Two-sided BGG resolutions of admissible representations

TL;DR

This paper proves the existence of two-sided BGG resolutions for admissible representations of affine Kac-Moody algebras at fractional levels, enabling new inductive methods for studying these representations.

Contribution

It confirms a longstanding conjecture and introduces a semi-infinite Borel-Weil theorem for minimal parabolic subalgebras, advancing the understanding of admissible representations.

Findings

Proved the conjecture of Frenkel, Kac, and Wakimoto.

Established a semi-infinite Borel-Weil theorem.

Enabled inductive analysis of admissible representations.

Abstract

We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of G-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infintie analogue of the generalized Borel-Weil theorem for mimimal parabolic subalgebras which enables an inductive study of admissible representations.