Extensions, Levi subgroups and character formulas

Abstract

This paper consists of three interconnected parts. Parts I, III study the relationship between the cohomology of a reductive group G and that of a Levi subgroup H. For example, we provide a sufficient condition, arising from Kazhdan–Lusztig theory, for a natural map Ext G • ( L , L ′ ) → Ext H • ( L H , L H ′ ) to be surjective, given irreducible G-modules L , L ′ and corresponding irreducible H-modules L H , L H ′ . In cohomological degree n = 1 , the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer [D. Hemmer, A row removal theorem for the Ext 1 quiver of symmetric groups and Schur algebras, Proc. Amer. Math. Soc. 133 (2) (2005) 403–414 (electronic)] for G = GL n , and both extend and improve upon the latter when our condition is met. Part II obtains results on Lusztig character formulas (LCFs) for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of G = GL n , these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group. This work on GL n improves upon [B. Parshall, L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (3) (2005) 655–688], which established only sufficient, rather than necessary and sufficient, conditions for the validity of the LCF in terms of the symmetric group.