2013/03/27 by Dan Ciubotaru, Ciubotaru, Dan, Xuhua He +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1303.6806
40 pages, added references, corrections to Appendix A
openalex publication_date 2013/03/27 · arxiv created 2013/05/07 · arxiv updated 2013/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover \wti W, a certain double cover of the Weyl group W, and an extended Dirac operator for graded Hecke algebras. Our approach leads to a new and uniform construction of the irreducible genuine \wti W-characters. In the process, we give a construction of the action by an outer automorphism of the Dynkin diagram on the cohomology groups of Springer theory, and 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 AW=\bC[W]\ltimes S(V).