扩张代数的代数研究
An algebraic study of extension algebras
摘要：
我们提出了简单的条件，保证几何卷积代数的行为像一个变种的拟遗传代数。特别地，证明了$\mathsf{BC}$型仿射Hecke代数的标准模，以及Schur-Schur代数的标准模满足Brauer-Humphreys型互易性和半正交性.此外，我们提出了一个新的准则的纯度的重量在几何方面。这证明了Shoji关于$\mathsf{B}$型极限符号的猜想[Shoji，Adv. Stud. Pure Math.40（2004）]，以及奇异Springer纤维的纯度[K，Duke Math.148（2009）]。利用这一点，我们描述了首项的$C^{\infty}$ -实现的解决方案的利勃-麦圭尔系统的附录。在[K，arXiv：1203.5254]中，我们将本文的结果应用于$\mathsf{ADE}$型KLR代数，建立了Kashwara问题和Lusztig猜想.
英文摘要：
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.
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