Generalized Springer correspondence and Green functions for type B/C graded Hecke algebras

This paper presents a combinatorial approach to the tempered representation theory of a graded Hecke algebra H of classical type B or C, with arbitrary parameters. We present various combinatorial results which together give a uniform combinatorial description of what becomes the Springer correspondence in the classical situation of equal parameters. More precisely, by using a general version of Lusztig's symbols which describe the classical Springer correspondence, we associate to a discrete series representation of H with central character W0c, a set Σ(W0c) of W0-characters (where W0 is the Weyl group). This set Σ(W0c) is shown to parametrize the central characters of the generic algebra which specialize into W0c. Using the parabolic classification of the central characters of on the one hand, and a truncated induction of Weyl group characters on the other hand, we define a set Σ(W0c) for any central character W0c of , and show that this property is preserved. We show that in the equal parameter situation we retrieve the classical Springer correspondence, by considering a set U of partitions which replaces the unipotent classes of SO2n+1(C) and Sp2n(C), and a bijection between U and the central characters of . We end with a conjecture, which basically states that our generalized Springer correspondence determines exactly as the classical Springer correspondence does in the equal label case. In particular, we conjecture that Σ(W0c) indexes the modules in with central character W0c, in the following way. A module M in has a natural grading for the action of W0, and the W0-representation χ(M) in its top degree is irreducible. When M runs through the modules in with central character W0c, χ(M) runs through Σ(W0c). Moreover, still in analogy with the equal parameter case, we conjecture that the W0-structure of the modules in can be computed using (generalized) Green functions.