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

TL;DR

This paper investigates the Ext-algebra of parabolic Verma modules in category O for a hermitian symmetric pair, describing its quiver structure and A infinity-structure, with explicit examples and combinatorial results.

Contribution

It provides a detailed description of the Ext-algebra and A infinity-structure for specific cases, using Kazhdan-Lusztig combinatorics and explicit computations.

Findings

Quiver with relations for n=1, 2 cases

Vanishing of higher multiplications in general

Explicit example of non-vanishing m_3

Abstract

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 $(\mathfrak{gl}_{n+m}, \mathfrak{gl}_{n} \oplus \mathfrak{gl}_m)$ 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 infinity-structure of a minimal model. An explicit example of the higher multiplications with non-vanishing $m_3$ is included.