Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 3

Special Issue in Honor of Claudio Procesi

Guest Editors: Luca Migliorini, Paolo Papi, and Mario Salvetti

The semi-infinite cohomology of Weyl modules with two singular points

Pages: 1251 – 1284

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n3.a6

Authors

Giorgia Fortuna (Machine Learning Reply, Torino, Italy)

Davide Lombardo (Dipartimento di Matematica, Università di Pisa, Italy)

Andrea Maffei (Dipartimento di Matematica, Università di Pisa, Italy)

Valerio Melani (Dipartimento di Matematica, Università di Firenze, Italy)

Abstract

In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module $\mathbb{V}^\lambda$ corresponding to a dominant weight $\lambda$. This object plays an important role in the theory. In $\href{ https://doi.org/10.1007/s00220-022-04430-w}{[4]}$, we introduced a possible analogue $\mathbb{V}^{\lambda,\mu}_{2}$ of the Weyl module in the setting of opers with two singular points, and in the case of $\mathfrak{sl}(2)$ we proved that it has the ‘correct’ endomorphism ring. In this paper, we compute the semi-infinite cohomology of $\mathbb{V}^{\lambda,\mu}_{2}$ and we show that it does not share some of the properties of the semi-infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this reason, we introduce a new module $\tilde{\mathbb{V}}^{\lambda,\mu}_{2}$ which, in the case of $\mathfrak{sl}(2)$, enjoys all the expected properties of a Weyl module.

Keywords

representations of affine Lie algebras, semi-infinite cohomology

2010 Mathematics Subject Classification

Primary 17B67, 17B69. Secondary 17B56.

Received 28 January 2023

Accepted 1 November 2023

Published 15 May 2024