On the Ext algebras of parabolic Verma modules and A∞-structures

We study the Ext-algebra of the direct sum of all parabolic Verma modules in the principal block of the Bernstein–Gelfand–Gelfand category O for the Hermitian symmetric pair (gln+m,gln⊕glm) and present the corresponding quiver with relations for the cases n=1,2. The Kazhdan–Lusztig combinatorics is used to deduce a general vanishing result for the higher multiplications in the A∞-structure of a minimal model. An example of higher multiplications with non-vanishing m3 is included.