Nonabelian theta functions of positive genus

Let $\mathcal{C}_g$ be a smooth projective irreducible curve over $\mathbb{C}$ of genus $g \geq 1$ and let $\{p_1,\dots, p_s\}$ be a set of distinct points on $\mathcal{C}_g$. We fix a nonnegative integer $\ell$ and denote by $M_g(\underline{p},\underline{\lambda})$ the moduli space of parabolic semistable vector bundles of rank $r$ on $\mathcal{C}_g$ with trivial determinant and fixed parabolic structure of type $\underline{\lambda}=(\lambda_1,\dots, \lambda_s)$ at $\underline{p}=(p_1,\dots, p_s)$, where each weight $\lambda_i$ is in $P_{\ell}(\mathrm{SL}(r))$. On $M_g(\underline{p},\underline{\lambda})$ there is a canonical line bundle $\mathcal{L}(\underline{\lambda}, \ell)$, whose global sections are called generalized parabolic $\mathrm{SL}(r)$-theta functions of order $\ell$. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.