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Green functions for groups of types \(E_ 6\) and \(F_ 4\) in characteristic \(2\). (English) Zbl 0815.20033
Let \(\mathbf G\) be a reductive algebraic group defined over \(F_ q\), \(F : {\mathbf G} \to {\mathbf G}\) a Frobenius morphism and \(G\) the finite group of \(F\)-fixed points of \(\mathbf G\). If \(\mathbf G\) is connected, the Green functions of \(G\) are functions \(Q_ w\) on the unipotent elements of \(G\), indexed by representatives \(w\) of \(F\)-conjugacy classes of the Weyl group \(W\). If \(\mathbf G\) is disconnected and obtained from \({\mathbf G}^ 0\) by adjoining a graph automorphism \(\sigma\) of prime order, the author [J. Algebra 159, 64-97 (1993; Zbl 0812.20024)] and F. Digne and J. Michel [Ann. Sci. Ec. Norm. Supér., IV. Sér. 27, 345-406 (1994)] have studied outer Green functions which are defined on the unipotent classes of \(G^ 0 \sigma\) and are indexed by the classes of \(W^ \sigma\). These outer Green functions have properties such as orthogonality relations analogous to the usual Green functions. The Green functions are known for connected exceptional groups in good characteristics, and these methods do not carry over to the bad characteristic case. In [loc.cit.] the author determined the outer Green functions for groups of types \(A_ \ell(2^ n) \cdot 2\) \((\ell \leq 5)\), \(D_ 4(2^ n) \cdot 2\) and \(D_ 4(3^ n) \cdot 3\). In this paper he computes the Green functions for the groups \(E_ 6(2^ n)\), \(^ 2E_ 6(2^ n)\) and \(F_ 4(2^ n)\) and the outer Green functions for the groups \(E_ 6(2^ n) \cdot 2\) and \(^ 2E_ 6(2^ n) \cdot 2\). Since giving the values of these Green functions is equivalent to giving the values of certain “almost characters” \(R_ \chi\) where \(\chi\) is an irreducible character of \(W\) in the connected case and an irreducible character of \(W^ \sigma\) in the disconnected case, he gives the values of the almost characters in tables. His method is to use the known Green functions on a parabolic subgroup \(Q\) and induce them to the group \(G\), and then to use the orthogonality relations for the missing functions. In the process of induction he uses an interpolation argument to avoid the calculation of the fusion of all the unipotent classes of \(Q\) in \(G\). In the case of the twisted disconnected groups he uses a Shintani descent argument. The Green functions that he computes satisfy properties such as Ennola duality that were known in the good characteristic case.

MSC:
20G05 Representation theory for linear algebraic groups
20C33 Representations of finite groups of Lie type
20G40 Linear algebraic groups over finite fields
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