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We define $\mathrm{SL}_{r}$ -opers in the setup of vector bundles on curves with a parabolic structure over a divisor. Basic properties of these objects are investigated.

Let $T$ be a maximal torus of $\operatorname{PSL}(n,\mathbb{C})$ . For $n\geq 4$ , we construct a smooth compactification of $\operatorname{PSL}(n,\mathbb{C})/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{\operatorname{PSL}(n,\mathbb{C})}$ for a suitable choice of $T$ -linearized ample line bundle on $\overline{\operatorname{PSL}(n,\mathbb{C})}$ . We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of $\operatorname{PSL}(n,\mathbb{C})/T$ is $\operatorname{PSL}(n,\mathbb{C})$ itself.

Let $E$ be a principal $G$--bundle over a smooth projective curve over an algebraically closed field $k$, where $G$ is a reductive linear algebraic group over $k$. We construct a canonical reduction of $E$. The uniqueness of canonical reduction is proved under the assumption that the characteristic of $k$ is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of $k$ is positive.