A boundedness theorem for principal bundles on curves

Let $G$ be a reductive group acting on an affine scheme $V$. We study the set of principal $G$-bundles on a smooth projective curve $\mathcal{C}$ such that the associated $V$-bundle admits a section sending the generic point of $\mathcal{C}$ into the GIT stable locus $V^{\mathrm{s}}(\theta )$. We show that after fixing the degree of the line bundle induced by the character $\theta$, the set of such principal $G$-bundles is bounded. The statement of our theorem is made slightly more general so that we deduce from it the boundedness for $\epsilon$-stable quasimaps and $\mathrm{\Omega }$-stable LG-quasimap.