Let G be a reductive connected algebraic group over an algebraically closed field, and let u be a unipotent element of G. Let \(A_ G(u)\) be the group of components of the centralizer \(Z_ G(u)\). \(A_ G(u)\) acts naturally by permutations on the set of irreducible components of the variety of Borel subgroups containing u and Springer has shown that (with some restrictions on the characteristic) the irreducible representations of \(A_ G(u)\) appearing in this permutation representation for various u (up to conjugacy) are in 1-1 correspondence with the irreducible representations of the Weyl group. However, in general, not all irreducible representations of \(A_ G(u)\) appear in this permutation representation. In this paper, the author investigates the missing representations.
Let P be a parabolic subgroup of G with Levi decomposition \(P=LU_ P\), and let v be a unipotent element in L. Let \(Y_{u,v}=\{gZ^ 0_ L(v)U_ p|\quad g\in G,\quad g^{-1}ug\in vU_ P\}.\) Then \(\dim Y_{u,v}\leq d=1/2(\dim Z_ G(u)-\dim Z_ L(v)).\) The group \(Z_ G(u)\) acts naturally on \(Y_{u,v}\) by left translation. This induces an action of the finite group \(A_ G(u)\) on the finite set \(S_{u,v}\) of irreducible components of dimension d of \(Y_{u,v}\). When P is a Borel subgroup and \(v=1\), this is just the action considered by Springer. An irreducible representation of \(A_ G(u)\) is said to be cuspidal if it does not appear in the permutation representation on \(S_{u,v}\) for any \(P\), \(v\) as above with \(P\neq G\). The author shows that very few representations of \(A_ G(u)\) are cuspidal. More precisely, for a fixed character \(\chi\) of the group \(\Gamma\) of components of the center of G, and for a field k of good characteristic, there is at most one pair \((u,\rho)\) with u unipotent in G (up to conjugacy) such that \(\rho\) is an irreducible cuspidal representation of \(A_ G(u)\) on which \(\Gamma\) acts according to \(\chi\). Given a pair \((u,\rho)\), the author defines a triple \((L,v,\rho')\) up to conjugacy, where \(L\) is the Levi subgroup of a parabolic subgroup of \(G\), \(v\) a unipotent element in \(L\), and \(\rho'\) is a cuspidal representation of \(A_ L(v)\), and he shows that the set of pairs \((u,\rho)\) giving rise to a fixed triple \((L,v,\rho')\) as above may be naturally put into 1-1 correspondence with the set of irreducible representations of the group of components of the normalizer of \(L\) which is shown to be a Coxeter group. It reduces to the correspondence described originally by Springer, in the case where \(L\) is a maximal torus, and it is called generalized Springer correspondence. The author determines in a combinatorial way this correspondence in the case of symplectic and special orthogonal groups in odd characteristic. This generalizes the main result of Shoji on the usual Springer correspondence for these groups. Throughout the paper, the intersection cohomology theory of Deligne-Goresky-MacPherson is used extensively. Using the result of this paper, by Spaltenstein, the correspondence for exceptional groups in arbitrary characteristic has also been determined explicitly in almost all cases.