Semi-infinite cohomology and the linkage principle for W-algebras

Fundamental local equivalences in quantum geometric Langlands

“…Via a standard construction, acts as an affine linear functional on , which we denote by , in such a way that, writing for the associated reflection in , one has Briefly, this arises via restricting a linear action of on to the affine hyperplane of functionals whose pairing with is 1. We refer the reader to [Dhi21, §§ 3.1 and 3.4], where this is reviewed in greater detail.…”

“…The projection of the sum of Verma modules (3.21) onto picks out the summand cf. [Dhi21, Lemma 8.7]. Accordingly, we consider the composition where the latter map is projection onto the block.…”

“…The proof of the lemma is straightforward; cf. Lemma 9.8 of [Dhi21] for the assertion regarding Verma modules.…”

“…It is therefore enough to show that is a parabolic subgroup of , which is straightforward; cf. the proof of Lemma 8.8 in [Dhi21].…”

“…Write for the Zhu algebra of and for the center of the universal enveloping algebra of . Let us normalize the identification as in [Dhi21]. Writing for the character of corresponding to , i.e.…”