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On the generalized Springer correspondence for classical groups. (English) Zbl 0579.20035
Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 289-316 (1985).
[For the entire collection see Zbl 0561.00006.]
Let k be a field of characteristic \(p>0\), \(\ell\) a prime distinct from p and G a connected reductive algebraic group defined over k. For any element \(x\in G\), let \(A(x)=C_ G(x)/C^ 0_ G(x)\). In [Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], T. A. Springer defined an action of the Weyl group W of G on the \(\ell\)-adic cohomology \(H^*_ c({\mathcal B}_ u,{\bar {\mathbb{Q}}}_{\ell})\) of \({\mathcal B}_ u=\{Borel\) subgroups \(B\subset G|\) \(u\in B\}\) where u is a unipotent element of G. The top cohomology has a (linear) basis, the set of irreducible components of \({\mathcal B}_ u\), and therefore A(u) acts on it. The A(u)- isotopic components of \(H_ c^{top}({\mathcal B}_ u,{\bar {\mathbb{Q}}}_{\ell})\) turn out to be irreducible \({\bar {\mathbb{Q}}}_{\ell}W\)-modules. This sets up the Springer correspondence \({\mathcal N}_ G\to W{\hat{\;}}\), where \({\mathcal N}_ G\) is the set of G- conjugacy classes of pairs (u,\(\phi)\) where u is a unipotent element of G and \(\phi\) is a character of A(u).
This correspondence is not bijective, and G. Lusztig [ibid. 75, 205-272 (1984; Zbl 0547.20032)] extended it to a bijection \(\rho^ G: {\mathcal N}_ G\to \amalg (N_ G(M)/M){\hat{\;}}\) where M runs over the conjugacy classes of Levi subgroups of parabolic subgroups of G, and the characters arising are those which come from ”cuspidal” pairs (u’,\(\phi\) ’) where u’ is a unipotent element of M and \(\phi '\in (A_ M(u')){\hat{\;}}.\)
The paper under review computes the correspondence \(\rho^ G\) explicitly for the only (classical) cases which remained viz. \(Sp_{2n}(k)\) and \(SO_{2n}(k)\) in characteristic 2 and \(Spin_ n(k)\) in odd characteristic.
The techniques revolve around the authors’ concept of induced unipotent classes and explicit knowledge of the inverse image of 1 and \(sgn\in (N_ G(M)/M){\hat{\;}}\). These, together with the restriction formula of Lusztig [loc. cit.] enable the authors to identify explicitly the characters which occur combinatorially.
Reviewer: G.I.Lehrer

20G05 Representation theory for linear algebraic groups
20G10 Cohomology theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
20G40 Linear algebraic groups over finite fields