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 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 B [T. Shoji, Adi). Stud. Pure Math. 40 (2004)], and the purity of the exotic Springer fibers [S. Kato, Duke Math. J. 148 (2009)]. Using this, we describe the leading terms of the C-infinity-realization of a solution of the Lieb-McGuire system in the appendix. In [S. Kato, Duke Math. J. 163 (2014)], we apply the results of this paper to the KLR algebras of type ADE to establish Kashwara's problem and Lusztig's conjecture.