There are many important recent works (for example, by Lusztig, Bezrukavnikov-Finkelberg-Ostrik, Ben-Zvi-Nadler, Boyarchenko-Drinfeld, Lusztig-Yun, Vilonen-Xue) on character sheaves (which are certain perverse sheaves on an algebraic group $G$ over an algebraically closed field that were originally introduced by Lusztig), where the structure and properties of various categories of character sheaves are studied. What are the established or conjectural applications (for example, to algebraic or geometric representation theory or link homology) of the theory of character sheaves (and in particular, the results on character sheaves mentioned above) other than Lusztig's original motivation of determining the characters of finite groups of Lie type $G(\mathbb F_q)$?

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    $\begingroup$ You can probably find some insights in the Math Reviews of Lusztig's papers by Bhama Srinivasan, a former student of J.A. Green; she got heavily involved with the finite groups of Lie type in particular. $\endgroup$ Jan 12, 2020 at 2:08
  • $\begingroup$ Relation of character sheaves to physics is discussed in Section 9.1.3 of arxiv.org/abs/1810.10652 $\endgroup$
    – Yellow Pig
    Feb 1, 2020 at 4:28


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