2.1 Prehomogeneous vector spaces

We gather together some basic facts about PHVSs. The main references are [Reference KimuraKim03, Reference KnappKna02, Reference MortajineMor91, Reference Sato and KimuraSK77].

A prehomogeneous vector space for $\mathrm {G}$ is a finite-dimensional complex vector space $V$ together with a holomorphic representation $\rho :\mathrm {G}\to \mathrm {GL}(V)$ such that $V$ has an open $\mathrm {G}$-orbit. Such an open orbit is necessarily unique and dense. If $V$ is a PHVS, let $\Omega$ denote the open orbit in $V$ and $S=V\setminus \Omega$ be the singular set. For $x\in V$, denote the $\mathrm {G}$-stabilizer of $x$ by $\mathrm {G}^x.$ A PHVS vector space $V$ is called regular if $\mathrm {G}^x$ is reductive for $x\in \Omega$, otherwise it is called nonregular.

We say that $(\mathrm {H},W)$ is a PHVSS of $(\mathrm {G},V)$ if $(\mathrm {H},W)$ is a PHVS, $\mathrm {H}\subset \mathrm {G}$ is a subgroup, $W\subset V$ is a vector subspace, and the action of $\mathrm {H}$ is the restriction of the action of $\mathrm {G}$.

Let $V$ be a PHVS for $\mathrm {G}$ with representation $\rho.$ A nonconstant rational function $F:V\to \mathbb {C}$ is called a relative invariant if there exists a Lie group character $\chi :\mathrm {G}\to \mathbb {C}^*$ such that

\[ F(\rho(g)\cdot x)=\chi(g)F(x)\quad \text{for all $g\in\mathrm{G}$ and $x\in V$}. \]

Here are some fundamental facts appearing in [Reference Sato and KimuraSK77, §4].

Example 2.3 The regular PHVS $M_{p,p}$ from Example 2.1(2) has a relative invariant $F:M_{p,p}\to \mathbb {C}$ given by $F(M)=\det (M).$ The associated character $\chi :\mathrm {G}\to \mathbb {C}^*$ is given by $\chi (A,B)=\det (A)\det (B)^{-1}$ since

\[ F((A,B)\cdot M)=\det(AMB^{-1})=\chi(A,B)F(M). \]

The regular PHVS $M_{p,q}\oplus M_{q,p}$ with $p\leqslant q$ from Example 2.1(3) has a relative-invariant given by $F(M,N)=\det (MN)$. The associated character is $\chi (A,B,C)=\det (A)\det (C^{-1})$ since

\[ F((A,B,C)\cdot(M,N))=\det(AMB^{-1}BNC^{-1})=\chi(A,B,C)F(M,N). \]

This relative invariant also defines a relative invariant for the $\mathrm {GL}_p\mathbb {C}\times \mathrm {SO}_q\mathbb {C}$ PHVSS given by $(M,N)=(M,-M^\mathrm {T})$. For $(A,B)\in \mathrm {GL}_p\mathbb {C}\times \mathrm {SO}_q\mathbb {C}$, the associated character is $\chi (A,B)=\det (AA^\mathrm {T})=\det (A)^2.$

2.2 Prehomogeneous vector spaces associated to $\mathbb {Z}$-gradings

A $\mathbb {Z}$-grading of a semisimple Lie algebra $\mathfrak {g}$ is a decomposition

\[ \mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_j\quad \text{such that}\ [\mathfrak{g}_i,\mathfrak{g}_j]\subset\mathfrak{g}_{i+j}. \]

The subalgebra $\mathfrak {p}=\bigoplus _{j\geqslant 0}\mathfrak {g}_j$ is a parabolic subalgebra with Levi subalgebra $\mathfrak {g}_0\subset \mathfrak {p}.$ There is an element $\zeta \in \mathfrak {g}_0$ such that $\mathfrak {g}_j=\{X\in \mathfrak {g}\,|\,[\zeta,x]=jx\}$; the element $\zeta$ is called the grading element of the $\mathbb {Z}$-grading.

Given a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$, let $\mathrm {G}_0<\mathrm {G}$ be the centralizer of $\zeta$; $\mathrm {G}_0$ acts on each factor $\mathfrak {g}_j$. The relation between $\mathbb {Z}$-gradings and PHVSs is given by the following theorem of Vinberg (see [Reference KnappKna02, Theorem 10.19]).

Prehomogeneous vector spaces arising from $\mathbb {Z}$-gradings of $\mathfrak {g}$ are said to be of parabolic type. From now on, we will consider only PHVSs of parabolic type.

Consider a nonzero nilpotent element $e\in \mathfrak {g}.$ By the Jacobson–Morozov theorem, $e$ can be completed to an $\mathfrak {sl}_2$-triple $\{f,h,e\}\subset \mathfrak {g}$, that is, a triple satisfying the bracket relations of $\mathfrak {sl}_2\mathbb {C}$:

\[ [h,e]=2e,\quad [h,f]=-2f,\quad[e,f]=h. \]

Moreover, given $\{h,e\}$ so that $[h,e]=2e$ and $h\in {\operatorname {ad}}_e(\mathfrak {g})$, there is a unique $f\in \mathfrak {g}$ such that $\{f,h,e\}$ is an $\mathfrak {sl}_2$-triple.

Given an $\mathfrak {sl}_2$-triple $\{f,h,e\}$, the semisimple element $h$ acts on $\mathfrak {g}$ with integral weights and thus defines a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$, where $\mathfrak {g}_j=\{x\in \mathfrak {g}\,|\, {\operatorname {ad}}_h(x)=jx\}.$ Note that $e\in \mathfrak {g}_2.$ We have the following result of Kostant and Malcev (see [Reference KnappKna02, Theorem 10.10]).

Let $B:\mathfrak {g}\times \mathfrak {g}\to \mathbb {C}$ denote the Killing form of $\mathfrak {g}.$ Given an $\mathfrak {sl}_2$-triple, $\{f,h,e\}\subset \mathfrak {g}$ with associated $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$, define the Lie algebra character $\chi _{h}:\mathfrak {g}_0\to \mathbb {C}$ by

(2.1)\begin{equation} \chi_{h}(x)=\tfrac{1}{2}B(h,x). \end{equation}

Proposition 2.8 There is a positive integer $q$ such that $q\cdot \chi _{h}$ exponentiates to a character

\[ \chi_{h,q}:\mathrm{G}_0\to \mathbb{C}^* \]

which has a relative invariant of degree $q\cdot B( {h}/{2}, {h}/{2})$.

Proof. Let $\mathfrak {g}_0^e\subset \mathfrak {g}_0$ be the Lie algebra of the $\mathrm {G}_0$-stabilizer of $e\in \mathfrak {g}_2.$ For $x\in \mathfrak {g}^e_0$, we have

\[ B(h,x)=B([e,f],x)=B(f,[e,x])=0. \]

Thus, $\chi _h(\mathfrak {g}_0^e)=0$, and there is a positive integer $q$ so that $q\cdot \chi _h$ exponentiates to a character which is trivial on the identity component of $\mathrm {G}_0^e.$ Since $\mathrm {G}_0^e$ has a finite number of components, we can choose $q$ so that $q\cdot \chi _h$ exponentiates to a character $\chi _{h,q}:\mathrm {G}_0\to \mathbb {C}^*$ whose restriction to $\mathrm {G}_0^e$ is trivial. Hence, by part (iii) of Proposition 2.2, the character $\chi _{h,q}$ has a relative invariant $F:\mathfrak {g}_2\to \mathbb {C}$ such that $F(e)\neq 0$. For the degree, we have $\operatorname {Ad}_{\exp (t h)}(e)=\exp (2t) e$. Thus,

(2.2)\begin{equation} F(\exp(t)e)=F\bigg(\exp\bigg(t\frac{h}{2}\bigg)\cdot e\bigg)= \chi_{h,q}\bigg(\exp\bigg(t\frac{h}{2}\bigg)\bigg)F(e)= \exp\bigg(tq\cdot B\bigg(\frac{h}{2},\frac{h}{2}\bigg)\bigg)F(e). \end{equation}

By Proposition 2.2 $F$ is homogeneous, and so has degree $q\cdot B( {h}/{2}, {h}/{2}).$

2.3 Canonical $\mathbb {Z}$-gradings associated to parabolics

In this section we fix some normalizations for the $\mathbb {Z}$-gradings we will consider.

Up to conjugation, all $\mathbb {Z}$-gradings of $\mathfrak {g}$ arise from labeling the nodes of the Dynkin diagram of $\mathfrak {g}$ with nonnegative integers [Reference VinbergVin94, Chapter 3.5]. This works as follows. Let $\mathfrak {t}\subset \mathfrak {g}$ be a Cartan subalgebra and $\Delta =\Delta (\mathfrak {g},\mathfrak {t})$ be the corresponding set of roots. Pick a set of simple roots $\Pi \subset \Delta$ and let $\Delta ^+\subset \Delta$ be the set of positive roots. Every root $\alpha \in \Delta ^+$ can be written as $\alpha =\sum _{\alpha _i\in \Pi }n_i\alpha _i$, where all $n_i$ are nonnegative integers. For each $\alpha _i\in \Pi$, choose nonnegative integers $p_i$, that is, label the nodes of the Dynkin diagram of $\Pi$ with nonnegative integers $p_i$. This choice defines a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$, where $\mathfrak {g}_j$ is the direct sum of all root spaces $\mathfrak {g}_\alpha$ such that $\alpha =\sum _{\alpha _i\in \Pi }n_i\alpha _i$ and $j=\sum _{i}n_ip_i.$

In the above construction, the associated parabolic $\mathfrak {p}=\bigoplus _{j\geqslant 0}\mathfrak {g}_j$ and the Levi subalgebra $\mathfrak {g}_0$ depend only on the labels $p_i\text { mod}~2$. Namely, up to conjugation, parabolic subalgebras $\mathfrak {p}$ are determined by subsets $\Theta \subset \Pi$, where

\[ \mathfrak{p}_\Theta=\mathfrak{t}\oplus\bigoplus_{\alpha\in span(\Theta)\cap \Delta^+} \mathfrak{g}_{-\alpha}\oplus \bigoplus_{\alpha\in\Delta^+}\mathfrak{g}_\alpha. \]

We define the canonical $\mathbb {Z}$-grading of $\mathfrak {p}_\Theta$ by labeling the $\alpha _i$-node of the Dynkin diagram $0$ if $\alpha _i\in \Theta$ and $1$ if $\alpha _i\in \Pi \setminus \Theta.$ That is, $\mathfrak {g}_j$ is given by

\[ \mathfrak{g}_j=\bigoplus\mathfrak{g}_\alpha,\quad \text{where $\alpha=\sum_{\alpha_i\in\Pi}n_i\alpha_i$ and $j=\sum_{\alpha_i\in\Pi\setminus\Theta}n_i$}. \]

An $\mathfrak {sl}_2$-triple $\{f,h,e\}\subset \mathfrak {g}$ is called even if ${\operatorname {ad}}_h$ has only even eigenvalues. Parabolic subalgebras arising from even $\mathfrak {sl}_2$-triples are called even Jacobson–Morozov parabolics. For such parabolics, the canonical grading is given by ${\operatorname {ad}}_{ {h}/{2}}$; the PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$ is regular by Corollary 2.7 and, by Proposition 2.8, there is a relative invariant associated to the character $\chi _h$ from (2.1). As a result we will call a PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$ which arises from an even $\mathfrak {sl}_2$-triple a JM-regular PHVS.

Example 2.13 Let $\eta =\sum _{\alpha _i\in \Pi }n_i\alpha _i$ be the longest root of $\Delta ^+.$ For $\mathfrak {g}$ not of type $\mathrm {E}_8,\mathrm {F}_4,\mathrm {G}_2,$ there is at least one simple root $\alpha _i$ with $n_i=1$. For such a root set $\Theta =\Pi \setminus \{\alpha _i\}.$ In this case, the canonical $\mathbb {Z}$-grading of the parabolic $\mathfrak {p}_\Theta$ is given by

\[ \mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1. \]

For these examples, $(\mathrm {G}_0,\mathfrak {g}_1)$ is regular if and only if it is JM-regular. The associated flag variety $\mathrm {G}/\mathrm {P}_\Theta$ is the compact dual of a Hermitian symmetric space and $(\mathrm {G}_0,\mathfrak {g}_1)$ is regular if and only if the associated Hermitian symmetric space is of tube type. An example of this is given in Example 2.1, in which case the symmetric space of $\mathrm {SU}(p,q)$ is the relevant Hermitian symmetric space; it is of tube type only when $q=p$.

Example 2.15 The simplest example of a PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$ which is regular but not JM-regular occurs in $\mathrm {SO}_7\mathbb {C}$ with $\Theta =\{\alpha _2\}$, that is, with labeled Dynkin diagram

Here $(\mathrm {G}_0,\mathfrak {g}_1)\cong (\mathrm {GL}_1\mathbb {C}\times \mathrm {GL}_2\mathbb {C}, \mathbb {C}^2\oplus \mathbb {C}^2)$ for the action $(\lambda,A)\cdot (v,w)=(\lambda v A^{-1},Aw^\mathrm {T}).$ A point in the open orbit is given by $v=(1,0)$ and $w=(1,0).$ For this point, the $\mathrm {G}_0$-stabilizer is given by $\lambda =1$ and for $\xi \in \mathbb {C}^*.$ Geometrically, this parabolic stabilizes an isotropic flag of the form $\mathbb {C}\subset \mathbb {C}^3\subset \mathbb {C}^7$.

2.4 Jacobson–Morozov regular prehomogeneous vector subspaces

Fix a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}} \mathfrak {g}_j$ with grading element $\zeta$ and consider a PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$. Following the proof [Reference Collingwood and McGovernCM93, Theorem 3.3.1] of the Jacobson–Morozov theorem, one can show that any nonzero element $e\in \mathfrak {g}_1$ can be completed to an $\mathfrak {sl}_2$-triple $\{f,h,e\}$ with $f\in \mathfrak {g}_{-1}$ and $h\in \mathfrak {g}_0.$

The semisimple element $h$ defines a new grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}} \tilde {\mathfrak {g}}_j$ where

\[ \tilde{\mathfrak{g}}_j=\{x\in\mathfrak{g}\,|\, {\operatorname{ad}}_h(x)=jx\}. \]

Define $\hat {\mathfrak {g}}_j=\mathfrak {g}_j\cap \tilde {\mathfrak {g}}_{2j}$ and the subalgebra $\hat {\mathfrak {g}}\subset \mathfrak {g}$ given by

(2.3)\begin{equation} \hat{\mathfrak{g}}=\bigoplus_{j\in\mathbb{Z}}\hat{\mathfrak{g}}_j. \end{equation}

Note that $h$ and $\zeta$ are both in $\hat {\mathfrak {g}}_0$ and $e\in \hat {\mathfrak {g}}_1.$ The difference $s=\zeta - {h}/{2}$ is semisimple or zero since $\zeta$ and $ {h}/{2}$ are semisimple and $[\zeta, {h}/{2}]=0$. Thus, $\hat {\mathfrak {g}}$ is reductive since $\hat {\mathfrak {g}}=\mathfrak {g}^{s}$ is the centralizer of $s.$ The following proposition is immediate.

Example 2.19 Consider the PHVS $(\mathrm {S}(\mathrm {GL}_{p}\mathbb {C}\times \mathrm {GL}_q\mathbb {C}), M_{p,q})$ from Example 2.1(2). If $p\neq q$, then the PHVS is not JM-regular. For $p>q,$ a maximal JM-regular PHVSS is isomorphic to $(\mathrm {S}(\mathrm {GL}_{q}\mathbb {C}\times \mathrm {GL}_q\mathbb {C}\times \mathrm {GL}_{p-q}\mathbb {C}),M_{q,q})$, for the action $(A,B,C)\cdot M=AMB^{-1}.$ In general, the maximal JM-regular PHVSS of the PHVSs $(\mathrm {G}_0,\mathfrak {g}_1)$ from Example 2.13 is related to the maximal subtube of the associated Hermitian symmetric space.

For the PHVS $(\mathrm {GL}_p\mathbb {C}\times \mathrm {SO}_q\mathbb {C},M_{p,q})$ from the first example of Example 2.1(3) with $p>q,$ the PHVS is not JM-regular and a maximal JM-regular PHVSS is isomorphic to $(\mathrm {GL}_q\mathbb {C}\times \mathrm {GL}_{p-q}\mathbb {C}\times \mathrm {SO}_q\mathbb {C}, M_{q,q})$, where the action is given by $(A,B,C)\cdot M=AMC^{-1}$. The maximal JM-regular PHVSS of $(\mathrm {S}(\mathrm {GL}_p\mathbb {C}\times \mathrm {GL}_q\mathbb {C}\times \mathrm {GL}_r\mathbb {C}),M_{p,q}\oplus M_{q,r})$ can be described similarly. For the regular (non-JM-regular) PHVS $(\mathrm {G}_0,\mathfrak {g}_1)=(\mathrm {GL}_1\mathbb {C}\times \mathrm {GL}_2\mathbb {C},\mathbb {C}^2\oplus \mathbb {C}^2)$ from Example 2.15, the maximal JM-regular PHVSS $(\hat {\mathrm {G}}_0,\hat {\mathfrak {g}}_1)$ containing $(v,w)=((1,0),(1,0))$ is

\[ \hat{\mathrm{G}}_0=\left\{ (\lambda,A)\in \mathrm{GL}_1\mathbb{C}\times\mathrm{GL}_2\mathbb{C}\,\Big|\,A=\left(\begin{pmatrix}a & 0\\0 & b\end{pmatrix}\right) \right\} \quad \text{and}\quad \hat{\mathfrak{g}}_1=\langle v\rangle\oplus \langle w\rangle. \]

The construction of a maximal JM-regular PHVSS of $(\mathrm {G}_0,\mathfrak {g}_1)$ containing $e\in \mathfrak {g}_1$ depends on $e$ and a choice of $\mathfrak {sl}_2$-triple $\{f,h,e\}$. However, if $e,e'\in \mathfrak {g}_1$ are in the same $\mathrm {G}_0$-orbit then any two maximal JM-regular PHVSSs containing $e$ and $e'$ are $\mathrm {G}_0$-conjugate.

Proposition 2.20 Let $e,e'\in \mathfrak {g}_1$ so that $e'\in \mathrm {G}_0\cdot e$, and let $\{f,h,e\}, \{f',h',e'\}$ be two $\mathfrak {sl}_2$-triples with $h,h'\in \mathfrak {g}_0$. Then there is $g\in \mathrm {G}_0$ so that

\[ \{f',h',e'\}=\{\operatorname{Ad}_g f, \operatorname{Ad}_g h,\operatorname{Ad}_g e\}. \]

In particular, the associated maximal JM-regular PHVSSs $(\hat {\mathrm {G}}_0,\hat {\mathfrak {g}}_1)$ and $(\hat {\mathrm {G}}_0',\hat {\mathfrak {g}}_1')$ containing $e$ and $e'$ respectively are $\mathrm {G}_0$-conjugate.

Finally, we note that every $e\in \mathfrak {g}_1$ defines a parabolic subgroup of $\mathrm {P}_{0,e}<\mathrm {G}_0.$ For $e\in \mathfrak {g}_1\setminus \{0\}$, let $\{f,h,e\}$ be an associated $\mathfrak {sl}_2$-triple with $h\in \mathfrak {g}_0.$ If $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\tilde {\mathfrak {g}}_j$ is the $\mathbb {Z}$-grading with grading element $h,$ then $\tilde {\mathfrak {p}}=\bigoplus _{j\geqslant 0}\tilde {\mathfrak {g}}_j$ is a parabolic subalgebra of $\mathfrak {g}.$ Moreover, $\tilde {\mathfrak {p}}$ depends only on $e\in \mathfrak {g}_1\setminus \{0\}$ and not on the $\mathfrak {sl}_2$-triple $\{f,h,e\}$ (see, for example, [Reference Collingwood and McGovernCM93, Remark 3.8.5]). Note that the parabolic $\tilde {\mathfrak {p}}$ can also be defined by

(2.4)\begin{equation} \tilde{\mathfrak{p}}=\{x\in\mathfrak{g}\,|\,\operatorname{Ad}_{\exp(-t({h}/{2}))}\ \text{is bounded as $t\to\infty$}\}. \end{equation}

The parabolic subalgebra $\mathfrak {p}_0(e)\subset \mathfrak {g}_0$ is defined by

\[ \mathfrak{p}_{0,e}=\mathfrak{g}_0\cap\tilde{\mathfrak{p}}. \]

We will denote the associated parabolic subgroup by $\mathrm {P}_{0,e}<\mathrm {G}_0.$ Note that $\hat {\mathfrak {g}}_0\subset \mathfrak {p}_{0,e}$ and $\hat {\mathrm {G}}_0<\mathrm {P}_{0,e}$ define a Levi subalgebra and subgroup, respectively.

Proposition 2.21 Let $e\in \mathfrak {g}_1\setminus \{0\}$ and $\{f,h,e\}$ be an $\mathfrak {sl}_2$-triple with $h\in \mathfrak {g}_0.$ If $s=\zeta - {h}/{2}$, then

\[ \mathfrak{p}_{0,e}=\{x\in\mathfrak{g}_0\,|\,\operatorname{Ad}_{\exp(ts)} x\ \text{is bounded as $t\to\infty$}\}. \]

3.1 The Toledo character and rank

Let $\mathrm {G}$ be a complex semisimple Lie group with Lie algebra $\mathfrak {g}$ and Killing form $B.$ Fix a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$ with grading element $\zeta.$ Recall that $(\mathrm {G}_0,\mathfrak {g}_1)$ is a PHVS, and let $\Omega \subset \mathfrak {g}_1$ be the open $\mathrm {G}_0$-orbit. Since $\mathfrak {g}_0$ is the centralizer of $\zeta,$ $B(\zeta,-):\mathfrak {g}_0\to \mathbb {C}$ defines a character.

Definition 3.1 The Toledo character $\chi _T:\mathfrak {g}_0\to \mathbb {C}$ is defined by

\[ \chi_T(x)=B(\zeta,x)B(\gamma,\gamma), \]

where $\gamma$ is the longest root such that $\mathfrak {g}_\gamma \subset \mathfrak {g}_1.$

Let $e,e'\in \mathfrak {g}_1\setminus \{0\}$ such that $e'\in \mathrm {G}_0\cdot e$ and let $\{f,h,e\}$, $\{f',h',e'\}$ be two $\mathfrak {sl}_2$-triples with $h,h'\in \mathfrak {g}_0.$ By Proposition 2.20, there is $g\in \mathrm {G}_0$ such that $\{\operatorname {Ad}_g f,\operatorname {Ad}_g h,\operatorname {Ad}_g e\}=\{f',h',e'\}$. Since $\operatorname {Ad}_g\zeta =\zeta,$ we have

(3.1)\begin{equation} \chi_T(h)=B(\zeta,h) B(\gamma,\gamma)=B(\operatorname{Ad}_g\zeta,\operatorname{Ad}_g h)B(\gamma,\gamma)=\chi_T(h'). \end{equation}

As a result, we make the following definition.

Definition 3.3 Let $e\in \mathfrak {g}_1$ and $\{f,h,e\}$ be an $\mathfrak {sl}_2$-triple with $h\in \mathfrak {g}_0.$ Define the Toledo rank of $e$ by

\[ \operatorname{rk}_T(e)=\tfrac{1}{2}\chi_T(h). \]

Define the Toledo rank of the PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$ by

\[ \operatorname{rk}_T(\mathrm{G}_0,\mathfrak{g}_1)=\operatorname{rk}_T(e)\quad \text{for $e\in\Omega$.} \]

Proposition 3.6 Let $e\in \mathfrak {g}_1$ and $\{f,h,e\}$ be an associated $\mathfrak {sl}_2$-triples with $h\in \mathfrak {g}_0.$ Then

(3.2)\begin{equation} \operatorname{rk}_T(e)=B\bigg(\frac{h}{2},\frac{h}{2}\bigg)B(\gamma,\gamma).\end{equation}

Proof. Write $\zeta = {h}/{2}+s$. Note that $s\in \mathfrak {g}^e$ and $ {h}/{2}\in {\operatorname {ad}}_e(\mathfrak {g}).$ Thus, $B( {h}/{2},s)=0$ and

\[ \operatorname{rk}_T(e)=B\bigg(\zeta,\frac{h}{2}\bigg)B(\gamma,\gamma)=B\bigg(\frac{h}{2}+s,\frac{h}{2}\bigg) B(\gamma,\gamma)=B\bigg(\frac{h}{2},\frac{h}{2}\bigg)B(\gamma,\gamma). \]

Example 3.7 We illustrate these notions in the case of Example 1.3, with $G=\mathrm {SO}_{2p+q}\mathbb {C}$ and $G_0=\mathrm {GL}_p\mathbb {C}\times \mathrm {SO}_q\mathbb {C}$. Taking an isotropic basis of $\mathbb {C}^{2p+q}=\mathbb {C}^p\oplus \mathbb {C}^q\oplus \mathbb {C}^p$, such that $\langle e_i,e_{2p+q+1-i}\rangle =1$, we have the grading element

\[ \zeta = \begin{pmatrix} \operatorname{Id}_p & & \\ & 0 & \\ & & - \operatorname{Id}_p \end{pmatrix}. \]

Then $\mathfrak {g}_1=\operatorname {Hom}(\mathbb {C}^p,\mathbb {C}^q)$, where $u\in \operatorname {Hom}(\mathbb {C}^p,\mathbb {C}^q)$ represents the matrix

\[ e =\left(\begin{array}{c|c|c} \phantom{u} & - u^\mathrm{T} & \\ \hline & & u \\ \hline & & \end{array}\right),\quad \text{in which } u^\mathrm{T}_{ij}=u_{p+1-j,q+1-i}. \]

The orbits are classified by two integers $(r_1,r_2)$, such that the image of $u\in \operatorname {Hom}(\mathbb {C}^p,\mathbb {C}^q)$ is the sum of a nondegenerate subspace of dimension $r_1$ and a totally isotropic subspace of dimension $r_2$. Let $J_r$ be the rank $r$ matrix

\[ J_r = \bigg(\begin{smallmatrix} & & 1 \\ & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \\ 1 & & \end{smallmatrix}\bigg). \]

Then one can take

\[ u = \begin{pmatrix} \operatorname{Id}_{r_2} & & \\ & \operatorname{Id}_{r_1} & \\ & 0 & \\ & J_{r_1} & \\ & & 0 \end{pmatrix},\quad h = \left(\begin{array}{ccc|ccc|ccc} 0 & & & & & & & & \\ & 2\operatorname{Id}_{r_1} & & & & & & & \\ & & \operatorname{Id}_{r_2} & & & & & & \\ \hline & & & \operatorname{Id}_{r_2} & & & & & \\ & & & & 0 & & & & \\ & & & & & -\operatorname{Id}_{r_2} & & & \\ \hline & & & & & & -\operatorname{Id}_{r_2} & & \\ & & & & & & & -2 \operatorname{Id}_{r_1} & \\ & & & & & & & & 0 \end{array}\right). \]

(The formula for $u$ is valid if $r_2+r_1\leqslant q/2$; the reader will modify $u$ accordingly if $r_2+r_1> {q}/{2}$. The formula for $h$ remains the same.) Take the standard invariant form $B(X,Y)=\operatorname {tr}(XY)$. If $q>1$ we have $B(\gamma,\gamma )=1$ and therefore $\operatorname {rk}_Te=\frac 14 B(h,h)=2r_1+r_2$. On the other hand $\chi _T(x)=B(\zeta,x)$, so that we will have $\tau (E,\varphi )=2\deg V$ if the Higgs bundle $(E,\varphi )$ is given as $V\oplus W\oplus V^*$ with $V$ a $\mathrm {GL}_p\mathbb {C}$-bundle and $W$ a $\mathrm {SO}_q\mathbb {C}$-bundle. This justifies inequality (1.2) as a consequence of Corollary 5.5.

3.2 Real forms and period domains

Let $\mathfrak {g}$ be a complex semisimple Lie algebra and consider a conjugate linear involution $\sigma :\mathfrak {g}\to \mathfrak {g}.$ The fixed point set $\mathfrak {g}^\sigma \subset \mathfrak {g}$ is a real subalgebra such that $\mathfrak {g}^\sigma \otimes \mathbb {C}=\mathfrak {g}.$ Such a subalgebra is called a real form of $\mathfrak {g}.$ On the level of groups, a real form $\mathrm {G}^\sigma <\mathrm {G}$ is the fixed point set of an anti-holomorphic involution $\sigma :\mathrm {G}\to \mathrm {G}.$ A real form is called compact if $\mathrm {G}^\sigma$ is compact or $\mathfrak {g}^\sigma$ is the Lie algebra of a maximal compact subgroup. Compact real forms exist and are unique up to conjugation.

Real forms of $\mathfrak {g}$ can be equivalently defined in terms of complex linear involutions $\theta :\mathfrak {g}\to \mathfrak {g}.$ Namely, Cartan proved that, given a real form $\sigma$, there is a compact real form $\tau :\mathfrak {g}\to \mathfrak {g}$ such that $\sigma \circ \tau =\tau \circ \sigma$, and that given a complex linear involution $\theta$, there is a compact real form $\tau :\mathfrak {g}\to \mathfrak {g}$ such that $\theta \circ \tau =\tau \circ \theta.$ The correspondence is then given by setting $\theta =\sigma \circ \tau$. A real subalgebra $\sigma :\mathfrak {g}\to \mathfrak {g}$ is said to be of Hodge type if $\sigma$ is an inner automorphism of $\mathfrak {g}$. If $\tau$ is a compact real form such that $\theta =\tau \circ \sigma =\sigma \circ \tau,$ then $\sigma$ is of Hodge type if and only if there is a Cartan subalgebra $\mathfrak {t}\subset \mathfrak {g}$ such that $\theta |_{\mathfrak {t}}=\operatorname {Id}.$

Given a complex linear involution $\theta :\mathfrak {g}\to \mathfrak {g}$, we will write $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ for $\pm 1$-eigenspaces of $\theta$, namely $\theta |_\mathfrak {h}=\operatorname {Id}$ and $\theta _\mathfrak {m}=-\operatorname {Id}$. We will call the decomposition $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ the complexified Cartan decomposition of a real form $\mathfrak {g}^\mathbb {R}\subset \mathfrak {g}$. There is a real form of Hodge type associated to every $\mathbb {Z}$-grading of $\mathfrak {g}$.

Proposition 3.8 Let $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$ be a $\mathbb {Z}$-grading, and define $\theta :\mathfrak {g}\to \mathfrak {g}$ by

\[ \theta|_{\mathfrak{g}_j}=(-1)^j\operatorname{Id}. \]

Then $\theta$ is a Lie algebra involution which defines a real form of Hodge type.

Let $\theta$ be the involution from Proposition 3.8 and $\tau$ be a compact real form such that $\sigma =\theta \circ \tau :\mathfrak {g}\to \mathfrak {g}$ is the associated real form of Hodge type. Let $\mathrm {G}^\mathbb {R}<\mathrm {G}$ be the associated real form of $\mathrm {G}$ and let $\mathrm {H}^\mathbb {R}<\mathrm {G}^\mathbb {R}$ be the associated maximal compact subgroup. Note that the real form $\sigma$ restricts to a compact real form on $\mathfrak {g}_0$. Set $\mathrm {H}_0^\mathbb {R}=\mathrm {G}_0\cap \mathrm {G}^\mathbb {R}.$

Consider the homogeneous space

\[ D=\mathrm{G}^\mathbb{R}/\mathrm{H}_0^\mathbb{R}. \]

Note that there is a fibration $D\to \mathrm {G}^\mathbb {R}/\mathrm {H}^\mathbb {R}$ over the Riemannian symmetric space of $\mathrm {G}^\mathbb {R}.$ In fact, the homogeneous space $D$ has a natural homogeneous complex structure. Indeed, the tangent bundle of $D=\mathrm {G}^\mathbb {R}/\mathrm {H}_0^\mathbb {R}$ is isomorphic to the associated bundle

\[ TD\cong \mathrm{G}^\mathbb{R}\times_{\mathrm{H}_0^\mathbb{R}}\mathfrak{g}^\mathbb{R}/\mathfrak{h}_0^\mathbb{R}. \]

Let $\mathfrak {g}^\mathbb {R}=\mathfrak {h}_0^\mathbb {R}\oplus \mathfrak {q}^\mathbb {R}$ be an orthogonal decomposition and $\mathfrak {q}=\mathfrak {q}^\mathbb {R}\otimes \mathbb {C}$. Then the complexified tangent bundle is $T_\mathbb {C} D\cong \mathrm {G}^\mathbb {R}\times _{\mathrm {H}_0^\mathbb {R}}\mathfrak {q},$ and a complex structure on $D$ is equivalent to an $\mathrm {H}_0^\mathbb {R}$-invariant decomposition $\mathfrak {q}=\mathfrak {q}_-\oplus \mathfrak {q}_+$ such that $\sigma (\mathfrak {q}_+)=\mathfrak {q}_-$ and $[\mathfrak {q}_\pm,\mathfrak {q}_\pm ]\subset \mathfrak {q}_\pm$. The decomposition $\mathfrak {q}=\mathfrak {q}_-\oplus \mathfrak {q}_+$ is given by setting

\[ \mathfrak{q}_+=\displaystyle\bigoplus_{j>0}\mathfrak{g}_j\quad\text{and}\quad\mathfrak{q}_-=\displaystyle\bigoplus_{j<0}\mathfrak{g}_j. \]

The complex manifold $D$ is called a period domain for $\mathrm {G}^\mathbb {R}$. Note that the holomorphic tangent bundle of a period domain decomposes as

\[ T^{1,0}_\mathbb{C} D\cong \mathrm{G}^\mathbb{R}\times \mathrm{H}_0^\mathbb{R}\mathfrak{q}_+=\bigoplus_{j>0} \mathrm{G}^\mathbb{R}\times_{\mathrm{H}_0^\mathbb{R}}\mathfrak{g}_j. \]

3.3 The Toledo character and holomorphic sectional curvature

Here we discuss how the choice of the Toledo character is related to a metric of minimal holomorphic sectional curvature $-1$ on $\mathrm {G}^\mathbb {R}/\mathrm {H}_0^\mathbb {R}$.

As is well known, one can define an invariant Hermitian metric on $\mathrm {G}^\mathbb {R}/\mathrm {H}_0^\mathbb {R}$ by using the Hermitian scalar product

\[ \langle x,y \rangle = B(x^*,y) \quad \text{if } x,y\in \mathfrak{g}_k, \]

where $x^*=-\tau (x)$ and $\tau$ is the compact conjugation fixed on $\mathfrak {g}$. This does not give a Kähler metric (the corresponding 2-form is not closed) but one can still define the holomorphic sectional curvature $K$ of its Chern connection. We have the well-known formula (see, for example, [Reference Carlson, Müller-Stach and PetersCMSP17])

\[ K(x)=- \frac{|[x,x^\ast]|^2}{|x|^4}. \]

We are interested in the function $K$ on $\mathfrak {g}_1$. It is known (and re-proved quickly below) that critical points of $K$ are obtained on elements $e$ such that $\{e^*,h=[e,e^*],e\}$ is an $\mathfrak {sl}_2$-triple. For such $e$, we have $|[e,e^*]|^2=\langle [[e,e^*],e],e \rangle =2|e|^2$, and $|e|^2=\langle [\zeta,e],e \rangle =\langle \zeta,[e,e^*]\rangle$, therefore

\[ K(e) = - \frac 2 {B(\zeta,h)}, \]

which is, up to a constant, the inverse of the Toledo rank defined earlier. Then we will prove the following proposition.

Proposition 3.12 The maximum of $K$ on $\mathfrak {g}_1$ is attained on the open orbit, and is equal to $- 2/{B(\zeta,h_{\rm reg})}$, where $e$ is in the open orbit and $\{e^*,h_{\rm reg}=[e,e^*],e\}$ is an $\mathfrak {sl}_2$-triple.

The minimum of $K$ on $\mathfrak {g}_1$ is attained on a minimal orbit, and is equal to $-B(\gamma,\gamma )$ for a long root $\gamma \in \Delta _1$ (the minimum is attained on $e_\gamma$, which belongs to a minimal orbit).

Therefore, after multiplying the Hermitian metric by $B(\gamma,\gamma )$, we obtain a normalized metric with normalized holomorphic sectional curvature

\[ -1 \leqslant K_{\rm norm} \leqslant - \frac1{\operatorname{rk}_T(G_0,\mathfrak{g}_1)}. \]

At a general critical point $e\in \mathfrak {g}_1$ of the curvature (therefore $\{e^*,[e,e^*],e\}$ is an $\mathfrak {sl}_2$-triple), we have the normalized value $K_{\rm norm}(e) = - 1/{\operatorname {rk}_T(e)}$.

From the Toledo character we obtain a $\mathrm {G}^\mathbb {R}$-invariant 2-form on $\mathrm {G}^\mathbb {R}/\mathrm {H}_0^\mathbb {R}$ by

\[ \omega(x,y)=i\chi_T([x,y]_0),\quad \text{for } x,y\in \mathfrak{q}^\mathbb{R}, \]

where $[x,y]_0$ is the projection of the bracket onto $\mathfrak {g}_0.$ This actually defines a pseudo-Hermitian metric, which coincides with the previous normalized metric in the horizontal directions (i.e., in $\mathfrak {g}_1$). This explains the choice of the normalization for the Toledo character.

Proof of Proposition 3.12 For completeness, we begin by proving that the critical points come from $\mathfrak {sl}_2$-triples $\{e^*,[e,e^*],e\}$. Since $K(e)$ is invariant by homothety, we can restrict to variations $\dot e\perp e$. Then

\[ \dot K = - \frac2{|e|^4} \langle [e,e^*],[\dot e,e^*]+[e,\dot e^*]\rangle = - \frac4{|e|^4} \Re \langle [[e,e^*],e], \dot e\rangle \]

It follows that for a critical point, $[[e,e^*],e]=\lambda e$, and up to renormalizing $e$ one can suppose $\lambda =2$.

We will now establish a second variation formula. Start from an $\mathfrak {sl}_2$-triple $(h=[e,e^*],e,e^*)$ and take a vector $x\perp e$ such that $|x|=|e|$. Consider

\[ e(t) = \cos(t) e + \sin(t) x = \bigg(1-\frac{t^2}2\bigg) e + t x + o(t^2). \]

Then $e(t)$ has constant norm, and, up to order 2,

\[ [e(t),e(t)^*] = [e,e^*] + t ( [x,e^*]+[e,x^*]) + t^2 ([x,x^*]-[e,e^*]) . \]

Therefore, again up to order 2,

\begin{align*} - |e|^4 K(e(t)) &= |[e(t),e(t)^*]|^2 \\ &= |[e,e^*]|^2 + t^2 ( |[x,e^*]+[e,x^*]|^2 + 2 \Re \langle [e,e^*],[x,x^*]-[e,e^*] \rangle). \end{align*}

We have already noticed above that $|[e,e^*]|^2=2|e|^2=2|x|^2$, and therefore

(3.3) \begin{align} \frac{d^2}{dt^2}\bigg|_{t=0}|e|^4 K(e(t)) &= 2( 4 |x|^2 - 2 \langle [h,x],x \rangle - |[x,e^*]+[e,x^*]|^2) \nonumber\\ &= 4 \langle (2- {\operatorname{ad}}_{h})x,x \rangle - 2 |[x,e^*]+[e,x^*]|^2. \end{align}

We first deduce that the maximum is attained on the open orbit. Observe that $e$ is in the open orbit if and only if $\mathfrak {g}_1 \cap \ker {\operatorname {ad}}_{e^*}=0$. Therefore, if $e$ is not in the open orbit, we can take $x\in \ker {\rm ad}_{e^*}$. Since ${\operatorname {ad}}_h \leqslant 0$ on $\ker {\operatorname {ad}}_{e^*}$, it follows from (3.3) that in the direction of $x$ we have

\[ \frac{d^2}{dt^2}\bigg|_{t=0}|e|^4 K(e(t)) \geqslant 8 |x|^2, \]

and therefore the maximum cannot be attained at $e$. This proves the claim that the maximum is obtained on the open orbit.

To find the minimum, we use the following fact [Reference ManivelMan13, Corollary 7, p. 100]. Denote $\mathfrak {u}$ the subspace of $\mathfrak {g}_1$ given as the sum of the eigenspaces of ${\operatorname {ad}}_h$ for the eigenvalues at least $2$. Then all orbits in the closure of the orbit of $e$ meet $\mathfrak {u}$. Therefore, suppose that the orbit of $e$ is not minimal. It follows that we can take $x\in \mathfrak {u}$, and (3.3) gives us, in the direction of $x$,

\[ \frac{d^2}{dt^2}\bigg|_{t=0}|e|^4 K(e(t)) \leqslant - 2 |[x,e^*]+[e,x^*]|^2. \]

Such $x$ cannot be in the kernel of ${\operatorname {ad}}_{e^*}$ (${\operatorname {ad}}_h\leqslant 0$ on this kernel). Therefore, the right-hand side is nonzero, so that the minimum cannot be attained at $e$. Therefore, the minimum can be attained only at a minimal orbit.

There can be several minimal orbits if the prehomogeneous space is not irreducible. They are the orbits of elements $e_\gamma$ for $\gamma \in \Delta$ the longest root. As we have seen, $K(e_\gamma )=- 2/{|e_\gamma |^2}=- 4/{|h_\gamma |^2}=- B(\gamma,\gamma )$. The result follows.

Remark 3.13 If we fix an orbit ${\mathcal {O}}$, then $K_{\rm norm}$ must have a maximum on $\overline {{\mathcal {O}}}$. It follows from the proof of the proposition, and in particular from the description of the orbits in the closure of ${\mathcal {O}}$, that the maximum on $\overline {{\mathcal {O}}}$ is achieved at some element $e\in {\mathcal {O}}$ such that $(e^*,[e,e^*],e)$ is an $\mathfrak {sl}_2$-triple; it is therefore equal to $K_{\rm norm}(e)=- 1/{\operatorname {rk}_T(e)}$. In particular, we obtain that for any $x\in \overline {{\mathcal {O}}}$,

(3.4)\begin{equation} -1 \leqslant K_{\rm norm}(x) \leqslant -\frac1{\operatorname{rk}_T(e)}. \end{equation}

Example 3.14 The case $\mathrm {G}=\mathrm {SL}_n\mathbb {C}$ is particularly simple to calculate. A first observation is that any nilpotent element $e\in \mathfrak {sl}_n\mathbb {C}$ belongs to the open orbit of the nilpotent subalgebra of a parabolic algebra of $\mathfrak {sl}_n\mathbb {C}$. It follows that $e$ can be considered as a point in the open orbit of a $\mathfrak {g}_1$ for some grading of $\mathfrak {sl}_n\mathbb {C}$. This observation is not essential for our calculation but relates every nilpotent $e$ to variations of Hodge structures.

We now calculate the corresponding bounds for the holomorphic sectional curvature,

\[ -1\leqslant K\leqslant -\frac1{\operatorname{rk}_Te}, \]

where, by (3.2), we have $\operatorname {rk}_Te=\frac 14 B(h,h)B(\gamma,\gamma )=\frac 12 B(h,h)$ since $B(\gamma,\gamma )=2$ in $\mathfrak {sl}_n$ for the standard invariant form $\operatorname {tr}(XY)$. If we have a Jordan block of size $k$, the eigenvalues of $h$ are $k-1$, $k-3$,…, $-(k-1)$, therefore $B(h,h)=(k-1)^2+(k-3)^2+\cdots +(-(k-1))^2=\frac 13 k(k^2-1)$. Finally, if $e$ has Jordan blocks of size $k_1, k_2,\ldots,k_j$ we obtain

\[ \operatorname{rk}_Te = \frac16 \sum_1^j k_i(k_i^2-1). \]

This gives immediately some curvature bounds in [Reference LiLi22].

Proposition 3.15 Let $e\in \mathfrak {g}_1$ and let $\Omega \subset \mathfrak {g}_1$ be the open $\mathrm {G}_0$-orbit of the PHVS $(\mathrm {G}_0,\mathfrak {g}_1)$. Then

\[ 0\leqslant \operatorname{rk}_T(e)\leqslant\operatorname{rk}_T(\mathrm{G}_0,\mathfrak{g}_1), \]

with equality if and only if $e\in \Omega$.

We end this section with a proposition which will be used often in subsequent sections.

Proof. By Proposition 2.8, there is a positive integer multiple $q\cdot \chi _T$ of the Toledo character which exponentiates to a character $\chi _{T,q}:\mathrm {G}_0\to \mathbb {C}^*$ which has a relative invariant $F$ of degree $q\cdot \operatorname {rk}_T(\mathrm {G}_0,\mathfrak {g}_1).$ We prove that $F$ is a polynomial by proving that it attains a finite value on any element $e'\in \mathfrak {g}_1$.

First note that if $e'=0,$ then (2.2) implies $F(0)=0.$ For $e'\neq 0,$ complete $e'$ into an $\mathfrak {sl}_2$-triple $\{f',h',e'\}$ with $f'\in \mathfrak {g}_{-1}$ and $h'\in \mathfrak {g}_0$. Then $e'+\ker ({\operatorname {ad}}_{f'})$ cuts all other $\mathrm {G}$-orbits, but this is not true in general for $\mathrm {G}_0$-orbits. Nevertheless, it remains true that $e'+\ker ({\operatorname {ad}}_{f'})$ cuts the open orbit, because $\mathfrak {g}_0\cdot e'+(\ker ({\operatorname {ad}}_{f'})\cap \mathfrak {g}_1)=\mathfrak {g}_1$, which implies that $\mathrm {G}_0\cdot (e'+\ker ({\operatorname {ad}}_{f'}))$ contains an open neighborhood of $e'$. So up to conjugating by some element of $\mathrm {G}_0$ we can suppose that $e=e'+X$ with $[f',X]=0$. Since $X$ belongs to the nonpositive eigenspaces of ${\operatorname {ad}}_{h'}$, it follows that $e'=\lim _{t\rightarrow +\infty }e^{t(h'-h)}\cdot e$. Therefore,

\[ F(e') = \lim_{t\rightarrow+\infty} \chi_{h,q}(e^{t(h'-h)}) F(e) = \lim_{t\rightarrow+\infty} e^{tq\chi_h(h'-h)} F(e). \]

This is finite if $\chi _h(h'-h)\leqslant 0$, that is, if $B(h,h')\leqslant B(h,h)$. Since $h'=[e',f']$ and $[h,e']=2e',$ we have $B(h,h')=2B(e',f')$ and $B(h,h)=2B(e,f)$, so the inequality to prove is

\[ B(e',f')\leqslant B(e,f). \]

As before, we may use the Hermitian scalar product from the associate real form and assume that $(e')^*=f'$. We claim that we can also assume $e^*=f$ while keeping the condition $e\in e'+\ker ({\operatorname {ad}}_{f'})$. Indeed, since $[f',X]=0$ we have $[e,e^*]=[e',f']\!+\![X,X^*]$ so we want $[X,X^*]\!=\!h-h'$; but $h-h'$ centralizes the triple $\{e',f',h'\}$ and preserves the slice $e'+\ker ({\operatorname {ad}}_{f'}).$ Thus, it is sufficient to conjugate $e$ by an element of the centralizer of $\{e',f',h'\}$ in $G_0$ to obtain $[e,e^*]=h$.

Then $\langle e',X \rangle =\frac 12 \langle [h',e'],X \rangle =\frac 12 \langle h',[X,f'] \rangle =0$. It follows that $B(e',f')=|e'|^2$ and $B(e,f)=|e|^2=|e'|^2+|X|^2\geqslant B(e',f')$. The proposition follows.

4.1 Higgs bundles and Hodge bundles

Let $\mathrm {G}$ be a complex reductive Lie group with Lie algebra $\mathfrak {g}$ and nondegenerate $\mathrm {G}$-invariant bilinear $\langle \cdot,\cdot \rangle$. Let $\rho :\mathrm {G}\to \mathrm {GL}(V)$ be a holomorphic representation. If $E\to X$ is a $\mathrm {G}$-bundle, we will denote the $V$-bundle $E\times _\mathrm {G} V$ associated to $E$ via the representation $\rho$ by $E(V).$

When $V=\mathfrak {g}$ and the representation $\rho$ is the adjoint representation, a $(\mathrm {G},\mathfrak {g})$-Higgs pair is called a $\mathrm {G}$-Higgs bundle. Suppose $\mathrm {G}^\mathbb {R}<\mathrm {G}$ is a real form with complexified maximal compact $\mathrm {H}<\mathrm {G}$ and complexified Cartan decomposition $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$. When $\rho :\mathrm {H}\to \mathrm {GL}(\mathfrak {m})$ is the restriction of the adjoint representation of $\mathrm {G}$, then an $(\mathrm {H},\mathfrak {m})$-Higgs pair is called a $\mathrm {G}^\mathbb {R}$-Higgs bundle.

If $E$ is a principal $\mathrm {G}$-bundle and $\hat {\mathrm {G}}<\mathrm {G}$ is a subgroup, then a structure group reduction of $E$ to $\hat {\mathrm {G}}$ is a section $\sigma$ of the bundle $E(\mathrm {G}/\hat {\mathrm {G}}).$ Associated to such a reduction is a principal $\hat {\mathrm {G}}$-subbundle $E_\sigma \subset E$ such that $E_\sigma (\mathrm {G})$ is canonically isomorphic to $E.$

For example, if $\mathrm {G}^\mathbb {R}<\mathrm {G}$ is a real form, $\mathrm {H}<\mathrm {G}$ is the complexification of a maximal compact of $\mathrm {G}^\mathbb {R}$ and $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ is complexified Cartan decomposition, then a $\mathrm {G}$-Higgs bundle $(E,\varphi )$ reduces to a $\mathrm {G}^\mathbb {R}$-Higgs bundle if there is a holomorphic reduction $E_{\mathrm {H}}$ of $E$ to $\mathrm {H}$ such that $\varphi \in H^0(E_{\mathrm {H}}(\mathfrak {m})\otimes K).$

Example 4.4 Let $\{f,h,e\}$ be a basis for $\mathfrak {sl}_2\mathbb {C}$, and let $\mathrm {T}<\mathrm {P}\mathrm {SL}_2\mathbb {C}$ be the subgroup with Lie algebra $\langle h\rangle.$ Note that $\mathrm {T}\cong \mathbb {C}^*$ and the adjoint action of $\mathrm {T}$ on $\langle e\rangle$ is given by $\lambda \cdot e=\lambda e.$ So, if $E_\mathrm {T}$ is the frame bundle of $K^{-1},$ then the associated bundle $E_\mathrm {T}(\langle e\rangle )\otimes K\cong {\mathcal {O}}_X$. Hence, we have a $\mathrm {P}\mathrm {SL}_2\mathbb {C}$-Higgs bundle

\[ (E_\mathrm{T}(\mathrm{P}\mathrm{SL}_2\mathbb{C}),e). \]

Since $\deg (K)$ is even, this example can be lifted to $\mathrm {SL}_2\mathbb {C}.$ The lifted action of the subgroup $\mathbb {C}^*\cong \hat {\mathrm {T}}<\mathrm {SL}_2\mathbb {C}$ with Lie algebra $\langle h\rangle$ is given by $\lambda \cdot e=\lambda ^2e.$ As a result, if $E_{\hat {\mathrm {T}}}$ is the frame bundle of a square root $K^{- {1}/{2}}$ of $K^{-1}$, then $(E_{\hat {\mathrm {T}}}(\mathrm {SL}_2\mathbb {C}),e)$ defines an $\mathrm {SL}_2\mathbb {C}$-Higgs bundle. These Higgs bundles will be the fundamental building blocks for the results of § 6.

Now suppose $\mathrm {G}$ is a complex reductive Lie group and $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$ is a $\mathbb {Z}$-grading of its Lie algebra with grading element $\zeta \in \mathfrak {g}_0.$ Let $\mathrm {G}_0<\mathrm {G}$ be the centralizer of $\zeta.$ Note that $\exp (\lambda \zeta )$ is in the center of $\mathrm {G}_0$ and $\operatorname {Ad}(\exp (\lambda \zeta ))$ acts on each $\mathfrak {g}_j$ by $\lambda ^j\cdot \operatorname {Id}$. Let $(E_{\mathrm {G}_0},\varphi )$ be a $(\mathrm {G}_0,\mathfrak {g}_k)$-Higgs pair. Note that the central element $\exp ( ({\lambda }/{k})\zeta )\in \mathrm {G}_0$ defines a holomorphic automorphism of $E_{\mathrm {G}_0}$ and acts on $\varphi$ by multiplication by $\lambda$. As a result we have an isomorphism of $(\mathrm {G}_0,\mathfrak {g}_k)$-Higgs pairs

(4.1)\begin{equation} (E_{\mathrm{G}_0},\varphi)\cong(E_{\mathrm{G}_0},\lambda\varphi)\quad \text{for all $\lambda\in\mathbb{C}^*$}. \end{equation}

Extending the structure group defines a $\mathrm {G}$-Higgs bundle $(E_\mathrm {G},\varphi )$,

\[ (E_\mathrm{G},\varphi)=(E_{\mathrm{G}_0}(\mathrm{G}),\varphi), \]

since $E_{\mathrm {G}_0}(\mathfrak {g}_k)\subset E_{\mathrm {G}_0}[\mathfrak {g}]\cong E_{\mathrm {G}}(\mathfrak {g})$. Moreover, $(E_\mathrm {G},\varphi )\cong (E_{\mathrm {G}},\lambda \varphi )$ for all $\lambda \in \mathbb {C}^*.$

Given a $\mathbb {Z}$-grading $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$, we can define a subalgebra $\tilde {\mathfrak {g}}$ consisting of summands $\mathfrak {g}_j$ with $j=0\text { mod}~k$. Note that $\tilde {\mathfrak {g}}$ has an associated $\mathbb {Z}$-grading with $\mathfrak {g}_k=\tilde {\mathfrak {g}}_1.$ Let $\tilde {\mathrm {G}}<\mathrm {G}$ be the associated reductive subgroup. The following proposition is immediate.

As a result, we will usually consider Hodge bundles of type $(\mathrm {G}_0,\mathfrak {g}_1)$. Recall from Proposition 3.8 that there is a canonical real form $\mathrm {G}^\mathbb {R}<\mathrm {G}$ of Hodge type associated to a $\mathbb {Z}$-grading. The complexified Cartan decomposition $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ satisfies $\mathfrak {g}_{2j}\subset \mathfrak {h}$ and $\mathfrak {g}_{2j+1}\subset \mathfrak {m},$ in particular $\mathrm {G}_0<\mathrm {H}.$ As a result, a $\mathrm {G}$-Higgs bundle which is a Hodge bundle of type $(\mathrm {G}_0,\mathfrak {g}_1)$ reduces to a $\mathrm {G}^\mathbb {R}$-Higgs bundle. Combining this observation with Proposition 4.7 gives the following result which was first observed by Simpson [Reference SimpsonSim92, § 4].

The above results also apply to $\mathrm {G}^\mathbb {R}$-Higgs bundles. Namely, fix a real form $\mathrm {G}^\mathbb {R}<\mathrm {G}$ with a maximal compact subgroup $\mathrm {H}^\mathbb {R}<\mathrm {G}^\mathbb {R}$. Let $\mathrm {H}<\mathrm {G}$ be the complexification of $\mathrm {H}^\mathbb {R}$ and $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ be a complexified Cartan decomposition. Consider a $\mathbb {Z}$-grading of $\mathfrak {g}$ given by $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {h}_j\oplus \mathfrak {m}_j$ with grading element $\zeta \in \mathfrak {h}_0.$ Let $\mathrm {H}_0<\mathrm {H}$ be the centralizer of $\zeta.$ Given an $(\mathrm {H}_0,\mathfrak {m}_k)$-Higgs pair $(E_{\mathrm {H}_0},\varphi )$, extending the structure group to $\mathrm {H}$ defines a $\mathrm {G}^\mathbb {R}$-Higgs bundle $(E_{\mathrm {H}_0}(\mathrm {H}),\varphi )$ such that $(E_{\mathrm {H}_0}(\mathrm {H}),\varphi )\cong (E_{\mathrm {H}_0}(\mathrm {H}),\lambda \varphi )$ for all $\lambda \in \mathbb {C}^*$.

A $\mathrm {G}^\mathbb {R}$-Higgs bundle $(E,\varphi )$ is called a Hodge bundle of type $(\mathrm {H}_0,\mathfrak {m}_k)$ if it reduces to a $(\mathrm {H}_0,\mathfrak {m}_k)$-Higgs pair. Consider the subalgebra $\tilde {\mathfrak {g}}=\tilde {\mathfrak {h}}\oplus \tilde {\mathfrak {m}}$ given by

\[ \tilde{\mathfrak{h}}=\displaystyle\bigoplus_{j=0\text{ mod}~2k}\mathfrak{h}_j\quad\text{and}\quad\tilde{\mathfrak{m}}=\displaystyle\bigoplus_{j=k\text{ mod}~2k}\mathfrak{m}_j. \]

Let $\tilde {\mathfrak {g}}^\mathbb {R}\subset \tilde {\mathfrak {g}}$ be the associated real form of Hodge type with complexified Cartan decomposition $\tilde {\mathfrak {g}}=\tilde {\mathfrak {h}}\oplus \tilde {\mathfrak {m}}.$ As in the complex case, we have the following result.

As a result, we will mostly consider Hodge bundles of type $(\mathrm {G}_0,\mathfrak {g}_1)$.

4.2 Moduli spaces and fixed points

To form a moduli space of Higgs pairs, we need to define suitable notions of stability. We describe this below and refer to [Reference García-Prada, Gothen and Mundet i RieraGGM09, Reference Bradlow, García-Prada and Mundet i RieraBGM03] for more details. Let $\mathrm {G}$ be a complex reductive Lie group and $\rho :\mathrm {G}\to \mathrm {GL}(V)$ be a holomorphic representation. Fix a maximal compact subgroup $\mathrm {K}^\mathbb {R}<\mathrm {G}$ and let $\mathfrak {k}^\mathbb {R}$ be its Lie algebra. An element $s\in i\mathfrak {k}^\mathbb {R}$ defines subspaces of $V$ via the representation $\rho$:

\[ V_s^0=\{v\in V\,|\,\rho(e^{ts})v= v \}\quad\text{and}\quad V_s=\{v\in V\,|\,\rho(e^{ts})(v) \text{ is bounded as $t\to\infty$}\}. \]

When $\rho :\mathrm {G}\to \mathrm {GL}(\mathfrak {g})$ is the adjoint representation, $\mathfrak {g}_s=\mathfrak {p}_s\subset \mathfrak {g}$ is a parabolic subalgebra with Levi subalgebra $\mathfrak {g}_s^0=\mathfrak {l}_s\subset \mathfrak {p}_s.$ The associated subgroups $\mathrm {L}_s<\mathrm {P}_s$ are given by

\[ \mathrm{L}_s=\{g\in\mathrm{G}\,|\,\operatorname{Ad}(g)s=s\}\quad\text{and}\quad\mathrm{P}_s=\{g\in\mathrm{G}\,|\,\operatorname{Ad}(e^{ts})(g)\ \text{is bounded as } t\to\infty\}. \]

Moreover, $s$ defines a character $\chi _s:\mathfrak {p}_s\to \mathbb {C}$ by

\[ \chi_s(x)=\langle s,x\rangle\quad \text{for $x\in\mathfrak{p}_s$.} \]

Let $E$ be a $\mathrm {G}$-bundle, $s\in i\mathfrak {k}^\mathbb {R}$ and $\mathrm {P}_s<\mathrm {G}$ be the associated parabolic subgroup. A reduction of structure group of $E$ to $\mathrm {P}_s$ is a $\mathrm {P}_s$-subbundle $E_{\mathrm {P}_s}\subset E$; this is equivalent to a section $\sigma \in \Gamma (E(\mathrm {G}/\mathrm {P}_s))$ of the associated bundle. We will denote the associated $\mathrm {P}_s$-subbundle by $E_\sigma.$ The degree of such a reduction will be defined using Chern–Weil theory. Since $\mathrm {P}_s$ is homotopy equivalent to the maximal compact $\mathrm {K}_s^\mathbb {R}=\mathrm {K}^\mathbb {R}\cap \mathrm {L}_s$ of $\mathrm {L}_s$, given a reduction of structure $\sigma$ of $E$ to $\mathrm {P}_s$, there is a further reduction of $E$ to $\mathrm {K}_s^\mathbb {R}$ which is unique up to homotopy. Let $E_{\sigma '}\subset E$ be the resulting $\mathrm {K}^\mathbb {R}_s$ principal bundle. The curvature $F_A$ of a connection $A$ on $E_{\sigma '}$ satisfies $F_A\in \Omega ^2(X,E_{\sigma '}(\mathfrak {k}_s^\mathbb {R})).$ Thus, evaluating the character $\chi _s$ on the curvature, we have $\chi _s(F_A)\in \Omega (X,i\mathbb {R}),$ and we define the degree of $\sigma$ as

(4.2)\begin{equation} \deg E(\sigma,s)=\frac{i}{2\pi}\int_X\chi_s(F_A). \end{equation}

If a multiple of $q\cdot \chi _s$ exponentiates to a character $\tilde \chi _s:\mathrm {P}_s\to \mathbb {C}^*$ and $\sigma \in \Gamma (E(\mathrm {G}/\mathrm {P}_s))$ is a reduction, then $E_{\sigma }\times _{\chi _s}\mathbb {C}^*=E_{\sigma }(\tilde \chi _s)$ is a line bundle and $\deg E(\sigma,s)= ({1}/{q})\deg E_\sigma (\tilde \chi _s).$ When $s$ is in the center $\mathfrak {z}$ of $\mathfrak {g}$ then $\mathrm {P}_s=\mathrm {G}$. In this case, the degree given in (4.2) is simply the degree of $E$ with respect to $\chi _s,$ and will be denoted by $\deg _\chi (E)$. Again, if a multiple $q\cdot \chi _s$ exponentiates to a character $\tilde \chi :\mathrm {G}\to \mathbb {C}^*$ we have

(4.3)\begin{equation} \deg_\chi(E)=\frac{1}{q}\deg E(\tilde\chi). \end{equation}

Let $d\rho :\mathfrak {g}\to \mathfrak {gl}(V)$ be the differential of $\rho$ and let $\mathfrak {z}^\mathbb {R}$ be the center of $\mathfrak {k}^\mathbb {R}$ and $\mathfrak {k}^\mathbb {R}=\mathfrak {k}_{ss}^\mathbb {R}=\mathfrak {z}^\perp$. Define

\[ \mathfrak{k}^\mathbb{R}_\rho=\mathfrak{k}^\mathbb{R}_{ss}\oplus (\ker(d\rho|_{\mathfrak{z}^\mathbb{R}}))^\perp. \]

We are now ready to define $\alpha$-stability notions for $\alpha \in i\mathfrak {z}^\mathbb {R}.$

The moduli space of $\alpha$-polystable $(\mathrm {G},V)$-Higgs pairs over $X$ is defined as the set of isomorphism classes of $\alpha$-polystable $(\mathrm {G},V)$-Higgs pairs and will be denoted by ${\mathcal {M}}^\alpha (\mathrm {G},V).$ A GIT construction of these spaces is given by Schmitt in [Reference SchmittSch08] and by Simpson for the moduli space of $0$-polystable $\mathrm {G}$-Higgs bundles [Reference SimpsonSim94].

There is a natural $\mathbb {C}^*$-action on the moduli spaces of $\alpha$-polystable Higgs pairs given by $\lambda \cdot (E,\varphi )=(E,\lambda \varphi ).$ If $\mathfrak {g}=\bigoplus _{j\in \mathbb {Z}}\mathfrak {g}_j$ is a $\mathbb {Z}$-grading, then the $\mathbb {C}^*$-action is trivial on the moduli space of $(\mathrm {G}_0,\mathfrak {g}_k)$-Higgs pairs by (4.1). As a result, polystable Higgs bundles which are Hodge bundles define fixed points of the $\mathbb {C}^*$-action on the moduli space of Higgs bundles. Simpson proved the converse [Reference SimpsonSim88, Reference SimpsonSim92]. Namely, all $\mathbb {C}^*$-fixed points in the moduli space of Higgs bundles are Hodge bundles. In fact, if a Higgs bundle is a Hodge bundle, then polystability of the Higgs bundle is equivalent to polystability of the associated pair. To explain why this is true, we need to use the correspondence between stability and solutions to gauge-theoretic equations.

We first describe this for $\mathrm {G}^\mathbb {R}$-Higgs bundles. Fix a maximal compact subgroup $\mathrm {K}^\mathbb {R}<\mathrm {G}$ and let $\tau :\mathfrak {g}\to \mathfrak {g}$ be the resulting conjugate linear involution. Now fix a real form $\mathrm {G}^\mathbb {R}<\mathrm {G}$ such that $\mathrm {K}^\mathbb {R}\cap \mathrm {G}^\mathbb {R}=\mathrm {H}^\mathbb {R}$ is a maximal compact subgroup of $\mathrm {G}^\mathbb {R}$ and let $\mathrm {H}<\mathrm {G}$ be the complexification of $\mathrm {H}^\mathbb {R}.$ Let $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ be the complexified Cartan decomposition of the real form. Consider a $\mathrm {G}^\mathbb {R}$-Higgs bundle $(E,\varphi )$. A metric on $E$ is by definition a structure group reduction $h\in \Gamma (E(\mathrm {H}/\mathrm {H}^\mathbb {R}))$ of $E$ to $\mathrm {H}^\mathbb {R}.$ Associated to a metric $h$, there is a unique connection $A_h$ (the Chern connection) which is compatible with the reduction and the holomorphic structure. Let $E_h$ be the $\mathrm {H}^\mathbb {R}$-subbundle associated to a metric h. Then $E(\mathfrak {m})$ is canonically identified with $E_h(\mathfrak {m})$. The compact real form $\tau$ defines an anti-holomorphic involution of $E_h(\mathfrak {m})$. Combining this with conjugation of 1-forms defines an involution