Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index

Dan Ciubotaru Xuhua He

Representation Theory mathscidoc:1912.43212

Advances in Mathematics, 283, 1-50, 2015.10
In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W of a Weyl group W, and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W, and an extended Dirac operator for graded Hecke algebras. We also introduce a q-elliptic pairing for W with respect to the reflection representation V. These constructions are of independent interest. The q-elliptic pairing is a generalization of the elliptic pairing of W introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product A W= C [W] S (V).
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@inproceedings{dan2015green,
  title={Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index},
  author={Dan Ciubotaru, and Xuhua He},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112955508865772},
  booktitle={Advances in Mathematics},
  volume={283},
  pages={1-50},
  year={2015},
}
Dan Ciubotaru, and Xuhua He. Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index. 2015. Vol. 283. In Advances in Mathematics. pp.1-50. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112955508865772.
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