An algebraic study of extension algebras

TL;DR

This paper establishes algebraic and geometric conditions under which certain convolution algebras behave like quasi-hereditary algebras, proving conjectures and properties related to affine Hecke algebras, Springer fibers, and KLR algebras.

Contribution

It introduces new criteria for algebraic structures to exhibit quasi-hereditary behavior and proves several conjectures in geometric representation theory.

Findings

Standard modules satisfy Brauer-Humphreys reciprocity.

Proved Shoji's conjecture on limit symbols of type B.

Established purity of exotic Springer fibers.

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.