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

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.