Topological invariants of parabolic G-Higgs bundles

Abstract

For a semisimple real Lie group G, we study topological properties of moduli spaces of polystable parabolic G-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichmüller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic G-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.

Appendix A: Stability condition for parabolic G-Higgs bundles

In this appendix, we review the definition by Biquard et al. [4] for a general parabolic G-Higgs bundle and the definition of parabolic degree for parabolic principal bundles. We start from the basic Lie algebra background and claim that for $G=\mathrm {U}(n)$ and $G^{\mathbb {C}}=\mathrm {GL}(n,\mathbb {C})$, the definition of parabolic degree coincides with the one defined earlier in §2.1. Moreover, for $G=\mathrm {Sp}(2n,\mathbb {R})$, the general definitions also reduce to the ones considered in §5.

Let G be a real reductive group, thus one has a Cartan decomposition of the Lie algebra $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$, where $\mathfrak {h}$ is the Lie algebra of the maximal compact subgroup H of G. This decomposition has the property that $[\mathfrak {h},\mathfrak {m}]\subset \mathfrak {m}$ and $[\mathfrak {m},\mathfrak {m}]\subset \mathfrak {h}$. Now, the right action of H defines the symmetric space $H\backslash G$. The stabilizer to $[1]\in H\backslash G$ is H, so we can identify $T_{[1]}H\backslash G$ with $\mathfrak {h}\backslash \mathfrak {g}\cong \mathfrak {m}$ and is stabilized by the adjoint action of H. Thus any metric on $\mathfrak {m}$ defines an H-invariant Riemannian metric on the symmetric space $H\backslash G$. This $H\backslash G$ is a symmetric of negative curvature, whose boundary could be defined by the geodesic rays denoted by $\partial _\infty (H\backslash G)$. This can be described in terms of a parabolic group as follows:

Definition A.1

A subgroup P of G is called parabolic, if there exists $s\in \mathfrak {m}$ such that

$$\begin{aligned} P= P_{s}:=\{g\in G|d([e^{ts}ge^{-ts}],[1]) \text{ is } \text{ bounded } \text{ when } t\rightarrow \infty \}. \end{aligned}$$

and the parabolic subalgebra of $\mathfrak {g}$ is then defined as

$$\begin{aligned} \mathfrak {p}_s:=\{x\in \mathfrak {g}| Ad(e^{ts})x \text{ is } \text{ bounded } \text{ when } t\rightarrow \infty \}. \end{aligned}$$

We call $s\in \mathfrak {m}$ to be the antidominant element for P, if $P=P_s$.

For a choice of $s\in \mathfrak {m}$ and $g\in G$, we may write $g=ph$ for some $p\in P_s$ and $h\in H$. We then set

$$\begin{aligned} s\cdot g:=Ad(h^{-1})(s). \end{aligned}$$

Note that the element coming from this action still stays in $\mathfrak {m}$.

We define the geodesic ray in $H\backslash G$ to be a morphism of the form $\gamma :[0,\infty )\rightarrow H\backslash G$ by $\gamma (t)= [ e^{t s}\cdot g]$ for some $s\in \mathfrak {h}$ and $g\in G$. Here $e^{ts}\cdot g$ is the ordinary product in G and [g] means the representative in $H\backslash G$.

Now we define the equivalence of two geodesics as follows:

Definition A.2

Let d be the distance function between points in $H\backslash G$. We say two geodesic rays $\gamma _1,\gamma _2$ are equivalent if $d(\gamma _1(t),\gamma _2(t))$ is bounded and independent of t.

Now the boundary at infinity of the symmetric space is defined by

$$\begin{aligned} \partial _\infty (H\backslash G)=\{\text{ geodesic } \text{ rays }\}/\sim . \end{aligned}$$

Remark A.3

The parabolic group $P_s$ is actually the stabilizer of the element $\gamma (t)=[e^{ts}\cdot g]$.

For any $s\in \mathfrak {m}\backslash \{0\}$, the ordinary geodesic $\eta _s(t):t\mapsto [ e^{ts}\cdot 1]$ provides an element in $\partial _\infty (H\backslash G)$. The claim is that for all possible s, then $\eta _s$ enumerates all elements in $\partial _\infty (H\backslash G)$. Indeed, one has the following lemma:

Lemma A.4

There is an equivalence between $[ e^{ts}\cdot g]$ and $[e^{tu}]$, where u is determined as follows: If g has the decomposition $g=ph$ with $p\in P_s$, $h\in H$, then $u=s\cdot g:=\mathrm {Ad}_{h^{-1}}(s)$.

Proof

One has $d([e^{ts}\cdot g],[e^{tu}])=d([e^{ts}ge^{-tu}],[1])=d([e^{ts}p e^{-ts} e^{ts}he^{-tu}],[1])$, thus the distance is bounded if and only if $e^{ts}he^{-tu}$ is bounded. On the other hand, we know that $e^{ts}h=he^{t \mathrm {Ad}_{h^{-1}}(s)}$ from the Baker-Hausdorff formula. It follows that in order to have boundedness, we may just let $u=\mathrm {Ad}_{h^{-1}}(s)$. This calculation also shows that the following limit exists:

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{t}\log (e^{ts}\cdot g)=\mathrm {Ad}_{h^{-1}}(s). \end{aligned}$$

Here the $\log$ is defined to be $\log (g)=v$ if g has the Cartan decomposition $g= ke^{v}$ for $k\in H$ and $v\in \mathfrak {m}$. $\square$

We deduce that for every element $\gamma$ in $\partial _\infty (H\backslash G)$ and for every element $x\in H\backslash G$, we can find an element s in $\mathfrak {m}$ such that $\gamma (t)=[x\cdot e^{ts}]$. We call $v(x,\gamma ):=s\in \mathfrak {m}\cong T_x(H\backslash G)$.

The Tits distance between $\gamma ,\gamma '\in \partial _\infty (H\backslash G)$ is defined by

$$\begin{aligned} d_{Tits}(\gamma ,\gamma ')=\sup _{x\in H\backslash G} \text {Angle}(v(x,\gamma ),v(x,\gamma ')), \end{aligned}$$

where the angle is in $[0,\pi ]$; it measures the maximal possible angle from the same initial point to the required boundary $\gamma$ and $\gamma '$.

Let P be the parabolic group $P_s$ associated to $s\in \mathfrak {m}$ and $Q=Q_\sigma$ the parabolic for $\sigma \in \mathfrak {m}$. Then we define the relative degree as

$$\begin{aligned} \deg ((P,s),(Q,\sigma )):=|s|\cdot |\sigma |\cdot \cos d_{Tits}(\eta (s),\eta (\sigma )). \end{aligned}$$

The definition can be seen from the topology of $\partial _\infty (H\backslash G)$. We are measuring the inner product of the tangent vector associated to $\eta (s):=\gamma _s(t)=[e^{ts}]$ and $\eta _\sigma :=\gamma _\sigma (t)=[e^{t\sigma }]$ and are trying to determine the maximum possible degree. An alternative characterization is given by the function

$$\begin{aligned} \mu _s:\mathfrak {m}\rightarrow \mathbb {R},\quad \mu _s(\sigma )=\lim _{t\rightarrow +\infty }\langle s\cdot e^{-t\sigma },\sigma \rangle . \end{aligned}$$

Lemma A.5

For $|s|=|\sigma |=1$, we can identify s and $\sigma$ with the corresponding element in $\partial _\infty (H\backslash G)$

$$\begin{aligned} \mu _s(\sigma )=\cos d_{Tits}(s,\sigma ). \end{aligned}$$

This lemma follows directly from our previous calculation identifying $[e^{ts}]$ with $[x\cdot e^{tu}]$. The proof can be found in [4] Proposition B.1.

Using these definitions, we can describe several examples in further detail; the following example is worked out based on [37] and [36].

Example A.6

($G=\mathrm {GL}(n,\mathbb {C}),H=\mathrm {U}(n)$) In this situation, $\mathfrak {h}$ is the set of anti-Hermitian matrices and $\mathfrak {m}$ is the set of Hermitian matrices. We also claim that the map $\mathfrak {m}\backslash \{0\}\rightarrow \partial _{\infty }(H\backslash G)$ sending $s\in \mathfrak {m}$ to the geodesic $\gamma (t)=[e^{ts}]$ defines a bijection. Indeed, we can describe the structure of parabolic group explicitly. For an element $s\in \mathfrak {m}$, s is a Hermitian matrix and thus it can be diagonalized. Moreover, the matrix s has real eigenvalues, say $\lambda _1<\lambda _2<\ldots <\lambda _r$. We denote $V_j=\ker (\lambda _j \mathrm {I}- s)$ to be the eigenspaces associated to each eigenvalue. We let $W_k=V_1\oplus \ldots V_k$ and now the group

$$\begin{aligned} P_s=\{g\in \mathrm {GL}(n,\mathbb {C})| g(W_k)\subset W_k,\forall k\in [1,n]\} \end{aligned}$$

is parabolic, considering $e^{ts} g e^{-ts}$. We can diagonalize $e^{s}$ and the eigenspaces are exactly the $V_j$’s and $e^{ts}$ acts on $V_j$ by multiplying by $e^{\lambda _j t}$. It follows that in order to have a bounded $e^{ts} g e^{-ts} v$ for $v\in V_j$, then it must be $g(v)\subset W_j$.

Now the relative degree can also be calculated by the choice of two antidominant elements. In fact, for two choices of $s,\sigma \in \mathfrak {m}$, we just need to calculate $\mu _s(\sigma )$.

For any $g\in \mathrm {GL}(n,\mathbb {C})$, we first compute $s\cdot g$. Recall that $s\cdot g=\mathrm {Ad}(h^{-1})(s)$ for $g=ph$, where $p\in P_s$ and $h\in H$. Since s can be diagonalized, $s(v)=\lambda _j v$ for $v\in V_j$. We also see that $g^{-1}(W_k)=h^{-1}(W_k)$ by preservation of flags. Then the claim is that

$$\begin{aligned} s\cdot g=\sum _{j=1}^k \lambda _j \pi _{g^{-1}(W_j)\cap g^{-1}(W_{j-1})^\perp }=\sum _{j=1}^k (\lambda _j -\lambda _{j+1})\pi _{g^{-1}W_j}, \end{aligned}$$

where $\pi _{W_j}$ is the projection of vector onto $W_j$.

Then we can use this claim to compute the relative degree, knowing that for two $s,\sigma$ there are $\lambda _1,\ldots ,\lambda _k$ for s and $\mu _1,\ldots ,\mu _l$ for $\sigma$, as well as eigenspaces $V_1,\ldots ,V_k$ for s and $A_1,\ldots ,A_l$ for $\sigma$. Also, we let $W_j=V_1\oplus \ldots \oplus V_j$ and $B_j=A_1\oplus \ldots \oplus A_j$ and then a calculation shows

$$\begin{aligned} \deg ((P_s,s),(P_\sigma ,\sigma ))=\sum _{i=1}^k \sum _{j=1}^l(\lambda _i-\lambda _{i+1})(\mu _j-\mu _{j+1})\dim (W_i\cap B_j). \end{aligned}$$

Here we assumed that $\lambda _{k+1}=0$.

We can now define the stability condition for parabolic principal bundles from the previous definition for the degree.

In [4] the authors introduce parabolic G-Higgs bundles over a punctured Riemann surface X for a non-compact real reductive Lie group G and establish a Hitchin-Kobayashi type correspondence for such pairs. This definition involves a choice for each puncture of an element in the Weyl alcove $\mathcal {A}$ of a maximal compact subgroup $H\subset G$.

Let $\left( X,D \right)$ be as earlier, a pair of a compact connected Riemann surface X and $D=\left\{ {{x}_{1}},\ldots ,{{x}_{s}} \right\}$ a divisor of s-many distinct points on X. Let also ${{H}^{\mathbb {C}}}$ be a reductive, complex Lie group. Fix a maximal compact subgroup $H\subset {{H}^{\mathbb {C}}}$, and a maximal torus $T\subset H$ with Lie algebra $\mathfrak {t}$. For an $H^{\mathbb {C}}$-principal holomorphic bundle E over X, and for any set W on which $H^{\mathbb {C}}$ acts on the left, we denote by E(W) the twisted product $E\times _{H^{\mathbb {C}}}W$, that is, the product quotient out the equivalence relation $E\times W/\sim$ with the latter defined by $(e,hw)\sim (eh,w)$. If W is the representation of $H^{\mathbb {C}}$, we can always associate a vector bundle E(W).

The following argument for the definition of a stability condition is analogous to the non-parabolic case (see [18]). For a representation into a Hermitian vector space $\rho :H\rightarrow U(B)$, let us denote the holomorphic extension still by $\rho :H^\mathbb {C}\rightarrow \mathrm {GL}(B)$. Given a parabolic subgroup $P_s$ we can define the subspace

$$\begin{aligned} B^-_s=\{v\in B| \rho (e^{t s})v \text{ is } \text{ bounded } \text{ when } t\rightarrow \infty \} \end{aligned}$$

(A.7)

and the invariant subspace under the Levi subalgebra

$$\begin{aligned} B^0_s=\{v\in B| \rho (e^{t s})v=v \text{ for } \text{ any } \text{ t }.\} \end{aligned}$$

(A.8)

We can also define the difference between these two, namely the bounded but not fixed part

$$\begin{aligned} B^{<0}_s=\{v\in B^-_s |v\not \in B^0_s\} \end{aligned}$$

(A.9)

Remark A.10

There is a one-to-one correspondence between an antidominant element s and an antidominant character $\chi$ (see [36]). In the sequel, we shall be making use of the subspaces $B_\chi ^-$ and $B_\chi ^0$ in accordance to [4].

A holomorphic reduction of structure group to a parabolic subgroup P is a section of the principal bundle $E(H^\mathbb {C}/P)$. Since canonically $E(H^\mathbb {C}/P)\cong E/P$, a holomorphic reduction $\sigma :X\rightarrow E/P$ together with the quotient map $E\rightarrow E/P$ gives a P-principle bundle $E_\sigma :=\sigma ^*(E)$. This is exactly where the term “reduction” is coming from. It then follows immediately that $E(B)\cong E_\sigma \times _P B$ and we have the vector bundle

$$\begin{aligned} E_{\sigma ,\chi }^-(B):=E_\sigma \times _{P_\chi } B_\chi ^-, \end{aligned}$$

where $P_\chi$ is the parabolic subgroup associated to the antidominant character $\chi$.

Following Lemma 2.12 from [18] we imply the definition for the degree of a principal bundle associated to a holomorphic reduction $\sigma$ and an antidominant character $\chi$.

Definition A.11

For a holomorphic reduction $\sigma \in \Gamma (E(H^\mathbb {C}/P))$ for a parabolic subgroup P and an antidominant character $\chi$ for P, we can associate the antidominant element $s_\chi$. Moreover, for a representation $\rho _W:H\rightarrow U(W)$ for some Hermitian vector space W, then $\rho _W(s_\chi )$ diagonalizes with real eigenvalues $\lambda _1<\lambda _2<\ldots <\lambda _r$. Let $V_j = \ker (\lambda _j \mathrm {Id}_W-\rho _W(s_\chi ))$ and $W_j=V_1\oplus \ldots \oplus V_j$, for $j=1, \ldots , r$ and $W=W_r$. Suppose that the associated vector bundles $\mathcal {W}_j=E(W_j)$ and $\mathcal {W}=E(W)$ are all holomorphic, then the degree of E is defined as

$$\begin{aligned} \deg E(\sigma ,\chi )=\lambda _k\deg \mathcal {W}+\sum _{i=1}^{r-1}(\lambda _i-\lambda _{i+1})\deg \mathcal {W}_j. \end{aligned}$$

We now include into our study the parabolic structure at each point. We fix an alcove ${\mathcal {A}}\subset \mathfrak {t}$ of H containing $0\in \mathfrak {t}$ and for ${{\alpha }_{i}}\in \sqrt{-1}{\bar{\mathcal {A}}}$ where $\bar{\mathcal {A}}$ is the closure of $\mathcal {A}$, and we let ${{P}_{{{\alpha }_{i}}}}\subset {{H}^{\mathbb {C}}}$ be the parabolic subgroup defined by the ${{\alpha }_{i}}$ coming from definition A.1.

Remark A.12

In fact, we can choose any element in $\mathfrak {t}$ and define the corresponding parabolic structure. But according to Lemma 3.3 in [4] one can always choose a shift in the cocharacter lattice $\Lambda _{cochar}$ and get a 1-1 correspondence between the local holomorphic sections. It is therefore more convenient to restrict to the alcoves, instead of the whole maximal toric algebra.

Definition A.13

A parabolic structure of weight ${{\alpha }_{i}}$ on E over a point ${{x}_{i}}$ is defined as the choice of a subgroup ${{Q}_{i}}\subset E{{\left( {{H}^{\mathbb {C}}} \right) }_{{{x}_{i}}}}$ with an antidominant character $\alpha _i$ for $Q_i$.

Remark A.14

In order to be compatible with Definition 2.1, a small modification for the choice of the $\alpha _i^j$ is necessary here. Indeed, we need a decreasing sequence of weights

$$\begin{aligned} \alpha _i^{r}>\alpha _i^{r-1}>\ldots >\alpha _i^{1}, \end{aligned}$$

so that the increasing filtration is then

$$\begin{aligned} E_r\subset E_{r-1}\subset \ldots E_1=E_{x_i}. \end{aligned}$$

If this is written as a decreasing filtration, it will recover the original filtration at the point. This now provides

$$\begin{aligned} \deg ((P_s,s),(Q_i,\alpha _i))&=\sum _{i=1}^s\sum _{j=1}^r(\lambda _i-\lambda _{i+1})(\alpha _i^{r+1-j}-\alpha _i^{r-j})\dim (W_i\cap E_{r+1-j}) \\&=\sum _{i=1}^s\sum _{j=1}^r(\lambda _i-\lambda _{i+1})(\alpha _i^{j}-\alpha _i^{j-1})\dim (W_i\cap E_{j}). \end{aligned}$$

Note that we have assumed here that $\alpha _i^0=0$.

The definition for the parabolic degree of a parabolic principal bundle now combines the last two definitions:

Definition A.15

Consider a parabolic principal $H^\mathbb {C}$-bundle E together with a holomorphic reduction $\sigma : X\rightarrow E/P_\chi$ and an antidominant character $\chi$. We define the parabolic degree of E with respect to $\sigma$ and $\chi$ to be the real number

$$\begin{aligned} \mathrm{pardeg}\,E(\sigma ,\chi ):=\deg E(\sigma ,\chi )+\sum _{i=1}^s \deg ((E_\sigma ,\chi ),(Q_i,\alpha _i)). \end{aligned}$$

(A.16)

Remark A.17

In [4] the authors take the minus sign in the term including the contribution from the weights in the expression of the parabolic degree. We take a plus sign here instead, coming from the increasing sequence of weights that we considered above; this shall ensure that when examining the case when $G=\mathrm {GL}(n,\mathbb {C})$ later on, the definition for the parabolic degree will coincide with Definition 2.3.

For a real reductive Lie group G with a maximal compact subgroup H, let $\mathfrak {g}=\mathfrak {h}\oplus \mathfrak {m}$ be the Cartan decomposition of the Lie algebra into its $\pm 1$-eigenspaces, where $\mathfrak {h}=\mathrm {Lie}\left( H \right)$ and let $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ be the bundle associated to E via the isotropy representation. Choose a trivialization $e\in E$ near the point ${{x}_{i}}$, such that near ${{x}_{i}}$ the parabolic weight lies in ${{\alpha }_{i}}\in \sqrt{-1}{\bar{\mathcal {A}}}$. In the trivialization e, we can decompose the bundle $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ under the eigenvalues of $\mathrm {ad}\left( {{\alpha }_{i}} \right)$ acting on ${{\mathfrak {m}}^{\mathbb {C}}}$ as

$$\begin{aligned} E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) =\underset{\mu }{\mathop {\oplus }}\,\mathfrak {m}_{\mu }^{\mathbb {C}}. \end{aligned}$$

In particular, take ${{\alpha }_{i}}\in \sqrt{-1}{{{{\mathcal {A}}'}}_{\mathfrak {g}}}$, where ${{{{\mathcal {A}}'}}_{\mathfrak {g}}}$ is the space of $\alpha \in {\bar{\mathcal {A}}}$ such that the eigenvalues of $\mathrm {ad}(\alpha )$ have modulus smaller than 1 on the entire $\mathfrak {g}$, and consider for $\alpha \in \sqrt{-1}\mathfrak {h}$ the subspaces of ${{\mathfrak {m}}^{\mathbb {C}}}$ similar to definition of $B_\alpha ^-$ and $B_\alpha ^0$. We can define $\mathfrak {m}_i:=(\mathfrak {m}^{\mathbb {C}})_{\alpha _i}^-$ as well as $\mathfrak {m}_i^0=(\mathfrak {m}^{\mathbb {C}})_{\alpha _i}^0$ according to equation A.7 and A.8. The decomposition implies that $\mathfrak {m}_i=\mathfrak {m}_i^0\oplus \mathfrak {n}_i$ for some $\mathfrak {n}_i$. We define the sheaf $PE\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ of parabolic sections of $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ as the sheaf of local holomorphic sections $\psi$ of $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ such that $\psi \left( {{x}_{i}} \right) \in {{\mathfrak {m}}_{i}}$. Similarly, the sheaf $NE\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ of strongly parabolic sections of $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ is defined as the sheaf of local holomorphic sections $\psi$ of $E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right)$ such that $\psi \left( {{x}_{i}} \right) \in {{\mathfrak {n}}_{i}}$. The following short exact sequences of sheaves are then realized

$$\begin{aligned}&0\rightarrow PE\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) \rightarrow E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) \rightarrow \underset{i}{\mathop {\bigoplus }}\,E{{\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) }_{{{x}_{i}}}}/{{\mathfrak {m}}_{i}}\rightarrow 0, \end{aligned}$$

(A.18)

$$\begin{aligned}&0\rightarrow NE\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) \rightarrow E\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) \rightarrow \underset{i}{\mathop {\bigoplus }}\,E{{\left( {{\mathfrak {m}}^{\mathbb {C}}} \right) }_{{{x}_{i}}}}/{{\mathfrak {n}}_{i}}\rightarrow 0. \end{aligned}$$

(A.19)

Definition A.20

Let $\left( X,D \right)$ be a pair of a compact connected Riemann surface X and $D=\left\{ {{x}_{1}},\ldots ,{{x}_{s}} \right\}$ a divisor of s-many distinct points on X. A parabolic G-Higgs bundle over (X, D) is a pair $(E,\Phi )$ such that E is an $H^\mathbb {C}$-principal bundle over X and $\Phi$ is a holomorphic section of $PE(\mathfrak {m}^\mathbb {C})\otimes K(D)$. A strongly parabolic G-Higgs bundle $(E,\Phi )$ over (X, D) involves $\Phi \in \Gamma (NE(\mathfrak {m}^\mathbb {C})\otimes K(D))$.

We have the following three notions of stability. Note that the parameter $\alpha$ in the following definition should not be confused with the parabolic structure $\alpha$ in previous settings (Definition 2.1).

Definition A.21

(Stability conditions) Let $(E,\Phi )$ be a parabolic G-Higgs bundle, and let $\alpha \in i\mathfrak {h}\cap \mathfrak {z}$, where $\mathfrak {z}$ is the center of $\mathfrak {h}^\mathbb {C}$. Also, let $\langle \cdot ,\cdot \rangle$ be a nondegenerate bilinear pairing on $\mathfrak {h}^{\mathbb {C}}$.

• 1. We say the parabolic G-Higgs bundle $(E,\Phi )$ is $\alpha$-semistable, if for any subgroup $P\subset H^\mathbb {C}$, any anti-dominant character $\chi$ of P and any holomorphic reduction $\sigma \in \Gamma (H^\mathbb {C}/P)$ such that $\Phi \in H^0(E(\mathfrak {m}^{\mathbb {C}})_{\sigma ,\chi }^-\otimes K(D))$, one has

  $$\begin{aligned} \mathrm{pardeg}\,E(\sigma ,\chi )-\langle \alpha ,\chi \rangle \ge 0. \end{aligned}$$

• 2. We say the parabolic G-Higgs bundle is $\alpha$-stable, if the above inequality is strict.

• 3. We say the parabolic G-Higgs bundle is $\alpha$-polystable, if it is semistable and if for $\sigma ,\chi$ as earlier such that $\Phi \in H^0(E(\mathfrak {m}^{\mathbb {C}})_{\sigma ,\chi }^-\otimes K(D))$, it is

  $$\begin{aligned} \mathrm{pardeg}\,E(\sigma ,\chi )-\langle \alpha ,\chi \rangle =0, \end{aligned}$$

  then there is a holomorphic reduction $\sigma _{L_s}\in \Gamma (E_\sigma (P/L_s))$ where $L_s$ is the Levi subgroup such that $\Phi \in H^0(E(\mathfrak {m}^\mathbb {C})_{\sigma _{L_s},\chi }^0\otimes K(D))\subset H^0(E(\mathfrak {m}^\mathbb {C})_{\sigma ,\chi }^-\otimes K(D))$.

• 4. If the parameter $\alpha =0$ above, we just call the corresponding Higgs bundle semistable, stable and polystable respectively.

This abstract definition can be unraveled considering particular examples, and indeed it turns out that it coincides with Definition 2.3 in the case $G=\mathrm {GL}(n,\mathbb {C})$. We describe this in detail in the following:

Example A.22

($G=\mathrm {GL}(n,\mathbb {C}),H=\mathrm {U}(n)$) For a vector space $\mathbb {C}^n$ and a representation $H^{\mathbb {C}}$ acting on V, there is an identification of a parabolic $G=\mathrm {GL}(n,\mathbb {C})$-bundle with a parabolic vector bundle W. Thus $E(\mathfrak {m}^{\mathbb {C}})$ is in fact $\mathrm {End}(W)$ and $\Phi \in H^0(\mathrm {End}(W)\otimes K(D))$. Moreover, one has to check that $\mathrm {Res}_{x_i}\Phi \in \mathfrak {m}_i:=\mathfrak {m}_{\alpha }^-$.

We already know that the parabolic structure $(Q_i,\alpha _i)$ involves a choice of real eigenvalues from 0 to 1 from Example A.22 and Remark A.12. Thus this corresponds to a choice of real numbers $-\frac{1}{2}\le \alpha _i^1\le \alpha _i^2\le \ldots \le \alpha _i^n\le \frac{1}{2}$ and thus the subspaces $A_j=\ker (\alpha _i^j \mathrm {Id}-\alpha _i)$ and also $B_{j}=A_n\oplus A_{n-1} \oplus \ldots \oplus A_j$. Moreover,

$$\begin{aligned} \Gamma (PE(\mathfrak {m}^{\mathbb {C}})\otimes K(D))=\{\Phi \in \mathrm {Hom}(E,E\otimes K(D))|\Phi (B_j)\subset B_j\otimes K(D)\}, \end{aligned}$$

which corresponds to the classical definition of a parabolic Higgs bundle.

We next check the $\alpha$-stability condition. For the group $G=\mathrm {GL}(n,\mathbb {C})$, its center is the set of diagonal matrices. Therefore, a choice of $\alpha$ in fact involves a choice of a real number. Indeed, since $\chi$ can be diagonalized with real eigenvalues $\lambda _1<\ldots <\lambda _r$ with eigenspaces of dimension $\dim V_1,\ldots \dim V_r$, then $\langle \alpha ,\chi \rangle =\alpha \sum _{i=1}^r \lambda _i \dim V_i$.

For every classical group G and any G-principal bundle P, one can associate a vector bundle W by considering the fundamental representation $\mathcal {W}=P\times _{G}W$ where W is the fundamental representation of G and the Higgs field will be a section of the vector bundle $P\times _{Ad}\mathfrak {m}^{\mathbb {C}}$. If $G=\mathrm {GL}(n,\mathbb {C})$, the associated vector bundle will be the vector bundle $\mathcal {W}$ and $\Phi$ will be section of $\mathrm {End}(\mathcal {W})\otimes K(D)$, thus we have the following claim:

Lemma A.23

With the same notation from Definition A.11, a parabolic $GL(n,\mathbb {C})$-Higgs bundle $(E,\Phi )$ is $\mu$-semistable/stable/polystable if and only if the associated parabolic Higgs bundle $(\mathcal {W},\Phi _{\mathcal {W}})$ is semistable/stable/polystable. Here $\mu :=\mu (\mathcal {W})=\frac{\mathrm{pardeg}\,\mathcal {W}}{n}$.

Proof

($\Leftarrow$) Let $P\subset H^\mathbb {C}$ be a parabolic subgroup, $\chi$ an antidominant character of P and $\sigma \in \Gamma (H^\mathbb {C}/P)$ a holomorphic reduction such that $\Phi \in H^0(E(\mathfrak {m}^{\mathbb {C}})_{\sigma ,\chi }^-\otimes K(D))$.

We may consider the case where there is only one point in the divisor D and then generalize the argument. Consider the antidominant character $\chi$ of P and the representation $\rho _W:H\rightarrow U(W)$ with the real eigenvalues $\lambda _1<\ldots <\lambda _r$. Then by Definition A.11, if we set $V_i=\ker (\lambda _j\mathrm {Id}_W-\rho _W(s_\chi ))$, $W_i=V_1\oplus \ldots \oplus V_r$ and $W=W_r$, one can calculate the parabolic degree as follows

$$\begin{aligned}&\mathrm{pardeg}\,E(\sigma ,\chi )-\langle \alpha ,\chi \rangle = \\&\quad =\sum _{k=1}^{r}(\lambda _k-\lambda _{k+1})(\deg \mathcal {W}_k-\mu \mathrm {rk}(\mathcal {W}_k)) + \sum _{k,j}(\lambda _k-\lambda _{k+1}) (\alpha _i^j-\alpha _i^{j-1})\dim (W_k\cap B_{j}) \\&\quad =\sum _{k=1}^{r}(\lambda _k-\lambda _{k+1}) \left( \deg \mathcal {W}_k-\mu \mathrm {rk}(\mathcal {W}_k)+\sum _{j=1}^{n}(\alpha _i^{j}-\alpha _i^{j-1})\dim (W_k\cap B_{j})\right) \\&\quad =\lambda _r (\mathrm{pardeg}\,\mathcal {W}-\mu \mathrm {rk}(\mathcal {W}))+\sum _{k=1}^{r-1} (\lambda _k-\lambda _{k+1}) (\mathrm{pardeg}\,\mathcal {W}_k-\mu \mathrm {rk}(\mathcal {W}_k)) \ge 0. \end{aligned}$$

The last inequality follows since the Higgs bundle preserves the flag, $\Phi (\mathcal {W}_j)\subset \Phi (\mathcal {W}_j)\otimes K(D)$, thus the slope inequality for stable/semistable parabolic Higgs bundle yields $\frac{\mathrm{pardeg}\,\mathcal {W}_k}{\mathrm {rank}(\mathcal {W}_k)}\le \mu (\mathcal {W})= \frac{\mathrm{pardeg}\,\mathcal {W}}{n}$, so $\mathrm{pardeg}\,\mathcal {W}_k\le \mu \mathrm {rk}(\mathcal {W}_k)$ if we rearrange the terms in the equation. Moreover, $\lambda _k-\lambda _{k+1}$ is less than 0 by definition.

Note that $\Phi \in H^0(E(\mathfrak {m}^\mathbb {C})^-_{\sigma ,\chi }\otimes K(D))$ actually says that $\Phi (\mathcal {W}_k)\subset \mathcal {W}_k\otimes K(D)$. Together with the assumption about parabolic structure from $\Phi (B_j)\subset B_j\otimes K(D)$ we derive a strong restriction on the possible W’s that are allowed.

Now we have to show that the polystability conditions are corresponding to each other. But this just comes from the fact that our Higgs bundle $\mathcal {W}=V_1\oplus V_2$ and $\Phi \in H^0(\mathrm {Hom}(V_1,V_1\otimes K(D))\oplus \mathrm {Hom}(V_2,V_2\otimes K(D)) )$, thus for any parabolic subgroup that preserves the flag of $\mathcal {W}$ we can reduce it to the subgroup $\mathrm {GL}(\dim V_1,\mathbb {C})\times \mathrm {GL}(\dim V_2,\mathbb {C})$, the Levi subgroup of parabolic reduction will be straightforward as well.

($\Rightarrow$) Let $(E(W),\Phi _W)$ be a $\mu$-semistable parabolic G-Higgs bundle. For any subbundle V of W, one can associate the filtration $0\subset V\subset W$ and the associated parabolic subgroup that preserves this flag. For any antidominant character $\chi$ of the associated parabolic and any holomorphic reduction $\sigma$ one has that

$$\begin{aligned} \mathrm{pardeg}\,E(\sigma ,\chi )-\langle \mu ,\chi \rangle =(\lambda _1-\lambda _2)(\mathrm{pardeg}\,V-\mu (\mathcal {W})\mathrm {rk}(V))\ge 0, \end{aligned}$$

(A.24)

which provides that $\frac{\mathrm{pardeg}\,V}{\mathrm {rk}(V)}\le \mu (\mathcal {W})$. The polystability condition is checked similarly. $\square$

We now come to our key example $G=\mathrm {Sp}(2n,\mathbb {R})$, for which we shall examine in detail what it means to be semistable, stable or polystable. The treatment is similar to Section 4.3 of [18].

Example A.25

($G=\mathrm {Sp}(2n,\mathbb {R}),H=U(n)$) For the group $G = \mathrm {Sp}(2n,\mathbb {R})$, it is $H^\mathbb {C}=\mathrm {GL}(n,\mathbb {C})$, and for the fundamental representation $\mathbb {V}$ of $\mathrm {GL}(n,\mathbb {C})$ one has the isotopy representation

$$\begin{aligned} \mathfrak {m}^\mathbb {C}=\mathrm {Sym}^2\mathbb {V}\oplus \mathrm {Sym}^2\mathbb {V}^*. \end{aligned}$$

Following Section 4.3 in [18] and replacing the degree by the parabolic degree, one has the following characterization of semistability, stability and polystability:

The parabolic $\mathrm {Sp}(2n,\mathbb {R})$-Higgs bundle consists of a parabolic vector bundle $V=E(\mathbb {V})$ and a section

$$\begin{aligned} \Phi =(\beta ,\gamma )\in H^0(K(D)\otimes \mathrm {Sym}^2V\oplus K(D)\otimes \mathrm {Sym}^2V^*). \end{aligned}$$

The stability condition turns out to be similar while we have a different Higgs field. For the parabolic vector bundle V we define the filtration of V of length $k-1$ to be a strictly increasing filtration of holomorphic subbundles

$$\begin{aligned} \mathcal {V}=(0=V_0\subsetneq V_1\subsetneq V_2\subsetneq \ldots \subsetneq V_{k-1}\subsetneq V_k=V). \end{aligned}$$

Let $\lambda =(\lambda _1<\lambda _2<\ldots <\lambda _k)$ be a strictly increasing sequence of k-real numbers, and we define

$$\begin{aligned} \mathcal {N}_\beta (\mathcal {V},\lambda )=\sum _{\begin{array}{c} \lambda _i+\lambda _j\le 0 \\ 1\le i,j\le k \end{array}}K(D)\otimes (V_i\otimes _S V_j) \text{ and } \mathcal {N}_\gamma (\mathcal {V},\lambda )=\sum _{\begin{array}{c} \lambda _i+\lambda _j\ge 0 \\ 1\le i,j\le k \end{array}}K(D)\otimes (V_{i-1}^\perp \otimes _S V_{j-1}^\perp ). \end{aligned}$$

Here, for $W_1,W_2$ subbundles of W, then $W_1\otimes _S W_2$ is the image of the map $W_1\otimes W_2\subset W\otimes W\rightarrow \mathrm {Sym}^2 W$. Also, $W_1^\perp$ is the kernel of restriction map $W^*\rightarrow W_1^*$. We also set

$$\begin{aligned} \mathcal {N}(\mathcal {V},\lambda ):=\mathcal {N}_\beta (\mathcal {V},\lambda )\oplus \mathcal {N}_\gamma (\mathcal {V},\lambda ) \end{aligned}$$

and define

$$\begin{aligned} pd(\mathcal {V},\lambda ,\alpha ):=\sum _{j=1}^k(\lambda _j-\lambda _{j+1})(\mathrm{pardeg}\,V_j-\alpha \mathrm {rk}(V_j)). \end{aligned}$$

Now we can define the subspaces associated to the parabolic structure and determine $PE(\mathfrak {m}^\mathbb {C})$. For any parabolic structure at $x_i$ we have a filtration $0=U_0\subsetneq U_1 \subsetneq \ldots \subsetneq U_{r-1}\subsetneq U_r=V_{x_i}$ with associated weights $\alpha _1<\alpha _2<\ldots <\alpha _r$. Define

$$\begin{aligned} U_\beta (V_{x_i},\alpha ):= & {} \sum _{\begin{array}{c} \alpha _i+\alpha _j\le 0 \\ 1\le i,j\le k \end{array}}K(D)\otimes (U_i\otimes _S U_j) \text{ and } \\ U_\gamma (V_{x_i},\alpha )= & {} \sum _{\begin{array}{c} \alpha _i+\alpha _j\ge 0 \\ 1\le i,j\le k \end{array}}K(D)\otimes (U_{i-1}^\perp \otimes _S U_{j-1}^\perp ), \end{aligned}$$

and analogously $U(V_{x_i},\alpha )=U_\beta (V_{x_i},\alpha )\oplus U_\gamma (V_{x_i},\alpha )$. Then the condition $\Phi \in H^0(PE(\mathfrak {m}^\mathbb {C})\otimes K(D))$ is nothing but $\Phi _{x_i}\in U(V_{x_i},\alpha )$.

Thus we have the following characterization of the stability condition by the same argument in [18]:

Lemma A.26

A parabolic $\mathrm {Sp}(2n,\mathbb {R})$-Higgs bundle $(E,\Phi )$ over (X, D) with parabolic structure $\alpha _i$ at each point $x_i\in D$ is a vector bundle V together with $\Phi =(\beta ,\gamma )\in H^0(K(D)\otimes \mathrm {Sym}^2V\oplus K(D)\otimes \mathrm {Sym}^2V^*)$ such that $\Phi |_{x_i}\in U(V_{x_i},\alpha _i)$. Moreover the Higgs bundle is $\alpha$-semistable if for a filtration of V and any strictly increasing sequence $\lambda$ we have $\Phi \in H^0(\mathcal {N}(\mathcal {V},\lambda ))$, then $pd(\mathcal {V},\lambda ,\alpha )\ge 0$. When the inequality is strict, we say the Higgs bundle is stable.

The pair is $\alpha$-polystable if for a nontrivial $\mathcal {V}$ and $\lambda$ such that $\Phi \in H^0(\mathcal {V},\lambda )$ and such that $pd(\mathcal {V},\lambda ,\alpha )=0$ there is an isomorphism of vector bundles

$$\begin{aligned} \sigma :V\rightarrow V_1\oplus V_2/V_1\oplus \ldots \oplus V_k/V_{k-1}, \end{aligned}$$

such that $V_j=\sigma ^{-1}(V_1\oplus V_2/V_1\oplus \ldots \oplus V_j/V_{j-1})$ and such that

$$\begin{aligned} \beta \in H^0\left( \sum _{\lambda _i+\lambda _j}K(D)\otimes \sigma ^{-1}(V_i/V_{i-1})\otimes _S \sigma ^{-1}(V_j/V_{j-1})\right) \end{aligned}$$

and

$$\begin{aligned} \gamma \in H^0\left( \sum _{\lambda _i+\lambda _j}K(D)\otimes \sigma ^{*}((V_i/V_{i-1})^*)\otimes _S \sigma ^{*}((V_j/V_{j-1})^*)\right) . \end{aligned}$$

Since $\mathrm {Sym}(V^*)\subset V^*\otimes V^*$ by sending $v\otimes _S w\rightarrow \frac{1}{2}(v\otimes w+ w\otimes v)$, we can write the $\mathrm {Sp}(2n,\mathbb {R})$-bundle as an $\mathrm {SL}(2n,\mathbb {C})$-bundle. Thus, we may write $E=V\oplus {{V}^{\vee }}$ and $\Phi =\left( \begin{matrix} 0 &{} \beta \\ \gamma &{} 0 \\ \end{matrix} \right) :E\rightarrow E\otimes K\left( D \right)$. Equipped with the stability condition and since $\beta ,\gamma$ are fixed under the change of basis $a\otimes b\mapsto b\otimes a$ in $V\otimes V$ and $V^*\otimes V^*$, we exactly revoke our Definition 5.2 of a parabolic $\mathrm {Sp}(2n,\mathbb {R})$-Higgs bundle.

The proof for the other examples of Sect. 8 is entirely analogous.