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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

MSC:
 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