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We study the spaces of twisted conformal blocks attached to a 
$\Gamma$-curve 
$\Sigma$ with marked 
$\Gamma$-orbits and an action of 
$\Gamma$ on a simple Lie algebra 
$\mathfrak {g}$, where 
$\Gamma$ is a finite group. We prove that if 
$\Gamma$ stabilizes a Borel subalgebra of 
$\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed 
$\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let 
$\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve 
$\Sigma /\Gamma$ obtained via the 
$\Gamma$-invariance of Weil restriction associated to 
$\Sigma$ and the simply connected simple algebraic group 
$G$ with Lie algebra 
$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic 
$\mathscr {G}$-torsors on 
$\Sigma /\Gamma$ when the level 
$c$ is divisible by 
$|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport).
