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

We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra g, the endomorphism algebra B of any projective-injective module of the parabolic BGG category O_λ ^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_λ ^p of the basic projective-injective module of O_λ ^p is a symmetric algebra.