An algebraic study of extension algebras

Abstract

We present simple conditions which guarantee a geometric convolution algebra to behave like a variant of the quasi-hereditary algebra. In particular, standard modules of the affine Hecke algebras of type $\mathsf{BC}$, and the quiver Schur algebras are shown to satisfy the Brauer-Humphreys type reciprocity and the semi-orthogonality property. In addition, we present a new criterion of purity of weights in the geometric side. This yields a proof of Shoji's conjecture on limit symbols of type $\mathsf{B}$ [Shoji, Adv. Stud. Pure Math. 40 (2004)], and the purity of the exotic Springer fibers [K, Duke Math. 148 (2009)]. Using this, we describe the leading terms of the $C^{\infty }$-realization of a solution of the Lieb-McGuire system in the appendix. In [K, arXiv:1203.5254], we apply the results of this paper to the KLR algebras of type $\mathsf{ADE}$ to establish Kashwara's problem and Lusztig's conjecture.

40pp, v1: separated out from arXiv:1203.5254, v2: major revision. title changed. v3: major revision. modified conditions, removed dg-algebra arguments, and Appendix B separated out. v4: minor revision. v5: assumption optimized, and explanation amplified

References in corpus (1)

• Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules

Cited by in corpus (10)

• Affine highest weight categories and affine quasihereditary algebras
• Equivariant coherent sheaves on the exotic nilpotent cone
• Geometric realizations of Lusztig's symmetries
• Peter-Weyl, Howe and Schur-Weyl theorems for current groups
• A categorical equivalence between affine Yokonuma-Hecke algebras and some quiver Hecke algebras
• $q$-Kreweras numbers for coincidental Coxeter groups attached to limit symbols
• Exotic Springer fibers for orbits corresponding to one-row bipartitions
• Springer correspndence for complex reflection groups
• Geometric realizations of Lusztig's symmetries of symmetrizable quantum groups
• Affine cellularity of quantum affine algebras