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Generalized Springer correspondence and Green functions for type \(B/C\) graded Hecke algebras. (English) Zbl 1151.20004

Summary: This paper presents a combinatorial approach to the tempered representation theory of a graded Hecke algebra \(\mathbb{H}\) 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 \(\mathbb{H}\) with central character \(W_0c\), a set \(\Sigma(W_0c)\) of \(W_0\)-characters (where \(W_0\) is the Weyl group). This set \(\Sigma(W_0c)\) is shown to parametrize the central characters of the generic algebra which specialize into \(W_0c\). Using the parabolic classification of the central characters of \(\widehat\mathbb{H}^t(\mathbb{R})\) on the one hand, and a truncated induction of Weyl group characters on the other hand, we define a set \(\Sigma(W_0c)\) for any central character \(W_0c\) of \(\widehat\mathbb{H}^t(\mathbb{R})\), 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 \(\mathcal U\) of partitions which replaces the unipotent classes of \(\text{SO}_{2n+1}(\mathbb{C})\) and \(\text{Sp}_{2n}(\mathbb{C})\), and a bijection between \(\mathcal U\) and the central characters of \(\widehat\mathbb{H}^t(\mathbb{R})\). We end with a conjecture, which basically states that our generalized Springer correspondence determines \(\widehat\mathbb{H}^t(\mathbb{R})\) exactly as the classical Springer correspondence does in the equal label case. In particular, we conjecture that \(\Sigma(W_0c)\) indexes the modules in \(\widehat\mathbb{H}^t(\mathbb{R})\) with central character \(W_0c\), in the following way. A module \(M\) in \(\widehat\mathbb{H}^t(\mathbb{R})\) has a natural grading for the action of \(W_0\), and the \(W_0\)-representation \(\chi(M)\) in its top degree is irreducible. When \(M\) runs through the modules in \(\widehat\mathbb{H}^t(\mathbb{R})\) with central character \(W_0c\), \(\chi(M)\) runs through \(\Sigma(W_0c)\). Moreover, still in analogy with the equal parameter case, we conjecture that the \(W_0\)-structure of the modules in \(\widehat\mathbb{H}^t(\mathbb{R})\) can be computed using (generalized) Green functions.

MSC:

20C08 Hecke algebras and their representations
22E35 Analysis on \(p\)-adic Lie groups
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20G05 Representation theory for linear algebraic groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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References:

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