An algebraic study of extension algebras

We present simple conditions which guarantee a geometric extension algebra to behave like a variant of quasi-hereditary algebras. In particular, standard modules of affine Hecke algebras of type $\sf{BC}$, 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 $\sf{B}$ [T. Shoji, {\it Adv. Stud. Pure Math.} 40 (2004)], and the purity of the exotic Springer fibers [S. Kato, {\it Duke Math. J.} 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 [S. Kato, {\it Duke Math. J.} 163 (2014)], we apply the results of this paper to the KLR algebras of type $\sf{ADE}$ to establish Kashwara's problem and Lusztig's conjecture.