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.


Additional Information

Print ISSN
pp. 567-615
Launched on MUSE
Open Access
Back To Top

This website uses cookies to ensure you get the best experience on our website. Without cookies your experience may not be seamless.