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Tensor product varieties, perverse sheaves, and stability conditions

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Abstract

The space spanned by the class of simple perverse sheaves in Zheng (2008) without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable highest weight modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. The two stability conditions coincide with the localization process in Zheng (2008), by using supports and singular supports of complexes of sheaves, respectively.

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Acknowledgments

We thank Catharina Stroppel for her invitation to University of Bonn, where this work was originated and sending us the paper [7]. We thank Wei Liang Gan for his invitation to present this work in the workshop of Lie groups, Lie algebras and their representations, University of California, Riverside, May 21-22, 2011. We also thank Igor B. Frenkel, Zongzhu Lin, Raphael Rouquier, Mark Shimozono, Josh Sussan, and Ben Webster for helpful conversations. We thank Jim Humphreys for sending the author a copy of [9]. We thank Yoshiyuki Kimura for pointing out a mistake in a previous version of this paper and sending us the paper  [14]. We are very grateful to the referee for the suggestions in improving this article. Partial support from National Science Foundation grant DMS 1101375/1160351 is acknowledged. Part of the work was done while the author was at Virginia Tech.

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  1. Department of Mathematics, University at Buffalo, State University of New York, 244 Mathematics Building, Buffalo, New York, 14260, USA

    Yiqiang Li

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  1. Yiqiang Li
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Li, Y. Tensor product varieties, perverse sheaves, and stability conditions. Sel. Math. New Ser. 20, 359–401 (2014). https://doi.org/10.1007/s00029-013-0121-y

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