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Let $X$ be an irreducible smooth projective curve over $\mathbb{C}$. Let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. Let ${\rm M}_{G,X}$ be the moduli space of semistable principal $G$-bundles on $X$. Are there any metrics on this moduli space?

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    $\begingroup$ What do you call a metric? (keep in mind that it is not a manifold). $\endgroup$ – abx May 23 '18 at 7:29
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Let $G$ is a reductive linear algebraic group over $\mathbb{C}$, and $X$ be a connected compact Riemann surface of genus $\geq 2$.

Fix a topological type $\tau$. Then the moduli space of semistable principal $G$-bundles over $X$ of type $\tau$, denoted $\mathcal{M}_\tau(X,G)$, is a projective variety, and also is in bijective correspondence with a semi-algebraic subspace of $\mathrm{Hom}(\Gamma, K)/K$ for appropriate discrete $\Gamma$, maximal compact $K\subset G$, and where $K$ acts on $\mathrm{Hom}(\Gamma, K)$ by conjugation.

See A. Ramanathan: Moduli for principal bundles over algebraic curves, I and II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996) 301–328 and 421–449.

If you don't fix the topological type, then I think you at best get an algebraic stack, so then I am not sure what "metric" means.

As it stands $\mathcal{M}_\tau(X,G)$ can be given a metric in a number of (probably not very interesting) ways:

  1. As a projective variety, it embeds into $\mathbb{CP}^n$ for some $n$. So you can restrict the Fubin-Study.

  2. As a subset of a semialgebraic space, you can embed it in some affine space and restrict the Euclidean metric.

Neither of these metrics are very natural, and are probably not what you are looking for (although they may answer the question as it is written).

For a "Weil–Petersson" like metric, which is what I guess you might be after, I found this paper of Biswas and Schumacher titled Kähler structure on moduli spaces of principal bundles:

https://www.sciencedirect.com/science/article/pii/S0926224506000441.

Here is an excerpt from the abstract, where I put in boldface the part you might be most interested in:

"Let $M$ be a moduli space of stable principal $G$-bundles over a compact Kähler manifold $(X, \omega_X)$, where $G$ is a reductive linear algebraic group defined over $\mathbb{C}$. Using the existence and uniqueness of a Hermite–Einstein connection on any stable $G$-bundle $P$ over $X$, we have a Hermitian form on the harmonic representatives of $H^1(X, ad(P ))$, where $ad(P )$ is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on $M$ is constructed; we call this the Petersson–Weil form."

Anyway, I hope this helps.

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