  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 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-Rapoport).
