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Let $p: Y \ra X$ be a Galois cover of smooth projective curves over $\CC$with Galois group $\Gamma$. This paper is devoted to the study of principalorthogonal and symplectic bundles $E$ on $Y$ to which the action of $\Gamma$ on$Y$ lifts. We notably describe them intrinsically in terms of objects definedon $X$ and call these objects parahoric bundles. We give necessary andsufficient conditions for the non-emptiness of the moduli of stable (andsemi-stable) parahoric special orthogonal, symplectic and spin bundles on theprojective line $\PP^1$.

Criteria for existence of stable parahoric $\SO_n$, $\Sp_n$ and $\Spin$ bundles on $\PP^1$