Parabolic semiinfinite sheaves on affine flag varieties

We study sheaves on affine flag varieties of reductive algebraic groups in parabolic semiinfinite settings, generalizing some constructions and results from the work of Gaitsgory and of Achar, Dhillon, and the first author (for the case of the Borel subgroup) and of Dhillon and Lysenko (for characteristic-$0$ coefficients and a particular case of parabolic semiinfinite condition). In particular we define perverse t-structures and Raskin equivalences in this context, and study an analogue of the Gaitsgory sheaf. In future work, these ∞-categories will be related to representation-theoretic ∞-categories associated with the Langlands dual group.