2·1. Satake diagrams

It is known that (for $\Bbbk={\mathbb C}$ ) the real forms of ${\mathfrak g}$ are represented by the Satake diagrams (see e.g. [ Reference Gorbatsevich, Onishchik and Vinberg22 , chapter 4, section 4·3]) and there is a one-to-one correspondence between the real forms and ${\mathbb Z}_2$ -gradings of ${\mathfrak g}$ . Thereby, one associates the Satake diagram to an involution (symmetric pair), cf. [ Reference Springer20 ]. The Satake diagram of $\sigma\in\textsf{Inv}({\mathfrak g})$ , denoted $\textsf{Sat}(\sigma)$ , is the Dynkin diagram of ${\mathfrak g}$ , with black and white nodes, where certain pairs of white nodes can be joined by an arrow. Following [ Reference De Concini and Procesi6 , section 1], we give an approach to constructing $\textsf{Sat}(\sigma)$ that does not refer to ${\mathbb C}$ and real forms. Let ${\mathfrak t}={\mathfrak t}_0\oplus{\mathfrak t}_1$ be a $\sigma$ -stable Cartan subalgebra of ${\mathfrak g}$ such that $\dim{\mathfrak t}_1$ is maximal, i.e., ${\mathfrak t}_1$ is a Cartan subspace of ${\mathfrak g}_1$ . Let $\Delta$ be the root system of $({\mathfrak g},{\mathfrak t})$ . Since ${\mathfrak t}$ is $\sigma$ -stable, $\sigma$ acts on $\Delta$ . One can choose the set of positive roots, $\Delta^+$ , such that if $\gamma\in\Delta^+$ and $\gamma\vert_{{\mathfrak t}_1}\ne 0$ , then $\sigma(\gamma)\in -\Delta^+$ . Let $\Pi$ be the set of simple roots in $\Delta^+$ and $\{\varpi_\alpha\mid \alpha\in\Pi\}$ the corresponding fundamental weights. We identify $\Pi$ with the nodes of the Dynkin diagram. Set $\Pi_0=\Pi_{0,\sigma}=\{\alpha\in \Pi\mid \alpha\vert_{{\mathfrak t}_1}\equiv 0\}$ and $\Pi_1=\Pi_{1,\sigma}=\Pi\setminus\Pi_0$ . Then:

1. (i) the node $\alpha$ is black if and only if $\alpha\in\Pi_0$ ;

2. (ii) if $\alpha\in\Pi_0$ , then $\sigma(\alpha)=\alpha$ and the root space ${\mathfrak g}_\alpha$ is contained in ${\mathfrak g}_0$ ;

3. (iii) if $\alpha\in\Pi_1$ , then $\sigma(\alpha)=-\beta+\sum_j a_j\nu_j$ for some $\beta\in\Pi_1$ and $\nu_j\in\Pi_0$ . Then also $\sigma(\varpi_\alpha)=-\varpi_\beta$ . In this case, if $\alpha\ne\beta$ , then the white nodes $\alpha$ and $\beta$ are joined by an arrow. If $\alpha=\beta$ , then there is no arrow attached to $\alpha$ .

If $\textsf{Sat}(\sigma)$ has k arrows, i.e., $\Pi_1$ has k pairs of simple roots $(\alpha_i,\beta_i)$ such that $\sigma(\varpi_{\alpha_i})=-\varpi_{\beta_i}$ , then $\dim{\mathfrak t}_0=\# \Pi_0+k$ and $\dim{\mathfrak t}_1=\#\Pi_1-k$ . More precisely, let $\{h_\alpha\mid \alpha\in\Pi\}\subset {\mathfrak t}$ be Chevalley generators, i.e., $[h_\alpha,e_\gamma]=(\alpha^\vee,\gamma)e_\gamma$ for $\gamma\in\Delta$ . Then

\begin{align*} {\mathfrak t}_0=\langle h_\alpha\mid \alpha\in\Pi_0\rangle \oplus \langle h_{\alpha_i}-h_{\beta_i} \mid i=1,\dots,k\rangle .\end{align*}

If $\Pi_1=\{\alpha_1,\beta_1,\dots,\alpha_k,\beta_k,\nu_1,\dots,\nu_s\}$ and we identify ${\mathfrak t}_1$ and ${\mathfrak t}_1^*$ via the Killing form, then ${\mathfrak t}_1=\langle \varpi_{\alpha_i}+\varpi_{\beta_i}\mid i=1,\dots,k\rangle\oplus\langle \varpi_{\nu_j}\mid j=1,\dots,s\rangle$ . Moreover, if ${\mathfrak b}$ is the Borel subalgebra corresponding to $\Delta^+$ , then ${\mathfrak b}+{\mathfrak g}_0={\mathfrak g}$ and ${\mathfrak b}\cap{\mathfrak g}_1={\mathfrak t}_1$ .

Let $x\in{\mathfrak t}_1$ be a generic semisimple element. Then ${\mathfrak g}^x_1={\mathfrak t}_1$ and $\textsf{Sat}(\sigma)$ encodes the structure of ${\mathfrak g}^x_0$ . Namely, ${\mathfrak g}^x$ is a Levi subalgebra such that the Dynkin diagram of $[{\mathfrak g}^x,{\mathfrak g}^x]=[{\mathfrak g}^x_0,{\mathfrak g}^x_0]$ is the subdiagram $\Pi_0$ . Here the number of arrows equals $\dim({\mathfrak g}^x_0/[{\mathfrak g}^x_0,{\mathfrak g}^x_0])$ .

Recall that an algebraic subgroup $H\subset G$ is said to be spherical, if B has an open orbit in $G/H$ . Then one also says that $G/H$ is a spherical homogeneous space and ${\mathfrak h}={\textrm{Lie }} H$ is a spherical subalgebra of ${\mathfrak g}$ . The above relation ${\mathfrak b}+{\mathfrak g}_0={\mathfrak g}$ exhibits the well-known fact that $G_0\subset G$ is spherical for any $\sigma$ .

3·1. Secant varieties

Let $\boldsymbol{{Sec}}(X)$ denote the secant variety of an irreducible projective variety $X\subset {\mathbb P}^N={\mathbb P}(V)$ . By definition, $\boldsymbol{{Sec}}(X)$ is the closure of the union of all secant lines to X. If $x,y\in X$ are different points and ${\mathbb P}^1_{xy}\subset {\mathbb P}(V)$ is the line through x and y, then

\begin{align*} \boldsymbol{{Sec}}(X)=\overline{\bigcup_{x,y\in X} {\mathbb P}^1_{x,y}} ,\end{align*}

see e.g. [ Reference Landsberg14 , chapter 5]. It follows that $\boldsymbol{{Sec}}(X)$ contains all tangent lines to X, too. If X is irreducible, then so is $\boldsymbol{{Sec}}(X)$ , and the expected dimension of $\boldsymbol{{Sec}}(X)$ is $\min\{2\dim X+1, N\}$ . If $\dim\boldsymbol{{Sec}}(X) < \min\{2\dim X+1, N\}$ , then $\boldsymbol{{Sec}}(X)$ is said to be degenerate. We say that $\delta=\delta_X=2\dim X+1-\dim\boldsymbol{{Sec}}(X)$ is the defect of $\boldsymbol{{Sec}}(X)$ or the secant defect of X.

Our primary goal is to use these notions for the projectivisation of minimal orbits, i.e., if $X={\mathbb P}({\mathcal O}_{\textsf{min}}(\unicode{x1D4B1}_\lambda))=G{\cdot}[v_\lambda]\subset {\mathbb P}(\unicode{x1D4B1}_\lambda)$ , where $[v_\lambda]$ is the image of $v_\lambda$ in ${\mathbb P}(\unicode{x1D4B1}_\lambda)$ .

Notation: if $v\in \unicode{x1D4B1}_\lambda$ , then [v] is a point in ${\mathbb P}(\unicode{x1D4B1}_\lambda)$ , whereas $\langle v\rangle$ is a 1-dimensional subspace of $\unicode{x1D4B1}_\lambda$ . More generally, $\langle v_1,\dots, v_k\rangle$ is the linear span of $v_1,\dots,v_k\in\unicode{x1D4B1}_\lambda$ .

For a projective variety $X\subset {\mathbb P}(V)$ , let $\widehat X\subset V$ denote the closed cone over X. That is, if $\pi\,:\,V\setminus\{0\}\to {\mathbb P}(V)$ is the canonical map, then $\widehat X=\pi^{-1}(X)\cup\{0\}$ . Then $\widehat {\boldsymbol{{Sec}}(X)}$ is called the conical secant variety of $\widehat X$ . We also write $\boldsymbol{{CS}}(\widehat X)$ or $\boldsymbol{{CS}}(\pi^{-1}(X))$ for this affine version of the secant variety. The formula for the defect of $\boldsymbol{{Sec}}(X)$ in terms of affine varieties reads

(3·2) \begin{equation} \delta_X=2\dim\widehat X- \dim \boldsymbol{{CS}}(\widehat X) .\end{equation}

If $Y\subset \unicode{x1D4B1}_\lambda$ is a subvariety, then we set $\unicode{x1D4A6}(Y)=\overline{\Bbbk^*{\cdot} Y}\subset \unicode{x1D4B1}_\lambda$ . By definition, $\unicode{x1D4A6}(Y)$ is a closed cone in $\unicode{x1D4B1}_\lambda$ . Here $\dim\unicode{x1D4A6}(Y)=\dim Y$ if and only if Y is already conical. Otherwise, $\dim\unicode{x1D4A6}(Y)=\dim Y+1$ . Clearly, if Y is G-stable, then so is $\unicode{x1D4A6}(Y)$ .

The conical secant variety of a minimal orbit has a simple explicit description. Let $\mu$ be the lowest weight of $\unicode{x1D4B1}_\lambda$ and $v_\mu$ a lowest weight vector. If $B^-$ is the Borel subgroup opposite to B (i.e., $B\cap B^-=T$ ), then $v_\mu$ is just a $B^-$ -eigenvector. If $w_0$ is the longest element of the Weyl group $W=N_G(T)/T$ , then $\mu=w_0(\lambda)$ and $\lambda^*\;:\!=\;-w_0(\lambda)$ is the highest weight of $(\unicode{x1D4B1}_\lambda)^*$ [ Reference Gorbatsevich, Onishchik and Vinberg22 , chapter 3, section 2·7]. Hence $\lambda=-\mu$ if and only if $\unicode{x1D4B1}_\lambda$ is a self-dual G-module.

The following assertion is not really new. Part (i) is implicit in [ Reference Chaput4 ] and part (ii) is stated in [ Reference Kaji9 , p.536]. However, to the best of my knowledge, an accurate proof is not easy to locate in the literature. For this reason, we have chosen to give a complete proof, which, incidentally, does not invoke Terracini’s lemma.

Proof. (i) It is clear that ${\mathbb P}(\unicode{x1D4A6}(G{\cdot}h_{\lambda}))=\overline{G{\cdot}[h_{\lambda}]}$ .

Note that $[h_\lambda]\in {\mathbb P}^1_{[v_\mu],[v_{\lambda}]}$ , hence $[h_\lambda]\in \boldsymbol{{Sec}}(G{\cdot}[v_\lambda])$ . Moreover, since $\lambda\ne\mu$ , the T-orbit of $[h_\lambda]$ is dense in ${\mathbb P}^1_{[v_\mu],[v_{\lambda}]}$ . Therefore, in order to prove that $G{\cdot}[h_{\lambda}]$ is dense in $\boldsymbol{{Sec}}(G{\cdot}[v_\lambda])$ , it suffices to show that a generic secant line ${\mathbb P}^1_{x,y}$ is G-conjugate to ${\mathbb P}^1_{[v_\mu],[v_{\lambda}]}$ .

Indeed, because $G{\cdot}[v_\lambda]$ is homogeneous, any secant line ${\mathbb P}^1_{x,y}$ is G-conjugate to a line of the form ${\mathbb P}^1_{x', [v_\lambda]}$ . Next, by the Bruhat decomposition, $C_{w_0}\;:\!=\;Bw_0 U$ is a dense open subset of G (the big cell in G). Hence $C_{w_0}{\cdot}[v_\lambda]=B{\cdot}[v_\mu]$ is dense in $G{\cdot}[v_\lambda]$ . Since the isotropy group of $[v_\lambda]$ contains B, we see that if $x'\in B{\cdot}[v_\mu]$ , then the secant line ${\mathbb P}^1_{x', [v_\lambda]}$ is B-conjugate to ${\mathbb P}^1_{[v_\mu], [v_\lambda]}$ , as required.

(ii) To compute the defect, we have to compare $\dim G{\cdot}[h_\lambda]$ and $\dim G{\cdot}[v_\lambda]$ . To this end, we use the triangular decomposition ${\mathfrak g}={\mathfrak u}^-\oplus{\mathfrak t}\oplus{\mathfrak u}$ , where ${\mathfrak b}={\mathfrak t}\oplus{\mathfrak u}$ and ${\mathfrak b}^-={\mathfrak t}\oplus{\mathfrak u}^-$ . One has ${\mathfrak u}{\cdot}v_\lambda=0$ and ${\mathfrak u}^-{\cdot}v_\mu=0$ . Hence

\begin{align*} {\mathfrak g}{\cdot}h_\lambda={\mathfrak u}^-{\cdot}v_\lambda + {\mathfrak u}{\cdot}v_\mu + {\mathfrak t}{\cdot}h_\lambda .\end{align*}

Here $\dim{\mathfrak u}^-{\cdot}v_\lambda=\dim{\mathfrak u}{\cdot}v_\mu=(\dim G{\cdot}v_\lambda)-1=\dim G{\cdot}[v_\lambda]$ . Set $p=\dim({\mathfrak u}{\cdot}v_{\mu}\cap {\mathfrak u}^-{\cdot}v_\lambda)$ . Then $\dim({\mathfrak u}^-{\cdot}v_\lambda + {\mathfrak u}{\cdot}v_\mu)=2\dim G{\cdot}[v_\lambda] -p$ . If $t\in{\mathfrak t}$ , then $t{\cdot}h_\lambda=\lambda(t)v_\lambda+ \mu(t)v_{\mu}$ . Hence

(3·3) \begin{equation} {\mathfrak t}{\cdot}h_\lambda=\begin{cases} \langle v_\lambda,v_\mu\rangle, & \text{ if } \lambda \ne -\mu, \\[5pt] \langle v_\lambda-v_\mu\rangle, & \text{ if } \lambda = -\mu .\end{cases}\end{equation}

(I) Suppose that $v_\mu\not\in {\mathfrak u}^-{\cdot}v_\lambda$ . Since ${\mathfrak g}{\cdot}v_\lambda={\mathfrak b}^-{\cdot}v_\lambda=\langle v_\lambda\rangle\oplus {\mathfrak u}^-{\cdot}v_\lambda$ , we also have $v_\mu\not\in {\mathfrak g}{\cdot}v_\lambda$ . (By symmetry, this also means that $v_\lambda\not\in {\mathfrak u}{\cdot}v_\mu$ , etc.) Then

(3·4) \begin{equation} {\mathfrak g}{\cdot}h_\lambda=({\mathfrak u}^-{\cdot}v_\lambda + {\mathfrak u}{\cdot}v_\mu) \oplus {\mathfrak t}{\cdot}h_\lambda\end{equation}

and ${\mathfrak g}{\cdot}v_{\mu}\cap {\mathfrak g}{\cdot}v_\lambda={\mathfrak b}{\cdot}v_{\mu}\cap {\mathfrak b}^-{\cdot}v_\lambda={\mathfrak u}{\cdot}v_{\mu}\cap {\mathfrak u}^-{\cdot}v_\lambda$ . Here one has two possibilities.

(a) if $\lambda\ne-\mu$ , then it follows from (3·3) and (3·4) that $\dim{\mathfrak g}{\cdot}h_\lambda=2\dim G{\cdot}[v_\lambda] -p +2$ . Moreover, here $h_\lambda\in{\mathfrak t}{\cdot}h_\lambda\subset {\mathfrak g}{\cdot}h_\lambda$ . Hence the orbit $G{\cdot}h_\lambda$ is conical and

\begin{equation*} \dim{\mathfrak g}{\cdot}[h_\lambda]=\dim{\mathfrak g}{\cdot}h_\lambda-1=2\dim G{\cdot}[v_\lambda] +1-p \end{equation*}

Thus, here $p=\dim({\mathfrak g}{\cdot}v_{\mu}\cap {\mathfrak g}{\cdot}v_\lambda)$ is the defect of $\boldsymbol{{Sec}}(G{\cdot}[v_\lambda])$ .

(b)if $\lambda=-\mu$ , then $\dim{\mathfrak g}{\cdot}h_\lambda=2\dim G{\cdot}[v_\lambda] -p +1$ . Here $h_\lambda\not\in{\mathfrak t}{\cdot}h_\lambda=\langle v_\lambda-v_\mu\rangle$ . Hence the orbit $G{\cdot}h_\lambda$ is not conical and $\dim{\mathfrak g}{\cdot}[h_\lambda]=\dim{\mathfrak g}{\cdot}h_\lambda=2\dim G{\cdot}[v_\lambda] +1-p$ , with the same conclusion as in (a).

(II) Suppose that $v_\mu\in {\mathfrak u}^-{\cdot}v_\lambda$ (equivalently, $v_\lambda\in {\mathfrak u}{\cdot}v_\mu$ ). Then

\begin{align*} {\mathfrak g}{\cdot}h_\lambda= {\mathfrak u}^-{\cdot}v_\lambda + {\mathfrak u}{\cdot}v_\mu\end{align*}

and the orbit $G{\cdot}h_\lambda$ is conical. Therefore,

\begin{equation*} \dim{\mathfrak g}{\cdot}[h_\lambda]=2\dim G{\cdot}[v_\lambda] -1-p=2\dim G{\cdot}[v_\lambda]+1-(p+2) \end{equation*}

But in this case, ${\mathfrak g}{\cdot}v_{\mu}\cap {\mathfrak g}{\cdot}v_\lambda=({\mathfrak u}{\cdot}v_{\mu}\cap {\mathfrak u}^-{\cdot}v_\lambda)\oplus\langle v_\lambda,v_\mu\rangle$ and $\dim({\mathfrak g}{\cdot}v_{\mu}\cap {\mathfrak g}{\cdot}v_\lambda)=p+2$ . Thus, one again obtains the required value for the defect.

3·2. Some invariant-theoretic consequences

If a reductive group G acts on an affine variety Z, then the algebra of invariants $\Bbbk[Z]^G$ is finitely-generated and $Z/\!\!/ G\;:\!=\;{\textsf{Spec }}(\Bbbk[Z]^G)$ is the categorical quotient, see [ Reference Popov and Vinberg24 ]. The inclusion $\Bbbk[Z]^G\subset\Bbbk[Z]$ yields the morphism $\pi_Z\,:\,Z\to Z/\!\!/ G$ , which is onto. Each fibre $\pi^{-1}_Z(\xi)$ , $\xi\in Z/\!\!/ G$ , contains a unique closed orbit, hence $Z/\!\!/ G$ parametrises the closed G-orbits in Z. If $Z=\unicode{x1D4B1}$ is a G-module, then the fibre ${\mathfrak N}_G(\unicode{x1D4B1})=\pi_\unicode{x1D4B1}^{-1}(\pi_\unicode{x1D4B1}(0))$ is called the null-cone. It is also true that

\begin{align*} {\mathfrak N}_G(\unicode{x1D4B1})=\{v\in\unicode{x1D4B1} \mid \overline{G{\cdot}v}\ni 0\}.\end{align*}

Recall that $\mu=-\lambda^*$ is the lowest weight of $\unicode{x1D4B1}_\lambda$ . In the proof of Theorem 3·2, we considered the condition that $v_{\mu}\not\in{\mathfrak g}{\cdot}v_\lambda$ . Actually, if ${\mathfrak g}$ is simple, then this condition is not satisfied only for two cases.

Using Theorem 3·2(i), we can characterise the cases in which $\boldsymbol{{CS}}({\mathcal O}_{\textsf{min}}(\unicode{x1D4B1}_\lambda))= \unicode{x1D4B1}_\lambda$ .

All such representations occur in the “Summary Table” in [ Reference Popov and Vinberg24 ], where the irreducible representations of simple Lie groups with polynomial algebras of invariants are listed. If $\unicode{x1D4B1}_\lambda$ satisfies either (a) or (b), then the condition $\unicode{x1D4A6}(G{\cdot}h_\lambda)=\unicode{x1D4B1}_\lambda$ means that $h_\lambda\in\unicode{x1D4B1}_\lambda$ is a point of general position in the sense of [ Reference Elashvili7 ]. Since table 1 in [ Reference Elashvili7 ] explicitly indicates such points for all representations in question, one easily obtains the following tables, where the relevant highest weights $\lambda$ and the defect $\delta=\delta_{{\mathbb P}({\mathcal O}_{\textsf{min}}(\lambda))}$ are given. If $\unicode{x1D4B1}_\lambda$ is self-dual, then we point out whether it is orthogonal or symplectic. The numbering of simple roots follows [ Reference Gorbatsevich, Onishchik and Vinberg22 ] and we write $\varpi_i$ in place of $\varpi_{\alpha_i}$ .

Table 1.

The representations with $\boldsymbol{{CS}}({\mathcal O}_{\textsf{min}}(\unicode{x1D4B1}_\lambda))=\unicode{x1D4B1}_\lambda$ : serial cases

Remark 3·7. The simple G-modules with $\delta>0$ have been listed in [ Reference Kaji9 ]. The representations in Tables 1 and 2 that do not occur in [ Reference Kaji9 ] are those with $\delta=0$ .

Table 2.

The representations with $\boldsymbol{{CS}}({\mathcal O}_{\textsf{min}}(\unicode{x1D4B1}_\lambda))=\unicode{x1D4B1}_\lambda$ : sporadic cases

We can also provide a quick approach to the classification in [ Reference Kaji9 ], which is outlined below. For $\alpha\in\Pi$ , let $[\theta\,:\,\alpha]$ denote the coefficient of $\alpha$ in the expression of $\theta$ via $\Pi$ . Set $r_\alpha={(\theta,\theta)}/{(\alpha,\alpha)}\in \{1,2,3\}$ . In particular, $r_\alpha=1$ for all $\alpha$ in the simply-laced case.

Proof. Write $\lambda=\sum_{i=1}^l a_i\varpi_i$ and $\theta=\sum_{i=1}^l n_i\alpha_i$ , i.e., $n_i=[\theta\,:\,\alpha_i]$ . (Here $l={\textsf{rk}}\,{\mathfrak g}$ .) Then

\begin{align*} (\lambda,\theta^\vee)=\frac{2}{(\theta,\theta)}\sum_{l=1}^l a_i n_i (\varpi_i,\alpha_i)= \sum_{l=1}^l a_i n_i \frac{(\alpha_i,\alpha_i)}{(\theta,\theta)}=\sum_{l=1}^l a_i \frac{n_i}{r_i} ,\end{align*}

where $r_i=r_{\alpha_i}$ . Since $\theta$ is a long root, we have $n_i/r_i\in{\mathbb N}$ for all i. The assertion follows.

Proof. (i) If $\delta=\dim({\mathfrak g}{\cdot}v_\lambda\cap{\mathfrak g}{\cdot}v_{-\lambda^*})>0$ , then there are $\nu,\nu'\in\Delta^+\cup\{0\}$ such that $\lambda-\nu=-\lambda^*+\nu'$ . Hence $\lambda+\lambda^*=\nu+\nu'$ is a dominant weight in the root lattice. Since $\theta=\theta^*$ , we have $(\lambda,\theta^\vee)=(\lambda^*,\theta^\vee)$ is a positive integer. On the other hand, $(\nu,\theta^\vee), (\nu',\theta^\vee)\in\{0,1,2\}$ . This implies that

\begin{align*} 2(\lambda,\theta^\vee)=(\lambda+\lambda^*,\theta^\vee)=(\nu+\nu',\theta^\vee)\in \{2,4\} .\end{align*}

(ii) If $(\lambda,\theta^\vee)=2$ , then $(\nu,\theta^\vee)=(\nu',\theta^\vee)=2$ . Hence $\nu=\nu'=\theta$ and $\lambda+\lambda^*=2\theta$ . If $\theta$ is a multiple of a fundamental weight, then the only possibility is $\lambda=\lambda^*=\theta$ . For ${\mathfrak g}={\mathfrak {sl}}_{n+1}$ , one has $\theta=\varpi_1+\varpi_n$ , which yields one more possibility. In either case, $\lambda-\theta=-\lambda^*+\theta$ is the only weight occurring in ${\mathfrak g}{\cdot}v_\lambda\cap{\mathfrak g}{\cdot}v_{-\lambda^*}$ , i.e., $\delta=1$ .

If $\lambda$ is dominant and $(\lambda,\theta^\vee)=1$ , then $\delta_{G{\cdot}[v_\lambda]}>0$ if and only if $\lambda+\lambda^*-\theta\in (\Delta^+\setminus\{\theta\})\cup\{0\}$ . Moreover, in this case $\delta\geqslant 2$ .

3·3. The minimal nilpotent orbit in ${\mathfrak g}$

For a simple Lie algebra ${\mathfrak g}\simeq\unicode{x1D4B1}_\theta$ , we usually write ${\mathcal O}_{\textsf{min}}$ in place of ${\mathcal O}_{\textsf{min}}({\mathfrak g})$ and say that ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ is the minimal nilpotent orbit. If $\gamma\in\Delta$ is a long root and $e_\gamma\in{\mathfrak g}^\gamma$ , then ${\mathcal O}_{\textsf{min}}=G{\cdot}e_\gamma$ . (In the simply-laced case, all roots are assumed to be long.)

Let $\{e,h,f\}$ be an ${\mathfrak{sl}}_2$ -triple with $e\in{\mathcal O}_{\textsf{min}}$ . Such an ${\mathfrak{sl}}_2$ -triple is said to be minimal. For the minimal ${\mathfrak{sl}}_2$ -triples, the $({\mathbb Z},h)$ -grading of ${\mathfrak g}$ has the following structure:

(3·5) \begin{equation} \text{${\mathfrak g}=\textstyle \bigoplus_{j=-2}^2 {\mathfrak g}(i)$, ${\mathfrak g}(2)=\langle e\rangle$, and ${\mathfrak g}({-}2)=\langle f\rangle$.}\end{equation}

Since $\dim{\mathfrak g}^e=\dim{\mathfrak g}(0)+\dim{\mathfrak g}(1)$ and ${\mathfrak g}^h={\mathfrak g}(0)$ , we have $\dim{\mathcal O}_{\textsf{min}}=\dim{\mathfrak g}(1)+2$ and $\dim G{\cdot}h=2\dim{\mathfrak g}(1)+2$ . Hence

(3·6) \begin{equation} \dim G{\cdot}h= 2\dim{\mathcal O}_{\textsf{min}} -2 .\end{equation}

Set ${\mathfrak s}=\langle e,h,f\rangle\simeq {\mathfrak{sl}}_2$ . Let $\unicode{x1D4E1}_i$ denote a simple ${\mathfrak{sl}}_2$ -module of dimension $i+1$ . It follows from (3·5) that

\begin{align*} {\mathfrak g}\vert_{\mathfrak s}=m_0\unicode{x1D4E1}_0+m_1\unicode{x1D4E1}_1+\unicode{x1D4E1}_2 ,\end{align*}

where $m_0=\dim{\mathfrak g}(0)-1$ and $m_1=\dim{\mathfrak g}(1)$ . The G-orbit ${\mathcal O}_{\textsf{min}}$ is characterised by the property that ${\mathfrak g}\vert_{\mathfrak s}$ contains no $\unicode{x1D4E1}_j$ with $j\geqslant 3$ and $\unicode{x1D4E1}_2$ occurs only once. A formulation that does not invoke ${\mathfrak{sl}}_2$ -triples and ${\mathbb Z}$ -gradings is

(3·7) \begin{equation} \text{ $e\in{\mathcal O}_{\textsf{min}}$ if and only if ${\textsf{ Im}}(({\textrm{ad }} e)^2)=\langle e\rangle$.}\end{equation}

Hence ${\textsf{ht}}({\mathcal O}_{\textsf{min}})=2$ . However, if ${\mathfrak g}\ne{\mathfrak{sl}}_2$ or $\mathfrak{sl}_3$ , then there are other G-orbits of height 2.

Let $\theta\in \Delta^+$ be the highest root. Then $e_\theta\in {\mathfrak g}^\theta$ is a highest weight vector and $f_{\theta}\in {\mathfrak g}^{-\theta}$ is a lowest weight vector. Under a suitable adjustment, three elements $e_\theta, h_\theta=[e_\theta, f_\theta], f_\theta$ form an ${\mathfrak{sl}}_2$ -triple. Here $e_\theta +f_\theta$ is conjugate to $h_\theta$ in $\langle e_\theta, h_\theta, f_\theta\rangle\simeq {\mathfrak{sl}}_2$ and hence in ${\mathfrak g}$ . Therefore, it follows from Theorem 3·2(i) that

(3·8) \begin{equation} \boldsymbol{{Sec}}({\mathbb P}({\mathcal O}_{\textsf{min}}))=G{\cdot}[h] \ \text{ and } \boldsymbol{{CS}}({\mathcal O}_{\textsf{min}})=\unicode{x1D4A6}(G{\cdot}h) ,\end{equation}

where h is the characteristic a minimal ${\mathfrak{sl}}_2$ -triple. This recovers the main result of [ Reference Kaji, Ohno and Yasukura10 ]. Since the orbit $G{\cdot}h$ is not conical, (3·6) can be written as $\dim G{\cdot}[h]=2\dim {\mathbb P}({\mathcal O}_{\textsf{min}})$ , which again shows that here $\delta_{{\mathbb P}({\mathcal O}_{\textsf{min}})}=1$ . It is also easily seen that $[{\mathfrak g},e_\theta]\cap [{\mathfrak g},f_\theta]=\langle h_\theta\rangle$ .

5·1. The case with ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$

There are two reasons why the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is important. First, one has ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing \ \Longleftrightarrow \ G{\cdot}h \cap{\mathfrak g}_1\ne\varnothing$ , which is a special case of [ Reference Antonyan1 , theorem 1], cf. Remark 2·3. The implication “ $\Rightarrow$ ” is easy (and will be presented in the next proof), while the converse is not. Second, ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ if and only if normal minimal ${\mathfrak{sl}}_2$ -triples exist.

5·1·1. Projections to ${\mathfrak g}_1$

Theorem 5·2. Suppose that ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ . Let $\{e,h,f\}$ be a normal minimal ${\mathfrak{sl}}_2$ -triple and $h_1=e-f \in{\mathfrak g}_1$ . Then:

1. (i) $h_1 \in \unicode{x1D4A6}(G{\cdot}h)\cap{\mathfrak g}_1$ ;

2. (ii) $\overline{\psi({\mathcal O}_{\textsf{min}})}=\unicode{x1D4A6}(G_0{\cdot}h_1)$ and $\dim \psi({\mathcal O}_{\textsf{min}})=\dim{\mathcal O}_{\textsf{min}}$ .

Proof. (i) If $i=\sqrt{-1}$ , then $ih_1=\left(\begin{smallmatrix} 0 & i \\[5pt] -i& 0 \end{smallmatrix}\right)$ is conjugate to $h=\left(\begin{smallmatrix} 1 & 0 \\[5pt] 0& -1\end{smallmatrix}\right)$ in $\langle e,h,f\rangle\simeq {\mathfrak{sl}}_2$ and thereby in ${\mathfrak g}$ . Hence $i h_1\in G{\cdot}h\cap{\mathfrak g}_1$ .

(ii) Since h is semisimple, $G{\cdot}h\cap{\mathfrak g}_1$ is a sole $G_0$ -orbit [ Reference Bulois3 , Proposition 6·6], hence $G{\cdot}h\cap{\mathfrak g}_1=G_0{\cdot}ih_1$ . Using Proposition 2·1 and (3·6), we obtain

\begin{align*} \dim G_0{\cdot}h_1=\dim G_0{\cdot}ih_1=\frac{1}{2}\dim G{\cdot}h= \dim{\mathcal O}_{\textsf{min}}-1 .\end{align*}

Then $\dim\unicode{x1D4A6}(G_0{\cdot}h_1)=\dim G_0{\cdot}h_1+1=\dim{\mathcal O}_{\textsf{min}}$ . Next step is to show that $h_1\in \psi({\mathcal O}_{\textsf{min}})$ . Indeed, one has

(5·2) \begin{equation} v\;:\!=\;\exp({\textrm{ad }} f){\cdot}e=e+[f,e]+\frac{1}{2}[f,[f,e]]=e-h-f\in{\mathcal O}_{\textsf{min}}\end{equation}

and $\psi(v)=e-f=h_1$ . Since $\overline{\psi({\mathcal O}_{\textsf{min}})}$ is a cone, the whole variety $\unicode{x1D4A6}(G_0{\cdot}h_1)$ is contained in $\overline{\psi({\mathcal O}_{\textsf{min}})}$ . Because the latter is irreducible and $\dim\overline{\psi({\mathcal O}_{\textsf{min}})}\leqslant\dim{\mathcal O}_{\textsf{min}}=\dim\unicode{x1D4A6}(G_0{\cdot}h_1)$ , the result follows.

Since $\dim{\mathcal O}_{\textsf{min}}=\dim\psi({\mathcal O}_{\textsf{min}})$ in Theorem 5·2, the degree of $\psi$ , $\deg(\psi)$ , is well-defined. By definition, $\deg(\psi)=\#\psi^{-1}(z)$ for generic $z\in\overline{\psi({\mathcal O}_{\textsf{min}})}$ . Here the structure of $\overline{\psi({\mathcal O}_{\textsf{min}})}$ shows that $\deg(\psi)=\#\psi^{-1}(h_1)$ .

Theorem 5·3. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , $\{e,h,f\}$ is a normal ${\mathfrak{sl}}_2$ -triple for ${\mathcal O}_{\textsf{min}}$ , and $h_1=e-f$ , then

\begin{align*} \psi^{-1}(h_1)=\{e-h-f, e+h-f\}=\{h_1-h, h_1+h\} .\end{align*}

In particular, $\deg(\psi)=2$ .

Proof. Computations in $\langle e,h,f\rangle\simeq{\mathfrak{sl}}_2$ show that $\{e-h-f, e+h-f\}\subset \psi^{-1}(h_1)=\psi^{-1}(e-f)$ . Conversely, suppose that $z\;:\!=\;h_1+x\in \psi^{-1}(h_1)$ for some $x\in{\mathfrak g}_0$ . Then ${\textsf{Im}}({\textrm{ad }} z)^2=\langle z\rangle$ , see (3·7). In particular, $({\textrm{ad }} z)^2(e)=\zeta z$ and $({\textrm{ad }} z)^2(f)=\eta z$ for some $\zeta,\eta\in\Bbbk$ .

Equating the ${\mathfrak g}_0$ - and ${\mathfrak g}_1$ -components in both relations yields two systems of equations:

\begin{align*} \begin{cases} [f,[e,x]]-[h,x]=\zeta x \\[5pt] ({\textrm{ad }} x)^2 e-2(e+f)=\zeta h_1=\zeta e-\zeta f \end{cases} \ \text{ and } \ \begin{cases} [e,[x,f]]-[h,x]=\eta x \\[5pt] ({\textrm{ad }} x)^2 f-2(e+f)=\eta h_1=\eta e-\eta f .\end{cases}\end{align*}

For computations with the ${\mathfrak g}_0$ -components, we use the fact that $[e,[x,e]]\in {\mathfrak g}_0\cap\langle e\rangle=\{0\}$ , and likewise for f. Then we rearrange the above relations as follows:

\begin{align*} (\textrm{I}): \begin{cases} [f,[e,x]]-[h,x]=\zeta x \\[5pt] -[e,[f,x]]-[h,x]=\eta x \end{cases} \ \text{ and } \ (\textrm{II})\,:\, \begin{cases} ({\textrm{ad }} x)^2 e=(\zeta+2)e+(2-\zeta)f \\[5pt] ({\textrm{ad }} x)^2 f= (\eta+2)e+(2-\eta)f .\end{cases}\end{align*}

1. (i) System (II) means that the plane $\langle e,f\rangle$ is $({\textrm{ad }} x)^2$ -stable and

   \begin{align*} (\textrm{II}')\,:\, \begin{cases} ({\textrm{ad }} x)^2 (e-f)=(\zeta-\eta)(e-f) \\[5pt] ({\textrm{ad }} x)^2 (e+f)= (\zeta+\eta)(e-f) +4(e+f) .\end{cases}\end{align*}

2. (ii) Taking the sum in (I) and using the Jacobi identity, we obtain $-3[h,x]=(\zeta+\eta)x$ .

3. (iii) Assume that $\zeta+\eta\ne 0$ . Then x is nilpotent and hence the determinant of (II) or (II)’ must be zero, i.e., $\zeta=\eta$ . But in this case we have $({\textrm{ad }} x)^2 (e-f)=0$ and $({\textrm{ad }} x)^2 (e+f)= 2\zeta(e-f) +4(e+f)$ . Therefore, $({\textrm{ad }} x)^{2k}(e+f)\ne 0$ for all $k\in{\mathbb N}$ , i.e., ${\textrm{ad }} x$ is not nilpotent. This contradiction means that $\zeta+\eta=0$ and $[h,x]=0$ .

4. (iv) For the minimal ${\mathfrak{sl}}_2$ -triples, one has ${\mathfrak g}^h=\langle h\rangle\dotplus{\mathfrak l}$ , where ${\mathfrak l}$ is the centraliser of the whole ${\mathfrak{sl}}_2$ -triple $\{e,h,f\}$ . Therefore,

   \begin{align*} x=ah + \tilde h,\end{align*}
   where $\tilde h\in {\mathfrak l}$ and $a\in\Bbbk$ . Substituting this x into system (I) and using the relations $[\tilde h,e]=[\tilde h,f]=0$ , one readily obtains $\tilde h=0$ , $\zeta=2$ , and $\eta=-2$ .

5. (v) Thus, $z=h_1+x=e-f+ah \in {\mathcal O}_{\textsf{min}}$ . But elements of this form are nilpotent in $\langle e,h,f\rangle\simeq {\mathfrak{sl}}_2$ if and only if $a=\pm 1$ .

By Theorem 4·3, there are two cases in which $\psi$ is finite (of degree 2). A more precise assertion for them is

Proposition 5·4. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_0=\varnothing$ , then $\psi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to \psi(\overline{{\mathcal O}_{\textsf{min}}})$ is the categorical quotient by the linear action of the cyclic group $\mathcal C_2$ with generator $({-}\sigma)\in GL({\mathfrak g})$ .

Proof. Let $\pi\,:\, \overline{{\mathcal O}_{\textsf{min}}} \to \overline{{\mathcal O}_{\textsf{min}}}/\mathcal C_2$ be the quotient morphism. Since the map

\begin{align*} (x\in\overline{{\mathcal O}_{\textsf{min}}}) \longmapsto (\psi(x)=(x-\sigma(x))/2\in {\mathfrak g}_1) \end{align*}

is $\mathcal C_2$ -equivariant, we obtain the commutative diagram

where $\psi$ is onto. Our goal is to prove that $\kappa$ is an isomorphism. Since $\deg(\psi)=\deg(\pi)=2$ , the map $\kappa$ is birational. Since $\overline{{\mathcal O}_{\textsf{min}}}$ is normal [ Reference Vinberg and Popov23 ], so is $\overline{{\mathcal O}_{\textsf{min}}}/\mathcal C_2$ . Therefore, it suffices to prove that $\psi(\overline{{\mathcal O}_{\textsf{min}}})$ is a normal variety. And this follows from the description of ${\textsf{Im}}(\psi)$ in Example 4·2.

5·1·2. Projections to ${\mathfrak g}_0$

To describe $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ , we need Lemma 5·1 and the relation $\unicode{x1D4A6}(G{\cdot}h)=\boldsymbol{{CS}}({\mathcal O}_{\textsf{min}})$ . Since $G{\cdot}h$ is a closed (hence non-conical) G-orbit, it follows from (3·6) that

\begin{align*} \dim\unicode{x1D4A6}(G{\cdot}h)=\dim G{\cdot}h +1=2\dim{\mathcal O}_{\textsf{min}}-1 .\end{align*}

Theorem 5·5. Suppose that ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , and let $\{e,h,f\}$ be a normal minimal ${\mathfrak{sl}}_2$ -triple. Then $\overline{\varphi({\mathcal O}_{\textsf{min}})}=\unicode{x1D4A6}(G_0{\cdot}h)$ and $\dim\varphi({\mathcal O}_{\textsf{min}})=\dim{\mathcal O}_{\textsf{min}}-1$ .

Proof. By (5·2), we have $e-h-f\in{\mathcal O}_{\textsf{min}}$ . Hence $-h\in\varphi({\mathcal O}_{\textsf{min}})$ and $\unicode{x1D4A6}(G_0{\cdot}h)\subset\overline{\varphi({\mathcal O}_{\textsf{min}})}$ . On the other hand, $\overline{\varphi({\mathcal O}_{\textsf{min}})}\subset \unicode{x1D4A6}(G{\cdot}h)\cap{\mathfrak g}_0$ (Lemma 5·1), and since $G_0{\cdot}h$ is an irreducible component of $G{\cdot}h \cap{\mathfrak g}_0$ [ Reference Richardson19 , theorem 3·1], it is clear that $\unicode{x1D4A6}(G_0{\cdot}h)$ is an irreducible component of $\unicode{x1D4A6}(G{\cdot}h)\cap{\mathfrak g}_0$ . Hence $\overline{\varphi({\mathcal O}_{\textsf{min}})}=\unicode{x1D4A6}(G_0{\cdot}h)$ .

To compute $\dim G_0{\cdot}h$ , we use the $({\mathbb Z},h)$ -grading related to our normal minimal ${\mathfrak{sl}}_2$ -triple, see (3·5). By the assumption, we have ${\mathfrak g}(2)\oplus{\mathfrak g}({-}2)=\langle e,f\rangle \subset{\mathfrak g}_1$ . Hence ${\mathfrak g}_0\subset {\mathfrak g}(1)\oplus{\mathfrak g}(0)\oplus{\mathfrak g}({-}1)$ . Furthermore, if $v\in {\mathfrak g}(1)\cap {\mathfrak g}_0$ , then $[f,v]\subset {\mathfrak g}({-}1)\cap{\mathfrak g}_1$ , and vice versa. (Note that $[e,[f,v]]=v$ for any $v\in{\mathfrak g}(1)$ .) This implies that $\dim \Bigl({\mathfrak g}_0\cap \bigl({\mathfrak g}(1)\oplus{\mathfrak g}({-}1)\bigr)\Bigr)=\dim{\mathfrak g}(1)$ . Therefore,

\begin{align*} \dim G_0{\cdot}h=\dim{\mathfrak g}_0-\dim({\mathfrak g}_0\cap{\mathfrak g}(0))=\dim{\mathfrak g}(1)=\dim {\mathcal O}_{\textsf{min}}-2 .\end{align*}

Thus, $\dim \unicode{x1D4A6}(G_0{\cdot}h)=\dim {\mathcal O}_{\textsf{min}}-1$ , and we are done.

Since $\dim{\mathcal O}_{\textsf{min}}-\dim\varphi({\mathcal O}_{\textsf{min}})=1$ , generic fibres of $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to \unicode{x1D4A6}(G_0{\cdot}h)$ are one-dimensional. Therefore, $\dim \varphi^{-1}(h)=1$ . Here a slight modification of (5·2) provides a clue about the structure of $\varphi^{-1}(h)$ . For any $t\in\Bbbk^*$ , one has

(5·3) \begin{equation} \exp({\textrm{ad }} {t}e){\cdot}\frac{1}{t}f=\frac{1}{t}f+[e,f]+\frac{1}{2}[te,[te,\frac{1}{t}f]]=\frac{1}{t}f+h-te \in \varphi^{-1}(h) .\end{equation}

We prove below that this actually yields the whole fibre $\varphi^{-1}(h)$ and thereby generic fibres of $\varphi$ are irreducible.

Theorem 5·6. Suppose that ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ . Let $\{e,h,f\}$ be a normal minimal ${\mathfrak{sl}}_2$ -triple and $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to \overline{\varphi({\mathcal O}_{\textsf{min}})}=\unicode{x1D4A6}(G_0{\cdot}h)$ . Then $\varphi^{-1}(h)=\{\frac{1}{t}f+h-te \in \varphi^{-1}(h)\mid t\in \Bbbk^*\}$ .

Proof. The inclusion ‘ $\supset$ ’ is shown in (5·3).

Our proof for ‘ $\subset$ ’ is close in spirit to the proof of Theorem 5·3, though some details are different. Suppose that $z=h+x\in \varphi^{-1}(h)$ for some $x\in{\mathfrak g}_1$ . Then ${\textsf{Im}}({\textrm{ad }} z)^2=\langle z\rangle$ , see (3·7). In particular, $({\textrm{ad }} z)^2(e)=\zeta z$ and $({\textrm{ad }} z)^2(f)=\eta z$ for some $\zeta,\eta\in\Bbbk$ .

Equating the ${\mathfrak g}_0$ - and ${\mathfrak g}_1$ -components in both relations yields two systems of equations:

\begin{align*} \begin{cases} [h,[x,e]]+2[x,e]=\zeta h \\[5pt] ({\textrm{ad }} x)^2 e+4e=\zeta x \end{cases} \ \text{ and } \ \begin{cases} [h,[x,f]]-2[x,f]=\eta h \\[5pt] ({\textrm{ad }} x)^2 f + 4f=\eta x .\end{cases}\end{align*}

Using the Jacoby identity and certain rearrangements, we obtain:

\begin{align*} (\textrm{I})\,:\, \begin{cases} [[h,x]+4x,e]=\zeta h \\[5pt] [[h,x]-4x,f]=\eta h \end{cases} \ \text{ and } \ (\textrm{II})\,:\, \begin{cases} ({\textrm{ad }} x)^2 e=-4e+\zeta x \\[5pt] ({\textrm{ad }} x)^2 f= -4f+\eta x .\end{cases}\end{align*}

For a while, we concentrate on system (I). Using the relations for the ${\mathfrak{sl}}_2$ -triple, one derives from (I) that

\begin{align*} (\textrm{I'})\,:\, \begin{cases} [h,x]+4x=-\zeta f +a_e \\[5pt] [h,x]-4x=\eta e+b_f , \end{cases}\end{align*}

where $a_e\in {\mathfrak g}^e$ and $b_f\in{\mathfrak g}^f$ . Taking the sum and difference in (I’), we obtain

(5·4) \begin{align} 2[h,x] = \eta e-\zeta f +a_e+b_f , \end{align}

(5·5) \begin{align} 8x= -\eta e-\zeta f +a_e-b_f .\end{align}

Next, we apply ${\textrm{ad }} h$ to equality (5·5) and then substitute [h,x] from (5·4) into the equality obtained. We get

(5·6) \begin{equation} 6\eta e-6\zeta f=-4a_e-4b_f +[h,a_e]-[h,b_f] .\end{equation}

Recall that associated with the (normal) ${\mathfrak{sl}}_2$ -triple $\{e,h,f\}$ , we have the $({\mathbb Z},h)$ -grading ${\mathfrak g}=\bigoplus_{i=-2}^2{\mathfrak g}(i)$ , see (3·5). Here $a_e\in {\mathfrak g}^e\subset {\mathfrak g}(\geqslant 0)$ and $b_f\in{\mathfrak g}^f\subset{\mathfrak g}(\leqslant 0)$ , i.e., $a_e=a_0+a_1+a_2$ and $b_f=b_0+b_{-1}+b_{-2}$ , where $a_i\in{\mathfrak g}(i)$ and $b_{-j}\in {\mathfrak g}({-}j)$ .

1. (i) Comparing the ${\mathbb Z}$ -homogeneous components of (5·6) in ${\mathfrak g}(i)$ with $i=-1,0,1$ , we conclude that $a_1=b_{-1}=0$ and $a_0=-b_0\;=\!:\;2r$ . This also means that $r\in{\mathfrak z}_{\mathfrak g}(e,h,f)$ .

2. (ii) Since ${\mathfrak g}(2)=\langle e\rangle$ , we have $a_2=ce$ and then comparing the ${\mathfrak g}(2)$ -components in (5·6) implies that $c=-3\eta$ . Likewise, $b_{-2}=\tilde c f$ and then $\tilde c=3\zeta$ .

Thus, our current achievement is that $a_e=a_0+ce=2r-3\eta e$ and $b_f=b_0+\tilde cf=-2r+3\zeta f$ . Substituting these $a_e$ and $b_f$ into (5·5), we obtain

\begin{align*} x=\frac{1}{2}({-}\eta e-\zeta f +r) .\end{align*}

Let us remember that there are also equations (II). Substituting this x into, say, the first equation in (II), we obtain

\begin{align*} \zeta\eta e=-4e +\zeta r/2 .\end{align*}

Therefore, $\zeta\eta=-4$ and $r=0$ . In other words, $x=-te+{f}/{t}$ , as required.

5·2. The case with ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$

If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then $G{\cdot}h\cap{\mathfrak g}_1=\varnothing$ and normal minimal ${\mathfrak{sl}}_2$ -triples do not exist. On the other hand, we have assertions of Theorem 4·6 at our disposal, and there are only six such cases (Example 4·8).