_{n}$
and
${\mathsf {\mathbf {{C}}}}_{n}$
, and (iv) if
${\mathsf {rk\,}}{\mathfrak g}\leqslant 3$
. It is not hard to check it for
${\mathsf {\mathbf {{D}}}}_{4}$
. Perhaps, the first step toward a general proof is to verify the conjecture for
${\mathfrak n}={\mathfrak u}$
. Since
${\mathfrak u}$
has a CP only for
${\mathsf {\mathbf {{A}}}}_{n}$
and
${\mathsf {\mathbf {{C}}}}_{n}$
, some fresh ideas are necessary here.
A The elements of
We list below the cascade roots (elements of 
) for all simple Lie algebras. The numbering of
$\Pi =\{\alpha _1,\dots ,\alpha _{{\mathsf {rk\,}} {\mathfrak g}}\}$
follows [Reference Onishchik and Vinberg17, Table 1] and, for roots of the classical Lie algebras, we use the standard
$\varepsilon $
-notation. The numbering of cascade roots yields a linear extension of the poset 
, i.e., it is not unique unless 
is a chain. In all cases,
$\beta _1=\theta $
and
$$\begin{align*}\Phi(\beta_i)=\{\alpha\in\Pi\mid \beta_i-\alpha\in \Delta^+\cup\{0\}\}. \end{align*}$$
In particular, if 
, then
$\Phi (\beta )=\{\beta \}$
and
$\beta $
is a minimal element of 
. Conversely, if
$\beta $
is a minimal element of 
and
$\Phi (\beta )=\{\alpha \}$
, a sole simple root, then
$\alpha =\beta $
.
The cascade elements for the classical Lie algebras :
-
${\mathsf {\mathbf {{A}}}}_{n}, n\geqslant 2$
$\beta _i=\varepsilon _i-\varepsilon _{n+2-i}=\alpha _i+\dots +\alpha _{n+1-i}$
$\Big (i=1,2,\dots ,\left [\frac {n+1}{2}\right ]\Big )$
;
-
${\mathsf {\mathbf {{C}}}}_{n}, n\geqslant 1$
$\beta _i=2\varepsilon _i=2(\alpha _i+\dots +\alpha _{n-1})+\alpha _n$
(
$i=1,2,\dots ,n-1$
) and
$\beta _n=2\varepsilon _n=\alpha _n$
;
-
${\mathsf {\mathbf {{B}}}}_{2n}$
,
${\mathsf {\mathbf {{D}}}}_{2n}$
,
${\mathsf {\mathbf {{D}}}}_{2n+1}$
(
$n\geqslant 2$
)
$\beta _{2i-1}=\varepsilon _{2i-1}+\varepsilon _{2i}$
,
$\beta _{2i}=\varepsilon _{2i-1}{-}\varepsilon _{2i}=\alpha _{2i-1}$
(
$i=1,2,\dots ,n$
);
-
${\mathsf {\mathbf {{B}}}}_{2n+1}, n\geqslant 1$
here
$\beta _1,\dots ,\beta _{2n}$
are as above and
$\beta _{2n+1}=\varepsilon _{2n+1}=\alpha _{2n+1}$
.
For all orthogonal series, we have
$\beta _{2i}=\alpha _{2i-1}$
,
$i=1,\dots ,n$
, while formulae for
$\beta _{2i-1}$
via
$\Pi $
slightly differ for different series. For example, for
${\mathsf {\mathbf {{D}}}}_{2n}$
one has
$\beta _{2i-1}=\alpha _{2i-1}+2(\alpha _{2i}+\dots +\alpha _{2n-2})+\alpha _{2n-1}+\alpha _{2n}$
(
$i=1,2,\dots ,n-1$
) and
$\beta _{2n-1}=\alpha _{2n}$
.
The cascade elements for the exceptional Lie algebras :
-
${\mathsf {\mathbf {{G}}}}_{2}$
$\beta _1=(32)=3\alpha _1+2\alpha _2, \ \beta _2=(10)=\alpha _1$
;
-
${\mathsf {\mathbf {{F}}}}_{4}$
$\beta _1=(2432)=2\alpha _1{+}4\alpha _2{+}3\alpha _3{+}2\alpha _4,\ \beta _2=(2210),\ \beta _3=(0210),\ \beta _4=(0010)=\alpha _3$
;
-
${\mathsf {\mathbf {{E}}}}_{6}$
, , , ;
-
${\mathsf {\mathbf {{E}}}}_{7}$
, , , , , , ;
-
${\mathsf {\mathbf {{E}}}}_{8}$
, , , , , , , .
, 
, 
, 
;
, 
, 
, 
, 
, 
, 
;
, 
, 
, 
, 
, 
, 
, 
.For the reader’s convenience, we provide the Hasse diagram of the cascade posets for
${\mathsf {\mathbf {{D}}}}_{n}$
and
${\mathsf {\mathbf {{E}}}}_{n}$
. The node “i” in the diagram represents 
, and we attach the set
$\Phi (\beta _i)\subset \Pi $
to it.
