Title: Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category,
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.