Graded extensions of Verma modules

Abstract

In this paper, we investigate extensions between graded Verma modules in the BGG category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the $R$-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan-Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for ${\mathfrak{sl}}_4$ and construct various ``unexpected'' higher extensions between Verma modules.

3 figures; color helps

References in corpus (2)

• Bigrassmannian permutations and Verma modules
• Extensions between Verma modules for dihedral groups