On adjoint orbits in nilpotent ideals of a Borel subalgebra

TL;DR

This paper investigates the structure of nilpotent elements within a nilpotent ideal of a Borel subalgebra in a complex semisimple Lie algebra, revealing a unique closed orbit under the Borel group action.

Contribution

It establishes the existence and uniqueness of a closed Borel orbit in the set of ad-nilpotent elements associated with a nilpotent ideal, characterized by minimal roots.

Findings

Unique closed Borel orbit in the set of ad-nilpotent elements.

The orbit corresponds to a nilpotent element supported on minimal roots.

Characterization of the orbit via root space decomposition.

Abstract

Let $\mathfrak{m}$ be a nilpotent ideal in the Borel subalgebra $\mathfrak{b}$ of a complex finite-dimensional semisimple Lie algebra, and $\mathfrak{m}^{\bullet}$ the subset of (ad-)nilpotent elements in $\mathfrak{b}$ such that $\mathfrak{m}$ is the minimal ideal containing them. This set is stable under the adjoint action of the corresponding Borel subgroup $B$. We prove that $\mathfrak{m}^{\bullet}$ contains a unique closed $B$-orbit which is the orbit of a nilpotent element whose support is the set of minimal roots associated to the root space decomposition of $\mathfrak{m}$.

Equations

$$\mathfrak{m}=\underset{\alpha \in {\Delta }_{\mathfrak{m}}^{+}}{⨁}{\mathfrak{g}}_{\alpha }.$$

$$\mathop{\mathrm{supp}}\nolimits(X)=\{\alpha;c_{\alpha}\neq 0\}\$$

$${\mathfrak{m}}^{\bullet }=\{X\in \mathfrak{n};{\Delta }_{\mathfrak{m}}^{+}\supset \mathrm{supp}(X)\supset {\mathcal{I}}_{\mathfrak{m}}\}=\mathfrak{m}\setminus {\mathfrak{m}}^{\circ }.$$

$${\mathfrak{m}}_0=\underset{\alpha \in {\Delta }_{{\mathcal{I}}_{\mathfrak{m}}}^{+}}{⨁}{\mathfrak{g}}_{\alpha }and{\mathfrak{b}}_0=\mathfrak{h}\oplus {\mathfrak{m}}_0$$

$$\mathbb{C}[\mathfrak{m}]=\mathbb{C}[{\xi }_{\alpha },\alpha \in {\Delta }_{\mathfrak{m}}^{+}]=\underset{\alpha \in -\mathbb{N}\Pi }{⨁}\mathbb{C}[\mathfrak{m}{]}_{\alpha }$$

$$\underset{\alpha \in -\mathbb{N}{\mathcal{I}}_{\mathfrak{m}}}{⨁}\mathbb{C}[\mathfrak{m}{]}_{\alpha }^{\mathfrak{n}}=\underset{\alpha \in -\mathbb{N}{\mathcal{I}}_{\mathfrak{m}}}{⨁}\mathbb{C}[\mathfrak{m}{]}_{\alpha }^{{\mathfrak{m}}_0}=\mathbb{C}[{\xi }_{\alpha },\alpha \in {\mathcal{I}}_{\mathfrak{m}}].$$

$$\mathfrak{c}=\underset{\alpha \in {\Delta }_{\mathfrak{m}}^{+}\setminus {\Delta }_{{\mathcal{I}}_{\mathfrak{m}}}^{+}}{⨁}{\mathfrak{g}}_{\alpha }.$$

$${\mathfrak{m}}_0\stackrel{\iota }{⟶}\mathfrak{m}={\mathfrak{m}}_0\oplus \mathfrak{c}\stackrel{\pi }{⟶}{\mathfrak{m}}_0$$

$$\mathbb{C}[{\mathfrak{m}}_0]\stackrel{{\pi }^{\ast }}{⟶}\mathbb{C}[\mathfrak{m}]\stackrel{{\iota }^{\ast }}{⟶}\mathbb{C}[{\mathfrak{m}}_0]$$

$$\mathbb{C}[\mathfrak{m}]=\mathbb{C}[{\mathfrak{m}}_0]\oplus \ker ({\iota }^{\ast })$$

$$\phi ={\psi }_n{\xi }_{\beta }^n+{\psi }_{n-1}{\xi }_{\beta }^{n-1}+\cdots +{\psi }_1{\xi }_{\beta }+{\psi }_0$$

$$0=Z(\phi )=Z({\psi }_n){\xi }_{\beta }^n+\eta$$

$$I_{+}=\{i;c_i-c_i^{\prime }>0\},I_0=\{i;c_i=c_i^{\prime }\},I_{-}=\{i;c_i-c_i^{\prime }<0\}.$$

$$\begin{array}[]{c}H=\sum\limits_{i\in I_{+}}(c_{-}^{\prime}-c_{-})H_{i}+\sum\limits_{i\in I_{-}}(c_{+}-c_{+}^{\prime})H_{i}+\sum\limits_{i\in I_{0}}H_{i}.\end{array}$$

$$\begin{array}[]{c}H=\sum\limits_{i=1}^{m-1}(n^{\prime}-\delta(H^{\prime}))H_{i}+0pt_{\Pi}(\beta)H^{\prime}.\end{array}$$

$${\alpha }_{i,j},{\alpha }_{k,\ell -2}+{\alpha }_{\ell -1},{\alpha }_{k,\ell -2}+{\alpha }_{\ell },{\alpha }_{k,\ell -2}+{\alpha }_{\ell -1}+{\alpha }_{\ell },{\alpha }_{i,\ell -2}+{\alpha }_{j,\ell -2}+{\alpha }_{\ell -1}+{\alpha }_{\ell }$$

$$\begin{array}[]{c}H^{\prime}=\sum\limits_{i=m}^{m^{\prime}-1}(n^{\prime\prime}-\delta^{\prime}(H^{\prime\prime}))H_{i}+(m^{\prime}-m)H^{\prime\prime}\end{array}$$

$$\begin{array}[]{c}H=\sum\limits_{i=1}^{m-1}\bigl{(}(m^{\prime}-m)n^{\prime\prime}-\delta(H^{\prime})\bigr{)}H_{i}+(m-1)H^{\prime},\end{array}$$

$$\begin{array}[]{c}H^{\prime\prime}=\sum\limits_{i=m}^{\ell-2}2b_{i}H_{i}+(b_{\ell-1}+b_{\ell})(H_{\ell-1}+H_{\ell}).\end{array}$$

$$\begin{array}[]{c}H=\sum\limits_{i=1}^{m-1}(2n^{\prime}-\delta_{1}(H^{\prime\prime}))H_{i}+(m-1)H^{\prime\prime}.\end{array}$$

$$\begin{array}[]{c|c}\Gamma&(H,n)\\ \hline\cr\hline\cr\{\alpha_{1}+\alpha_{2},\alpha_{2}+\alpha_{3},\alpha_{3}+\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+H_{4},2)\\ \hline\cr\{\alpha_{1}+\alpha_{2},\alpha_{2}+2\alpha_{3},\alpha_{3}+\alpha_{4}\}&(2H_{1}+H_{2}+H_{3}+2H_{4},3)\\ \hline\cr\{\alpha_{1}+\alpha_{2},\alpha_{2}+2\alpha_{3},\alpha_{2}+\alpha_{3}+\alpha_{4}\}&(2H_{1}+H_{2}+H_{3}+H_{4},3)\\ \hline\cr\{\alpha_{1}+\alpha_{2}+\alpha_{3},\alpha_{2}+2\alpha_{3},\alpha_{3}+\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+2H_{4},3)\\ \hline\cr\{\alpha_{1}+\alpha_{2}+\alpha_{3},\alpha_{2}+2\alpha_{3},\alpha_{2}+\alpha_{3}+\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+H_{4},3)\\ \hline\cr\{\alpha_{1}+\alpha_{2}+2\alpha_{3},\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\alpha_{2}+2\alpha_{3}+\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+H_{4},4)\\ \hline\cr\{\alpha_{1}+\alpha_{2}+2\alpha_{3},\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\alpha_{2}+2\alpha_{3}+2\alpha_{4}\}&(2H_{1}+H_{2}+H_{3}+H_{4},5)\\ \hline\cr\{\alpha_{1}+2\alpha_{2}+2\alpha_{3},\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\alpha_{2}+2\alpha_{3}+\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+2H_{4},5)\\ \hline\cr\{\alpha_{1}+2\alpha_{2}+2\alpha_{3},\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\alpha_{2}+2\alpha_{3}+2\alpha_{4}\}&(3H_{1}+H_{2}+H_{3}+2H_{4},7)\\ \hline\cr\{\alpha_{1}+2\alpha_{2}+2\alpha_{3},\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},\alpha_{2}+2\alpha_{3}+2\alpha_{4}\}&(H_{1}+H_{2}+H_{3}+H_{4},5)\end{array}$$

$${\Omega }_{\mathbf{T}}(X)=\{Y\in \mathfrak{m};\mathrm{supp}(Y)={\mathcal{I}}_{\mathfrak{m}}\}.$$

$$\dim {\Omega }_{\mathbf{B}}(X)=\dim {\Omega }_{\hat{\mathbf{B}}}(X)<\dim \overline{{\Omega }_{\hat{\mathbf{B}}}(Y)}⩽\dim {\Omega }_{\mathbf{B}}(Y)+1.$$

Full text

1. Introduction

Let $\mathfrak{b}$ be a Borel subalgebra of a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$, $\mathbf{G}$ the adjoint algebraic group of $\mathfrak{g}$ and $\mathbf{B}$ the Borel subgroup of $\mathbf{G}$ whose Lie algebra is $\mathfrak{b}$. We fix a decomposition $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$ where $\mathfrak{n}$ is the nilradical of $\mathfrak{b}$ and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{b}$. Observe that $\mathfrak{n}$ is also the set of (ad-)nilpotent elements in $\mathfrak{b}$.

Adjoint nilpotent $\mathbf{G}$-orbits in $\mathfrak{g}$ are classified by many combinatorial objects. They are well studied, and have many nice characterisations and properties (finiteness, algebraic and geometric). In contrast, $\mathbf{B}$-orbits of $\mathfrak{n}$ are less well understood. There are in general an infinite number of them, and they are not necessary conical. Kashin [5] determined exactly when there is a finite number of $\mathbf{B}$-orbits in $\mathfrak{n}$. Hille and Röhrle [4] obtained a more general result using a quiver model. Boos [1] and Melnikov [6] studied $\mathbf{B}$-orbits in certain subvarieties of $\mathfrak{n}$ in type $A$. Note also that connected components of the intersection of a nilpotent $\mathbf{G}$-orbit with $\mathfrak{b}$ are called orbital varieties which are related to irreducible components of the corresponding Springer fibers [9, 3], and relations between nilpotent $\mathbf{G}$-orbits and nilpotent ideals in $\mathfrak{b}$ are studied in [11].

Since $\mathfrak{g}$ is semisimple, an ideal in $\mathfrak{b}$ is nilpotent if and only if it is contained in $\mathfrak{n}$. For any nilpotent ideal $\mathfrak{m}$ of $\mathfrak{b}$, we denote by $\mathfrak{m}^{\bullet}$ the set of $X\in\mathfrak{n}$ such that $\mathfrak{m}$ is the minimal nilpotent ideal in $\mathfrak{b}$ containing $X$. Thus $\mathfrak{n}$ is the disjoint union of $\mathfrak{m}^{\bullet}$ where $\mathfrak{m}$ runs through all nilpotent ideals of $\mathfrak{b}$.

Let $\Delta\supset\Delta^{+}\supset\Pi$ be respectively the root system of $\mathfrak{g}$, the set of positive roots and the set of simple roots relative to $(\mathfrak{b},\mathfrak{h})$. Denote by $\mathfrak{g}_{\alpha}$ the root subspace relative to $\alpha$ in $\mathfrak{g}$. For any nilpotent ideal $\mathfrak{m}$ of $\mathfrak{b}$, there exists a unique subset $\Delta_{\mathfrak{m}}^{+}$ of $\Delta^{+}$ such that

[TABLE]

Denote by $\mathcal{I}_{\mathfrak{m}}$ the set of minimal roots in $\Delta_{\mathfrak{m}}^{+}$ with respect to the usual partial order $\prec$ on $\Delta$ given by $\alpha\prec\beta$ if $\beta-\alpha$ is a sum of simple roots. The main result of the paper is the following theorem.

Theorem 1.1**.**

Let $\mathfrak{m}$ be a nilpotent ideal of $\mathfrak{b}$ and $X\in\mathfrak{m}^{\bullet}\cap\bigoplus_{\alpha\in\mathcal{I}_{\mathfrak{m}}}\mathfrak{g}_{\alpha}$. Then the $\mathbf{B}$-orbit of $X$ is the unique closed $\mathbf{B}$-orbit in the $\mathbf{B}$-stable subvariety $\mathfrak{m}^{\bullet}$.

The key observation here is that being a set of pairwise non comparable positive roots, a result of Sommers [10] says that $\mathcal{I}_{\mathfrak{m}}$ is conjugate under the Weyl group of $\mathfrak{g}$ to a subset of $\Pi$. This allows us in section 3 to obtain sufficient information on $\mathfrak{n}$-invariant functions on $\mathfrak{m}$ in order to prove that $\mathfrak{m}^{\bullet}$ contains a unique closed $\mathbf{B}$-orbit. In section 4, we prove a nice geometrical property (Theorem 4.1) for sets consisting of pairwise non comparable positive roots which allows us to precise exactly this closed orbit by studying orbits under the Cartan subgroup $\mathbf{T}$ in $\mathbf{B}$ whose Lie algebra is $\mathfrak{h}$.

Notations.

We shall conserve the notations above in the rest of the paper. Denote by $\mathbf{U}$ the unipotent radical of $\mathbf{B}$ which is the subgroup of $\mathbf{B}$ generated by $\exp(\mathrm{ad}(X))$, $X\in\mathfrak{n}$. Recall that $\mathfrak{n}$ is the Lie algebra of $\mathbf{U}$ and $\mathbf{B}=\mathbf{T}\ltimes\mathbf{U}$.

For any linearly independent subset $\mathcal{I}$ of $\Delta$ and $\mathbb{A}\subset\mathbb{C}$, we shall denote by $\mathbb{A}\mathcal{I}$ the set of elements of $\mathfrak{h}^{*}$ of the form $\sum_{\alpha\in\mathcal{I}}c_{\alpha}\alpha$ where $c_{\alpha}\in\mathbb{A}$ for all $\alpha\in\mathcal{I}$. Given $\beta=\sum_{\alpha\in\mathcal{I}}c_{\alpha}\alpha\in\mathbb{A}\mathcal{I}$, we shall denote by $0pt_{\mathcal{I}}(\beta)=\sum_{\alpha\in\mathcal{I}}c_{\alpha}$ the height of $\beta$ with respect to $\mathcal{I}$.

2. Nilpotent elements, nilpotent ideals and antichains

Let us fix a basis $(X_{\alpha})_{\alpha\in\Delta^{+}}$ of $\mathfrak{n}$ where $X_{\alpha}\in\mathfrak{g}_{\alpha}$. For $X=\sum_{\alpha\in\Delta^{+}}c_{\alpha}X_{\alpha}\in\mathfrak{n}$, we set

[TABLE]

the support of $X$, $\mathcal{I}_{X}$ the set of minimal roots in $\mathop{\mathrm{supp}}\nolimits(X)$ and $\mathfrak{m}_{X}$ the minimal nilpotent ideal in $\mathfrak{b}$ containing $X$. The following lemmas are immediate from the definitions.

Lemma 2.1**.**

Let $X\in\mathfrak{n}$ and $\Delta_{X}^{+}=\{\alpha\in\Delta^{+};\alpha\succcurlyeq\beta$ for some $\beta\in\mathcal{I}_{X}\}$.

1. (1)

We have $\mathfrak{m}_{X}=\bigoplus_{\alpha\in\Delta_{X}^{+}}\mathfrak{g}_{\alpha}$. In particular, $\mathcal{I}_{X}=\mathcal{I}_{\mathfrak{m}_{X}}$. 2. (2)

For any $\sigma\in\mathbf{B}$, we have $\mathfrak{m}_{X}=\mathfrak{m}_{\sigma(X)}$ and $\mathcal{I}_{X}=\mathcal{I}_{\sigma(X)}$.

Lemma 2.2**.**

Let $\mathfrak{m}$ be a nilpotent ideal of $\mathfrak{b}$ and $\mathfrak{m}^{\circ}$ be the union of all nilpotent ideals in $\mathfrak{b}$ contained strictly in $\mathfrak{m}$. Then

[TABLE]

is an open $\mathbf{B}$-stable subset of $\mathfrak{m}$.

Let $\mathfrak{m}$ be a nilpotent ideal of $\mathfrak{b}$. Then it is completely determined by $\mathcal{I}_{\mathfrak{m}}$. Being minimal elements of $\Delta_{\mathfrak{m}}^{+}$, the set $\mathcal{I}_{\mathfrak{m}}$ contains pairwise non comparable roots, also called an antichain. Conversely, any antichain $\mathcal{I}$ in $\Delta^{+}$ defines a nilpotent ideal $\mathfrak{m}_{X}$ where $X$ is any element in $\mathfrak{n}$ verifying $\mathop{\mathrm{supp}}\nolimits(X)=\mathcal{I}$. Thus the map $\mathfrak{m}\mapsto\mathcal{I}_{\mathfrak{m}}$ is a bijection between the set of nilpotent ideals in $\mathfrak{b}$ and the set of antichains in $\Delta^{+}$.

Theorem 2.3** (Sommers [10]).**

Let $\mathcal{I}$ be an antichain in $\Delta^{+}$. There exists an element $w$ in the Weyl group of $\mathfrak{g}$ such that $w(\mathcal{I})\subset\Pi$.

Remark 2.4*.*

We obtain immediately from Theorem 2.3 that if $\mathcal{I}$ is a non empty antichain in $\Delta^{+}$, then the set $\Delta_{\mathcal{I}}=\mathbb{Z}\mathcal{I}\cap\Delta=\mathbb{Q}\mathcal{I}\cap\Delta$ is a reduced root system, $\mathcal{I}$ is a set of simple roots of $\Delta_{\mathcal{I}}$ and $\Delta_{\mathcal{I}}^{+}=\Delta_{\mathcal{I}}\cap\Delta=\mathbb{N}\mathcal{I}\cap\Delta$ is the set of positive roots with respect to $\mathcal{I}$.

3. Invariant functions and closed $\mathbf{B}$-stable subsets

Let $\mathfrak{m}$ be a nilpotent ideal in $\mathfrak{b}$. By Remark 2.4, $\Delta_{\mathcal{I}_{\mathfrak{m}}}$ is a reduced root system. Therefore $\mathfrak{g}_{0}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta_{\mathcal{I}_{\mathfrak{m}}}}\mathfrak{g}_{\alpha}$ is a reductive Lie subalgebra in $\mathfrak{g}$. Let

[TABLE]

which is a Borel subalgebra of $\mathfrak{g}_{0}$ whose nilradical is $\mathfrak{m}_{0}$. Denote by $\mathbf{B}_{0}$ the closed connected subgroup of $\mathbf{G}$ whose Lie algebra is $\mathfrak{b}_{0}$.

Let $(\xi_{\alpha})_{\alpha\in\Delta_{{\mathfrak{m}}}^{+}}$ be the dual basis of the basis $(X_{\alpha})_{\alpha\in\Delta_{{\mathfrak{m}}}^{+}}$ of $\mathfrak{m}$. The ring of regular functions on $\mathfrak{m}$ decomposes as a direct sum

[TABLE]

where $\mathbb{C}[\mathfrak{m}]_{\alpha}$ denotes the vector space span of monomials $\xi_{\alpha_{1}}\cdots\xi_{\alpha_{r}}$ where $r\in\mathbb{N}$, $\alpha_{1},\dots,\alpha_{r}\in\Delta_{\mathfrak{m}}^{+}$ and $\alpha_{1}+\cdots+\alpha_{r}=-\alpha$. Since $H(\xi_{\alpha})=-\alpha(H)\xi_{\alpha}$ for any $H\in\mathfrak{h}$, $\mathbb{C}[\mathfrak{m}]_{\alpha}$ is also the $\mathfrak{h}$-weight subspace of $\mathbb{C}[\mathfrak{m}]$ of weight $\alpha$.

Note also that for $\alpha,\beta\in\Delta_{\mathfrak{m}}^{+}$, $X_{\alpha}(\xi_{\beta})$ is a non zero multiple of $\xi_{\beta-\alpha}$ if $\beta-\alpha\in\Delta_{\mathfrak{m}}^{+}$, and is zero otherwise.

Proposition 3.1**.**

We have

[TABLE]

Proof.

Let

[TABLE]

Thus we have $\mathfrak{m}=\mathfrak{m}_{0}\oplus\mathfrak{c}$ as $\mathfrak{b}_{0}$-modules. As the composition of the natural morphisms of $\mathfrak{b}_{0}$-modules

[TABLE]

is the identity map, the composition of the comorphisms

[TABLE]

is also the identity map. We shall identify $\mathbb{C}[\mathfrak{m}_{0}]$ with its image under $\pi^{*}$ which is precisely $\mathbb{C}[\xi_{\alpha},\alpha\in\Delta_{\mathcal{I}_{\mathfrak{m}}}^{+}]$. Hence

[TABLE]

as $\mathfrak{b}_{0}$-modules. Consequently, $\mathbb{C}[\mathfrak{m}]^{\mathfrak{m}_{0}}=\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}\oplus\ker(\iota^{*})^{\mathfrak{m}_{0}}$.

Suppose that $\ker(\iota^{*})^{\mathfrak{m}_{0}}_{\lambda}\neq 0$ for some $\lambda\in-\mathbb{N}\mathcal{I}_{\mathfrak{m}}$. Fix such a $\lambda=\sum_{\alpha\in\Pi}c_{\alpha}\alpha$ such that $0pt_{\Pi}(\lambda)$ is maximal. Let $\varphi\in\ker(\iota^{*})^{\mathfrak{m}_{0}}_{\lambda}$ be non zero. There exist $\beta\in\Delta_{\mathfrak{m}}^{+}$ and $n>0$ such that

[TABLE]

where $\psi_{k}\in\mathbb{C}[\xi_{\alpha},\alpha\not\succcurlyeq\beta]_{\lambda+k\beta}$ and $\psi_{n}\neq 0$.

For any $Z\in\mathfrak{m}_{0}$, we have

[TABLE]

where by our choice of $\beta$, the degree of $\eta$ in $\xi_{\beta}$ is at most $n-1$. So $Z(\psi_{n})=0$. Thus $\psi_{n}\in\mathbb{C}[\mathfrak{m}]^{\mathfrak{m}_{0}}_{\lambda+n\beta}$, and our choice of $\lambda$ implies that $\psi_{n}\in\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}$, and hence $\lambda+n\beta\in-\mathbb{N}\mathcal{I}_{\mathfrak{m}}$. It follows that $\beta\in\mathbb{Q}\mathcal{I}_{\mathfrak{m}}\cap\Delta_{\mathfrak{m}}^{+}=\mathbb{N}\mathcal{I}_{\mathfrak{m}}\cap\Delta_{\mathfrak{m}}^{+}=\Delta_{\mathcal{I}_{\mathfrak{m}}}^{+}$ (Remark 2.4), and so $\psi_{n}\xi_{\beta}^{n}\in\mathbb{C}[\mathfrak{m}_{0}]$ which is impossible since $\varphi\in\ker(\iota^{*})^{\mathfrak{m}_{0}}_{\lambda}$. We have therefore established that $\bigoplus_{\alpha\in-\mathbb{N}\mathcal{I}_{\mathfrak{m}}}\mathbb{C}[\mathfrak{m}]_{\alpha}^{\mathfrak{m}_{0}}=\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}$.

Now, we check readily that $\mathbb{C}[\xi_{\alpha},\alpha\in\mathcal{I}_{\mathfrak{m}}]$ is contained in both $\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}$ and $\mathbb{C}[\mathfrak{m}]^{\mathfrak{n}}$. We claim that $\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}=\mathbb{C}[\xi_{\alpha},\alpha\in\mathcal{I}_{\mathfrak{m}}]$.

Let us prove our claim. Observe that the ring of $\mathfrak{m}_{0}$-invariant regular functions on $\mathfrak{m}_{0}$ is the ring of $\mathfrak{b}_{0}$-semi-invariant regular functions on $\mathfrak{m}_{0}$. By Richardson’s dense orbit theorem [7], $\mathfrak{m}_{0}$ contains an open $\mathbf{B}_{0}$-orbit. It follows from a theorem of Rosenlicht [8] that the field of $\mathbf{B}_{0}$-invariant rational functions is $\mathbb{C}$. This implies that $\dim\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}_{\alpha}\leqslant 1$ for any $\alpha\in-\mathbb{N}\mathcal{I}_{\mathfrak{m}}$. Since $\mathbb{C}[\xi_{\alpha},\alpha\in\mathcal{I}_{\mathfrak{m}}]\subset\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}$, we have our claim.

We deduce immediately from our claim the required equalities since $\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}=\mathbb{C}[\xi_{\alpha},\alpha\in\mathcal{I}_{\mathfrak{m}}]\subset\mathbb{C}[\mathfrak{m}]^{\mathfrak{n}}\subset\mathbb{C}[\mathfrak{m}]^{\mathfrak{m}_{0}}$. ∎

Theorem 3.2**.**

The intersection of two non empty closed $\mathbf{B}$-stable subsets of $\mathfrak{m}^{\bullet}$ is non empty.

Proof.

Let us conserve the notations in the proof of Proposition 3.1. Set $\mathbf{M}_{0}$ to be the closed connected subgroup of $\mathbf{B}_{0}$ whose Lie algebra is $\mathfrak{m}_{0}$. In particular, $\mathbf{M}_{0}$ is the subgroup generated by $\exp(\mathrm{ad}(X))$ where $X\in\mathfrak{m}_{0}$.

By Proposition 3.1 and Lemma 2.2, $\varphi=\prod_{\alpha\in\mathcal{I}_{\mathfrak{m}}}\xi_{\alpha}\in\mathbb{C}[\mathfrak{m}_{0}]^{\mathfrak{m}_{0}}$ and $\mathfrak{m}^{\bullet}=\{X\in\mathfrak{m};\varphi(X)\neq 0\}$. So $\mathbb{C}[\mathfrak{m}^{\bullet}]$ is the localization of $\mathbb{C}[\mathfrak{m}]$ by the multiplicative subset generated by $\varphi$.

Let $\mathfrak{m}_{0}^{\bullet}=\mathfrak{m}_{0}\cap\mathfrak{m}^{\bullet}$. Similarly, $\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]$ is the localization of $\mathbb{C}[\mathfrak{m}_{0}]$ by the multiplicative subset generated by $\varphi$. As in the proof of Proposition 3.1, we obtain from the composition $\mathfrak{m}_{0}^{\bullet}\stackrel{{\scriptstyle\iota}}{{\rightarrow}}\mathfrak{m}^{\bullet}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathfrak{m}_{0}^{\bullet}$ of $\mathbf{B}_{0}$-equivariant morphisms that $\mathbb{C}[\mathfrak{m}^{\bullet}]=\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]\oplus\ker(\iota^{*})$ as $\mathbf{B}_{0}$-modules.

Let $J$ be a proper $\mathbf{B}$-stable ideal in $\mathbb{C}[\mathfrak{m}^{\bullet}]$. Then $J+\ker(\iota^{*})$ is $\mathbf{B}_{0}$-stable, and hence $(J+\ker(\iota^{*}))/\ker(\iota^{*})\simeq\iota^{*}(J)$ is a $\mathbf{B}_{0}$-stable ideal in $\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]$.

As mentioned in the proof of Proposition 3.1, $\mathfrak{m}_{0}$ contains an open $\mathbf{B}_{0}$-orbit. Since $\mathfrak{m}_{0}^{\bullet}$ is open in $\mathfrak{m}_{0}$ (Lemma 2.2), we deduce that either $\iota^{*}(J)=0$ or $\iota^{*}(J)=\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]$.

Suppose that $\iota^{*}(J)=\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]$. Then there exists $(f,g)\in J_{0}\times\ker(\iota^{*})_{0}$ such that $f+g=1$. In particular, $f$ and $g$ are non constant since $J$ is proper.

Let $V$ be the $\mathbf{M}_{0}$-submodule generated by $f$ in $\mathbb{C}[\mathfrak{m}^{\bullet}]$. Then $V$ is finite-dimensional, and we have $V^{\mathbf{M}_{0}}\neq 0$ because $\mathbf{M}_{0}$ is unipotent. Let $\sigma_{1},\dots,\sigma_{r}\in\mathbf{M}_{0}$ and $c_{1},\dots,c_{r}\in\mathbb{C}^{*}$ be such that $\tilde{f}=\sum_{i=1}^{r}c_{i}\sigma_{i}(f)$ is a non zero element of $V^{\mathbf{M}_{0}}=V^{\mathfrak{m}_{0}}$.

Set $\tilde{g}=\sum_{i=1}^{r}c_{i}\sigma_{i}(g)$. Then $\tilde{f}+\tilde{g}=\sum_{i=1}^{r}c_{i}$. As $J$ and $\ker(\iota^{*})$ are $\mathbf{B}_{0}$-stable, we deduce that $\tilde{f}\in J^{\mathfrak{m}_{0}}$, and hence $\tilde{g}\in\ker(\iota^{*})^{\mathfrak{m}_{0}}$. Moreover, $\tilde{f}$ and $\tilde{g}$ are both non constant because $J$ is proper, and by construction, they are sums of vectors of weights in $-\mathbb{N}\mathcal{I}_{\mathfrak{m}}$ because $f$ and $g$ are of weight [math].

Since $\tilde{g}\in\ker(\iota^{*})^{\mathfrak{m}_{0}}$ and $\varphi\in\mathbb{C}[\mathfrak{m}]^{\mathfrak{m}_{0}}$ (Proposition 3.1), there exists $n>0$ such that $\varphi^{n}\tilde{g}\in\bigoplus_{\alpha\in-\mathbb{N}\mathcal{I}_{\mathfrak{m}}}\mathbb{C}[\mathfrak{m}]_{\alpha}^{\mathfrak{m}_{0}}=\mathbb{C}[\xi_{\alpha},\alpha\in\mathcal{I}_{\mathfrak{m}}]$ (Proposition 3.1). This implies that $\tilde{g}\in\mathbb{C}[\mathfrak{m}_{0}^{\bullet}]$ which is absurd because $\tilde{g}\in\ker(\iota^{*})$ and $\tilde{g}\neq 0$.

We conclude that $\iota^{*}(J)=0$, and hence $J\subset\ker(\iota^{*})\neq\mathbb{C}[\mathfrak{m}^{\bullet}]$.

Consequently, the sum of two proper $\mathbf{B}$-stable ideals can never be $\mathbb{C}[\mathfrak{m}^{\bullet}]$. So the intersection of two non empty closed $\mathbf{B}$-stable subsets of $\mathfrak{m}^{\bullet}$ is non empty. ∎

Corollary 3.3**.**

There is a unique closed $\mathbf{B}$-orbit in $\mathfrak{m}^{\bullet}$.

Proof.

Any $\mathbf{B}$-orbit of minimal dimension is closed, so a closed $\mathbf{B}$-orbit exists and Theorem 3.2 says that it is unique. ∎

4. Description of the unique closed orbit

We shall show in this section that the unique closed $\mathbf{B}$-orbit in Corollary 3.3 is the orbit of any element in $\mathfrak{m}$ whose support is $\mathcal{I}_{\mathfrak{m}}$. Our approach is to use the following geometrical property concerning antichains of positive roots to study the action of the torus $\mathbf{T}$ on the projective variety $\mathbb{P}(\mathfrak{m})$.

Theorem 4.1**.**

Let $\Gamma\subset\Delta^{+}$ be a non empty antichain. There exists $(H,n)\in\mathfrak{h}\times\mathbb{N}^{*}$ such that $\alpha(H)\in\mathbb{N}^{*}$ for all $\alpha\in\Pi$ and $\gamma(H)=n$ for all $\gamma\in\Gamma$.

Proof.

Let $\ell=\sharp\Pi$ be the rank of $\Delta$, $\Pi=\{\alpha_{1},\dots,\alpha_{\ell}\}$, $(H_{1},\dots,H_{\ell})$ the basis of $\mathfrak{h}$ whose dual basis is $\Pi$. Given $\alpha=\sum_{i=1}^{\ell}c_{i}\alpha_{i}\in\Delta$, we set $\Pi_{\alpha}=\{\alpha_{i}\in\Pi\,;c_{i}\neq 0\}=\{\alpha_{i}\in\Pi\,;\alpha(H_{i})\neq 0\}$.

We shall prove the theorem by induction on $\ell$. If $\ell=1$, then $\Gamma=\Pi$. So $(H_{1},1)$ verifies the required conditions. Let us suppose that $\ell>1$.

1. Suppose that $\sharp\Gamma\leqslant 2$.

If $\Gamma=\{\gamma\}$, then $(H_{1}+\cdots+H_{\ell},0pt(\gamma))$ verifies the required conditions.

If $\Gamma=\{\gamma,\gamma^{\prime}\}$ where $\gamma=\sum_{i=1}^{\ell}c_{i}\alpha_{i}$ and $\gamma^{\prime}=\sum_{i=1}^{\ell}c_{i}^{\prime}\alpha_{i}$ are distinct, set

[TABLE]

Since $\gamma$ and $\gamma^{\prime}$ are non comparable, both $I_{+}$ and $I_{-}$ are non empty. Let $c_{\pm}=\sum_{i\in I_{\pm}}c_{i}$, $c_{\pm}^{\prime}=\sum_{i\in I_{\pm}}c_{i}^{\prime}$, $c=\sum_{i\in I_{0}}c_{i}=\sum_{i\in I_{0}}c_{i}^{\prime}$ and

[TABLE]

Since $c_{-}^{\prime}>c_{-}>0$ and $c_{+}>c_{+}^{\prime}>0$, $(H,c_{-}^{\prime}c_{+}-c_{-}c_{+}^{\prime}+c)$ verifies the required conditions.

2. Suppose that $\Pi_{\Gamma}=\bigcup_{\gamma\in\Gamma}\Pi_{\gamma}\neq\Pi$.

Let $\mathfrak{h}_{\Gamma}=\mathrm{Vect}(H_{i},\alpha_{i}\in\Pi_{\Gamma})$. Being a subset of $\Pi$, $\Pi_{\Gamma}$ is an antichain. If $\Pi_{\Gamma}\neq\Pi$, then $\mathop{\mathrm{rk}}\nolimits\Delta_{\Pi_{\Gamma}}<\mathop{\mathrm{rk}}\nolimits\Delta$. By induction, there exists $(H^{\prime},n)\in\mathfrak{h}_{\Gamma}\times\mathbb{N}^{*}$ verifying the required conditions for $\Gamma$ as an antichain in $\Delta_{\Pi_{\Gamma}}^{+}$. The pair $(H^{\prime}+\sum_{\alpha_{i}\in\Pi\setminus\Pi_{\Gamma}}H_{i},n)$ verifies the required conditions.

3. Suppose that $\Pi_{\Gamma}=\Pi$ and $\Delta$ is not irreducible.

There is a partition $\Pi_{1}\cup\cdots\cup\Pi_{r}$ of $\Pi$ such that $\Delta=\Delta_{\Pi_{1}}\cup\cdots\cup\Delta_{\Pi_{r}}$ is the disjoint union of irreductible root systems of strictly lower rank. As $\Pi_{\Gamma}=\Pi$, $\Gamma=(\Gamma\cap\Delta_{\Pi_{1}})\cup\cdots\cup(\Gamma\cap\Delta_{\Pi_{r}})$ is the disjoint union of non empty antichains. By induction, for $1\leqslant i\leqslant r$, there exists $(K_{i},n_{i})\in\mathfrak{h}_{\Pi_{i}}\times\mathbb{N}^{*}$ verifying the required conditions for $\Gamma\cap\Delta_{\Pi_{i}}$ as an antichain in $\Delta_{\Pi_{i}}^{+}$. Then $(\sum_{i=1}^{r}\frac{n_{1}\cdots n_{r}}{n_{i}}K_{i},n_{1}\cdots n_{r})$ verifies the required conditions.

4. We are left with situation where $\sharp\Gamma\geqslant 3$, $\Pi_{\Gamma}=\Pi$ and $\Delta$ is irreducible.

We shall use the numbering of simple roots of irreducible root systems in [12, Chapter 18]. Let $\Delta_{1}^{+}=\{\alpha\in\Delta^{+};\alpha_{1}\in\Pi_{\alpha}\}$, $\Gamma_{1}=\Gamma\cap\Delta_{1}^{+}$ and $\Gamma^{\prime}=\Gamma\setminus\Gamma_{1}$. Since $\Pi_{\Gamma}=\Pi$, the set $\Gamma_{1}$ is non empty. Let us fix an element $\gamma\in\Gamma_{1}$.

The main observation here is that when $\Gamma^{\prime}$ is non empty, the integer $m=\min\{k\,;\alpha_{k}\in\Pi_{\Gamma^{\prime}}\}>1$, and as $\Pi_{\Gamma^{\prime}}\subset\Pi(m)=\{\alpha_{m},\dots,\alpha_{\ell}\}$, we may apply the induction hypothesis on the non empty antichain $\Gamma^{\prime}$ in $\Delta_{\Pi(m)}$.

$\triangleright$ Case 1 : $\Delta$ is of type $A_{\ell}$, $B_{\ell}$ or $C_{\ell}$.

We have the following properties for these irreducible root systems.

• (P1)

For $1\leqslant i<j\leqslant\ell$, let $\alpha_{i,j}=\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j}$. Non simple positive roots of $\Delta$ are of the form :

Type $A_{\ell}$ : $\alpha_{i,j}$, where $1\leqslant i<j\leqslant\ell$.

Type $B_{\ell}$ : $\alpha_{i,j}$ or $\alpha_{i,\ell}+\alpha_{j,\ell}$, where $1\leqslant i<j\leqslant\ell$.

Type $C_{\ell}$ : $\alpha_{i,j}$ or $\alpha_{i,\ell}+\alpha_{j-1,\ell-1}$, where $1\leqslant i<j\leqslant\ell$.

In particular, for $\alpha\in\Delta^{+}$, $\Pi_{\alpha}$ consists of simple roots with consecutive indices, and we have either $\alpha=\sum_{\beta\in\Pi_{\alpha}}\beta$ or $\alpha_{\ell}\in\Pi_{\alpha}$.

We obtain by (P1) that $\Delta_{1}^{+}$ is totally ordered with respect to $\prec$ in these cases. So $\Gamma_{1}=\{\gamma\}$. By induction, there exists $(H^{\prime},n^{\prime})\in\mathfrak{h}_{\Pi(m)}\times\mathbb{N}^{*}$ verifying the required conditions for $\Gamma^{\prime}$ as an antichain in $\Delta_{\Pi(m)}$.

Let us fix $\gamma^{\prime}\in\Gamma^{\prime}$ such that $\alpha_{m}\in\Pi_{\gamma^{\prime}}$, and write $\gamma=\beta+\delta$ where $\beta=\sum_{i=1}^{m-1}c_{i}\alpha_{i}$ and $\delta=\sum_{i=m}^{\ell}c_{i}\alpha_{i}$. Since $\gamma$ and $\gamma^{\prime}$ are non comparable, we check using (P1) that $\gamma^{\prime}-\delta$ is a non zero sum of simple roots in $\Pi(m)$. So $0\leqslant\delta(H^{\prime})<\gamma^{\prime}(H^{\prime})=n^{\prime}$. Set

[TABLE]

Since $\alpha_{1}\in\Pi_{\gamma_{1}}$, we have $0pt_{\Pi}(\beta)\in\mathbb{N}^{*}$, and therefore $(H,0pt_{\Pi}(\beta)n^{\prime})$ verifies the required conditions.

$\triangleright$ Case 2 : $\Delta$ is of type $D_{\ell}$, $\ell\geqslant 4$.

For $1\leqslant i<j\leqslant\ell-2$, let $\alpha_{i,j}=\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j}$.

• (P2)

Non simple positive roots in $\Delta$ are of the form

[TABLE]

where $1\leqslant i<j\leqslant\ell-2$ and $1\leqslant k\leqslant\ell-2$. In particular, if $\alpha=\sum_{i=1}^{\ell}c_{i}\alpha_{i}$ is a positive root, then $c_{i}\in\{0,1,2\}$ for all $i$, and if $c_{i}=2$, then $2\leqslant i\leqslant\ell-2$ and $c_{i-1}=c_{\ell-1}=c_{\ell}=1$. Also, if $c_{\ell-1}=c_{\ell}=1$, then $c_{\ell-2}\neq 0$.

The set $\Delta_{1}^{+}$ is not totally ordered with respect to $\prec$. However, only $\alpha_{1,\ell-2}+\alpha_{\ell-1}$ and $\alpha_{1,\ell-2}+\alpha_{\ell}$ are non comparable in $\Delta_{1}^{+}$. So $\sharp\Gamma_{1}\leqslant 2$. This implies that $\Gamma^{\prime}$ is non empty because $\sharp\Gamma\geqslant 3$.

We shall adapt the arguments in Case 1 according to $\sharp\Gamma_{1}$.

$\circ$ Subcase (i) : $\sharp\Gamma_{1}=1$.

Let $\gamma^{\prime}$, $\beta$, $\delta$ be as in Case 1. If $\sharp(\Pi_{\gamma}\cap\{\alpha_{\ell-1},\alpha_{\ell}\})\neq 1$, then by (P2), $\gamma^{\prime}-\delta$ is a non zero sum of simple roots of $\Pi(m)$. So we may apply the same arguments as in Case 1 to obtain $(H,n)$ verifying the required conditions.

Suppose that $\sharp(\Pi_{\gamma}\cap\{\alpha_{\ell-1},\alpha_{\ell}\})=1$. Then either $\gamma=\alpha_{1,\ell-2}+\alpha_{\ell-1}$ or $\gamma=\alpha_{1,\ell-2}+\alpha_{\ell}$. By symmetry of the Dynkin diagram of $\Delta$, we may assume that $\gamma=\alpha_{1,\ell-2}+\alpha_{\ell-1}$.

Since $\gamma$ and $\gamma^{\prime}$ are non comparable, (P2) implies that $\alpha_{\ell}\in\Pi_{\gamma^{\prime}}$. If $\alpha_{\ell-1}\in\Pi_{\gamma^{\prime}}$, then again by (P2), $\gamma^{\prime}-\delta$ is a non zero sum of simple roots in $\Pi(m)$. So we may apply the same arguments as in Case 1 to obtain $(H,n)$ verifying the required conditions.

Suppose now that $\alpha_{\ell-1}\not\in\Pi_{\gamma^{\prime}}$. This implies that $\gamma^{\prime}$ is either $\alpha_{\ell}$ or $\alpha_{m,\ell-2}+\alpha_{\ell}$. Since $\sharp\Gamma\geqslant 3$, the set $\Gamma^{\prime\prime}=\Gamma\setminus\{\gamma,\gamma^{\prime}\}$ is non empty. By (P2) and the fact that $\Gamma$ is an antichain, for any $\theta\in\Gamma^{\prime\prime}$, we have $\alpha_{\ell-1},\alpha_{\ell}\in\Pi_{\theta}\subset\{\alpha_{m+1},\dots,\alpha_{\ell}\}$. Let $m^{\prime}$ be minimal such that $\alpha_{m^{\prime}}\in\Pi_{\Gamma^{\prime\prime}}$. Then $m<m^{\prime}\leqslant\ell-2$. Fix $\gamma^{\prime\prime}\in\Gamma^{\prime\prime}$ such that $\alpha_{m^{\prime}}\in\Pi_{\gamma^{\prime\prime}}$.

By induction, there exist $H^{\prime\prime}=\sum_{i=m^{\prime}}^{\ell}b_{i}H_{i}\in\mathfrak{h}_{\Pi(m^{\prime})}$ and $n^{\prime\prime}\in\mathbb{N}^{*}$ such that $(H^{\prime\prime},n^{\prime\prime})$ verifies the required conditions for $\Gamma^{\prime\prime}$ as an antichain in $\Delta_{\Pi(m^{\prime})}$. Since $\alpha_{\ell}\in\Pi_{\theta}$ for any $\theta\in\Gamma^{\prime\prime}$, $(H^{\prime\prime}+kH_{\ell},n^{\prime\prime}+k)$ verifies also the required conditions for any $k\in\mathbb{N}$. Thus we may assume that $b_{\ell}>b_{\ell-1}$.

We have $\gamma^{\prime}=\beta^{\prime}+\delta^{\prime}$ where $\beta^{\prime}=\alpha_{m,m^{\prime}-1}$ and $\delta^{\prime}=\alpha_{m^{\prime},\ell-2}+\alpha_{\ell}$. By (P2), $\gamma^{\prime\prime}-\delta^{\prime}$ is a non zero sum of simple roots in $\Pi(m^{\prime})$. It follows that $0\leqslant\delta^{\prime}(H^{\prime\prime})<\gamma^{\prime\prime}(H^{\prime\prime})=n^{\prime\prime}$. Set

[TABLE]

Since $m^{\prime}>m$, $(H^{\prime},(m^{\prime}-m)n^{\prime\prime})$ verifies the required conditions for $\Gamma^{\prime}$ as an antichain in $\Delta(m)$.

Now $0\leqslant\delta(H^{\prime})=(m^{\prime}-m)(n^{\prime\prime}-\delta^{\prime}(H^{\prime\prime})+\delta(H^{\prime\prime}))=(m^{\prime}-m)(n^{\prime\prime}-b_{\ell}+b_{\ell-1})$. Thus $\delta(H^{\prime})<(m^{\prime}-m)n^{\prime\prime}$ because $b_{\ell}>b_{\ell-1}$. Consequently, by setting

[TABLE]

the pair \bigl{(}H,(m-1)(m^{\prime}-m)n^{\prime\prime}\bigr{)} verifies the required conditions.

$\circ$ Subcase (ii) : $\sharp\Gamma_{1}=2$.

In this case, $\Gamma_{1}=\{\alpha_{1,\ell-2}+\alpha_{\ell-1},\alpha_{1,\ell-2}+\alpha_{\ell}\}$. The fact that $\Gamma$ is an antichain implies that $\alpha_{\ell-1},\alpha_{\ell}\in\Pi_{\theta}$ for any $\theta\in\Gamma^{\prime}$. Let us fix $\gamma^{\prime}\in\Gamma^{\prime}$ such that $\alpha_{m}\in\Pi_{\gamma^{\prime}}$.

By induction, there exist $H^{\prime}=\sum_{i=m}^{\ell}b_{i}H_{i}\in\mathfrak{h}_{\Pi(m)}$ and $n^{\prime}\in\mathbb{N}^{*}$ such that $(H^{\prime},n^{\prime})$ verifies the required conditions for $\Gamma^{\prime}$ as an antichain in $\Delta_{\Pi(m)}$. Set

[TABLE]

Since $\alpha_{\ell-1},\alpha_{\ell}\in\Pi_{\beta}$ for any $\beta\in\Gamma^{\prime}$, it follows from (P2) that $(H^{\prime\prime},2n^{\prime})$ verifies also the required conditions for $\Gamma^{\prime}$ as an antichain in $\Delta_{\Pi(m)}$.

We have $\alpha_{1,\ell-2}+\alpha_{\ell-1}=\alpha_{1,m-1}+\delta_{1}$ and $\alpha_{1,\ell-2}+\alpha_{\ell}=\alpha_{1,m-1}+\delta_{2}$ where $\delta_{1}=\alpha_{m,\ell-2}+\alpha_{\ell-1}$ and $\delta_{2}=\alpha_{m,\ell-2}+\alpha_{\ell}$. As in Subcase (i), we check readily that $\gamma^{\prime}-\delta_{1}$ and $\gamma^{\prime}-\delta_{2}$ are non zero sums of simple roots in $\Pi(m)$. It follows that $0\leqslant\delta_{1}(H^{\prime\prime})=\delta_{2}(H^{\prime\prime})<\gamma^{\prime}(H^{\prime\prime})=2n^{\prime}$. Set

[TABLE]

Then $(H,2(m-1)n^{\prime})$ verifies the required conditions.

$\triangleright$ Case 3 : $\Delta$ is of exceptional type.

The case $\Delta$ is of type $G_{2}$ is void since $\sharp\Gamma\geqslant 3$. For the other exceptional types, $\sharp\Delta_{1}^{+}$ is $16,33,78,15$ respectively for $E_{6}$, $E_{7}$, $E_{8}$ and $F_{4}$, the maximal cardinality of $\Gamma_{1}$ is $2,3,5,2$ and the number of possibilities for $\Gamma_{1}$ is $26,119,1348,22$. So there are too many cases to consider for $\Gamma_{1}$ to try to adapt the arguments in case 1.

Observe that if $\Gamma$ contains a simple root $\alpha_{i}$, then $\Pi_{\Gamma\setminus\{\alpha_{i}\}}=\Pi\setminus\{\alpha_{i}\}$ because $\Pi_{\Gamma}=\Pi$. By induction, there exists $(H^{\prime},n^{\prime})\in\mathfrak{h}_{\Pi\setminus\{\alpha_{i}\}}\times\mathbb{N}^{*}$ verifying the required conditions for $\Gamma\setminus\{\alpha_{i}\}$ as an antichain in $\Delta_{\Pi\setminus\{\alpha_{i}\}}$. So $(H^{\prime}+n^{\prime}H_{i},n^{\prime})$ verifies the required conditions. Thus we are left to prove the result for $\Gamma$ verifying $\sharp\Gamma\geqslant 3$, $\Pi_{\Gamma}=\Pi$, $\Gamma\cap\Pi=\emptyset$ and $\Gamma$ is maximal by inclusion. The number of such $\Gamma$ is $91$ for $E_{6}$, $512$ for $E_{7}$, $3289$ for $E_{8}$ and $10$ for $F_{4}$.

Computations were carried out using Gap4 to obtain in all these cases $(H,n)$ verifying the required conditions. The table below gives $(H,n)$ for the $10$ cases in $F_{4}$.

[TABLE]

Note that the values $n$ obtained in our computations are minimal. For the other types, the largest $n$ found is $9$ for $E_{6}$, $14$ for $E_{7}$ and $25$ for $E_{8}$. ∎

Remarks 4.2*.*

1. (1)

In the proof of Theorem 4.1, we have $\Pi_{\Gamma^{\prime}}=\Pi(m)$ in Case 1 and most cases in Case 2. 2. (2)

Our proof relies on a type by type analysis of positive roots. As this is a geometrical property, it would be nice to have a more geometric proof.

Observe that the natural $\mathbb{C}^{*}$-action on $\mathfrak{n}$ commutes with the action of $\mathbf{B}$. We have therefore an action of $\widehat{\mathbf{B}}=\mathbf{B}\times\mathbb{C}^{*}$ (resp. $\widehat{\mathbf{T}}=\mathbf{T}\times\mathbb{C}^{*}$) on $\mathfrak{n}$ given by $(\sigma,\lambda)(X)=\sigma(\lambda X)=\lambda\sigma(X)$ for $(\sigma,\lambda)\in\widehat{\mathbf{B}}$ and $X\in\mathfrak{n}$. For any group $\mathbf{H}$ acting on $\mathfrak{n}$ and $X\in\mathfrak{n}$, we shall denote by $\Omega_{\mathbf{H}}(X)$ the $\mathbf{H}$-orbit of $X$.

Corollary 4.3**.**

Let $\mathfrak{m}$ be a nilpotent ideal of $\mathfrak{b}$ and $X\in\mathfrak{m}$ be such that $\mathop{\mathrm{supp}}\nolimits(X)=\mathcal{I}_{\mathfrak{m}}$.

1. (1)

The $\mathbf{B}$-orbit $\Omega_{\mathbf{B}}(X)$ is conical. 2. (2)

For any $Y\in\mathfrak{m}^{\bullet}$, we have $X\in\overline{\Omega_{\widehat{\mathbf{T}}}(Y)}$. 3. (3)

$\Omega_{\mathbf{B}}(X)$* is the unique closed $\mathbf{B}$-orbit of $\mathfrak{m}^{\bullet}$.*

Proof.

1. By Theorem 2.3, the set $\mathcal{I}_{\mathfrak{m}}$ is linearly independent. Therefore

[TABLE]

Hence $\Omega_{\mathbb{C}^{*}}(X)\subset\Omega_{\mathbf{T}}(X)$, which in turn implies that $\Omega_{\widehat{\mathbf{B}}}(X)=\Omega_{\mathbf{B}}(X)$.

2. By Theorem 4.1, $\mathcal{I}_{\mathfrak{m}}$ is exactly the set of elements in $\mathop{\mathrm{supp}}\nolimits(Y)$ lying on a face of the convex hull of $\mathop{\mathrm{supp}}\nolimits(Y)$. We obtain from [2, Proposition 2.18] that $\overline{\Omega_{\widehat{\mathbf{T}}}(Y)}$ contains an element in $\mathfrak{m}$ whose support is $\mathcal{I}_{\mathfrak{m}}$. So the result follows by (4.1).

3. By Corollary 3.3, it suffices to show that $\Omega_{\mathbf{B}}(X)$ is closed in $\mathfrak{m}^{\bullet}$. Now points 1 and 2 say that for any $Y\in\mathfrak{m}^{\bullet}$, we have $\Omega_{\mathbf{B}}(X)=\Omega_{\widehat{\mathbf{B}}}(X)\subset\overline{\Omega_{\widehat{\mathbf{B}}}(Y)}.$ It follows that if $Y\not\in\Omega_{\mathbf{B}}(X)$, then

[TABLE]

Thus $\Omega_{\mathbf{B}}(X)$ is a $\mathbf{B}$-orbit of minimal dimension in $\mathfrak{m}^{\bullet}$, so it is closed. ∎

Remarks 4.4*.*

1. (1)

It follows immediately from Corollary 4.3 that an element $X\in\mathfrak{b}$ is nilpotent if and only if $0\in\overline{\Omega_{\mathbf{B}}(X)}$. 2. (2)

$\mathbf{B}$-orbits of nilpotent elements in $\mathfrak{b}$ are not in general conical. Here are examples of non conical $\mathbf{B}$-orbits which are minimal with respect to the rank of a given type where we use the numbering of simple roots in [12].

• $A_{6}$ :

$X_{\alpha_{1}}+X_{\alpha_{3}}+X_{\alpha_{4}}+X_{\alpha_{6}}+X_{\alpha_{1}+\alpha_{2}}+X_{\alpha_{5}+\alpha_{6}}+X_{\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}}$

• $B_{4}$ :

$X_{\alpha_{2}}+X_{\alpha_{4}}+X_{\alpha_{1}+\alpha_{2}}+X_{\alpha_{2}+\alpha_{3}}+X_{\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}}$

• $C_{4}$ :

$X_{\alpha_{1}}+X_{\alpha_{3}}+X_{\alpha_{4}}+X_{\alpha_{1}+\alpha_{2}}+X_{2\alpha_{2}+2\alpha_{3}+\alpha_{4}}$

• $D_{5}$ :

$X_{\alpha_{2}}+X_{\alpha_{4}}+X_{\alpha_{5}}+X_{\alpha_{1}+\alpha_{2}}+X_{\alpha_{2}+\alpha_{3}}+X_{\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4}+\alpha_{5}}$

• $E_{6}$ :

$X_{\alpha_{1}}+X_{\alpha_{2}}+X_{\alpha_{3}}+X_{\alpha_{5}}+X_{\alpha_{6}}+X_{\alpha_{2}+\alpha_{4}}+X_{\alpha_{1}+\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}}$

• $F_{4}$ :

$X_{\alpha_{1}}+X_{\alpha_{3}}+X_{\alpha_{4}}+X_{\alpha_{1}+\alpha_{2}}+X_{\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}}$