I am looking at Usha Bhosle's paper on Moduli of Parabolic $G$-bundles. She calculated the dimension of the moduli space.

Now I want to know the the dimension of the moduli of parabolic $G$-Higgs bundles over a curve of genus $g$. I think this should be $\dim M = \dim Z(G) + (g-1)\dim G + 2\dim G/P$.


Let $G$ be a connected reductive complex affine algebraic group. Let $\mathcal{N}$ be the moduli space of holomophic $G$-bundles over a Riemann surface of genus $g\geq 2$ with $n$ punctures (and fixed generic boundary data). Let $\mathcal{M}$ be the moduli space of $G$-Higgs bundles over a Riemann surface of genus $g\geq 2$ with $n$ punctures (and fixed generic boundary data).

Intuitively, the expected dimension of $\mathcal{M}$ is 2 times that of $\mathcal{N}$ assuming the boundary data is compatible (and the notions of stability). The reason for this is that one can generally think of $\mathcal{M}$ as a partial completion of the cotangent bundle of $\mathcal{N}$.

In Moduli Spaces of Parabolic Higgs Bundles and Parabolic K(D) Pairs over Smooth Curves: I, by H. U. Boden, K. Yokogawa, one finds this philosophy holds true for $G=SL_m$ or $G=GL_m$.

More generally, in Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group, by O. Biquard, O. Garcia-Prada, & I. Mundet i Riera, one finds a general correspondence to the Betti moduli space of representations.

Let's compute the dimension (for generic semisimple boundary data) on the Betti side to see the above philosophy about the expected dimension.

A presentation for the surface is $$\langle a_1,b_1,....,a_g,b_g,c_1,...,c_n\ |\ [a_1,b_1]\cdots [a_g,b_g]c_1\cdots c_n=1\rangle.$$ Fix semisimple conjugacy classes $\mathcal{C}_1,...,\mathcal{C}_n$ and assume that $C_1\cdots C_n\in DG$ for all $C_i\in \mathcal{C}_i$, where $DG$ is the derived subgroup of $G$. Denote by $PG=G/Z(G)$ where $Z(G)$ is the center of $G$.

Let $\mu:G^{2g}\times \mathcal{C}_1\times\cdots \times\mathcal{C}_n\to DG$ be the map that sends $$(A_1,B_1,...,A_g,B_g,C_1,...,C_n)\mapsto [A_1,B_1]\cdots [A_g,B_g]C_1\cdots C_n.$$ Then $\mathfrak{X}_G:=\mu^{-1}(I)/\!/PG$ is the parabolic $G$-character variety that corresponds to $\mathcal{M}$, under appropriate assumptions.

For sufficiently generic boundary data, the dimension of $\mu^{-1}(I)$ is $$2g\dim G + \sum_{i=1}^n\dim \mathcal{C}_i-\dim(DG)=(2g-1)\dim G +\dim Z(G)+ \sum_{i=1}^n\dim \mathcal{C}_i.$$ Consequently, the dimension of the quotient is \begin{eqnarray}\dim \mu^{-1}(I)-\dim PG&=&2(g-1)\dim G+2\dim Z(G) + \sum_{i=1}^n\dim \mathcal{C}_i\\&=&2(g-1)\dim G+2\dim Z(G) + \sum_{i=1}^n\dim G/H_i,\end{eqnarray} where $H_i$ are the stabilizers of the conjugacy classes $\mathcal{C}_i$. Note: since we chose the conjugacy classes to be of semisimple elements they are closed orbits and so the stabilizers are reductive subgroups.

Now if we did the same calculation on the Betti side with respect to holomorphic $G$-bundles, then we want to consider representations into a maximal compact subgroup of $G$ (call it $K$). Thus, we obtain the expected dimension (for generic boundary data) of $\mathfrak{X}_K$ to be: $$2(g-1)\dim K+2\dim Z(K) + \sum_{i=1}^n\dim K/J_i.$$ And so since $2\dim K=\dim G$ we conclude: $\dim \mathfrak{X}_G=2\dim \mathfrak{X}_K$, if we have corresponding boundary data (like the fixed conjugacy classes have the property that $J_i$ is a maximal compact in $H_i$ for all $i$).

In short, you don't have enough 2's in your guessed formula.


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