Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

TL;DR

This paper proves that the endomorphism algebra of projective-injective modules in parabolic category O for singular weights is symmetric, extending previous results and confirming a conjecture by Khovanov.

Contribution

It establishes the symmetry of endomorphism algebras for singular weights in parabolic category O, generalizing prior work for regular weights and providing explicit symmetrizing forms.

Findings

Endomorphism algebra is symmetric for singular weights.

Extension of Mazorchuk and Stroppel's results to singular weights.

Existence of a homogeneous symmetrizing form.

Abstract

We show that for any singular dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category $\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra $B$ is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra $B_\lambda^{\mathfrak{p}}$ of the basic projective-injective module of $\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra.