On the generalized Springer correspondence for exceptional groups.

*(English)*Zbl 0574.20029
Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 317-338 (1985).

[For the entire collection see Zbl 0561.00006.]

A connected reductive group G over an algebraically closed field is the union of its Borel subgroups, each one being its own normalizer. The variety \({\mathcal B}\) of Borel subgroups is complete and smooth. For each x in G, its centralizer \(Z_ G(x)\) acts on the subvariety \({\mathcal B}_ x\) of fixed points of x acting on \({\mathcal B}\), hence on the \(\ell\)-adic cohomology \(H^*({\mathcal B},{\mathbb{Q}}_{\ell})\); another action on this cohomology is due to the Weyl group W of G, and has been defined by T. A. Springer and G. Lusztig [Invent. Math. 44, 279-293 (1978; Zbl 0376.17002), and Adv. Math. 42, 169-178 (1981; Zbl 0473.20029), resp.]. These two actions commute. In particular, there is an action of \(W\times Z_ G(x)\) on \(H(x)=H^{top}({\mathcal B}_ x,{\mathbb{Q}}_{\ell})\), the action of the identity component \(Z_ G(x)^ 0\) being trivial. The Springer correspondence is the following: for any simple character \(\rho\) of W, there exists a unipotent u in G such that \(\rho\) appears in H(u), defined up to conjugacy; for such u, the isotypic component of \(\rho\) in H(u) is irreducible under \(W\times Z_ G(u)/Z_ G(u)^ 0\) and this injects the set of simple characters of W in the set of orbits by G in \(\{\) (u,\(\phi)\), \(\phi\) simple character of \(Z_ G(u)/Z_ G(u)^ 0\) such that \(\rho\) \(\otimes \phi\) appears in H(u)\(\}\). Moreover, each such (u,\(\phi)\) defines up to the action of G a (v,\(\psi)\) for some Levi subgroup of G and a simple character of the associated little Weyl group (generalized Springer correspondence). In this article, the author deals with the case of exceptional groups and gives an explicit description of this correspondence, except in the case when the little Weyl group is diheral of order 12 and its degree two representations.

A connected reductive group G over an algebraically closed field is the union of its Borel subgroups, each one being its own normalizer. The variety \({\mathcal B}\) of Borel subgroups is complete and smooth. For each x in G, its centralizer \(Z_ G(x)\) acts on the subvariety \({\mathcal B}_ x\) of fixed points of x acting on \({\mathcal B}\), hence on the \(\ell\)-adic cohomology \(H^*({\mathcal B},{\mathbb{Q}}_{\ell})\); another action on this cohomology is due to the Weyl group W of G, and has been defined by T. A. Springer and G. Lusztig [Invent. Math. 44, 279-293 (1978; Zbl 0376.17002), and Adv. Math. 42, 169-178 (1981; Zbl 0473.20029), resp.]. These two actions commute. In particular, there is an action of \(W\times Z_ G(x)\) on \(H(x)=H^{top}({\mathcal B}_ x,{\mathbb{Q}}_{\ell})\), the action of the identity component \(Z_ G(x)^ 0\) being trivial. The Springer correspondence is the following: for any simple character \(\rho\) of W, there exists a unipotent u in G such that \(\rho\) appears in H(u), defined up to conjugacy; for such u, the isotypic component of \(\rho\) in H(u) is irreducible under \(W\times Z_ G(u)/Z_ G(u)^ 0\) and this injects the set of simple characters of W in the set of orbits by G in \(\{\) (u,\(\phi)\), \(\phi\) simple character of \(Z_ G(u)/Z_ G(u)^ 0\) such that \(\rho\) \(\otimes \phi\) appears in H(u)\(\}\). Moreover, each such (u,\(\phi)\) defines up to the action of G a (v,\(\psi)\) for some Levi subgroup of G and a simple character of the associated little Weyl group (generalized Springer correspondence). In this article, the author deals with the case of exceptional groups and gives an explicit description of this correspondence, except in the case when the little Weyl group is diheral of order 12 and its degree two representations.

Reviewer: P.GĂ©rardin

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

20G15 | Linear algebraic groups over arbitrary fields |