By Mochizuki S.

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**Example text**

G(x)O in this way. When U EGis unipotent the groups ~G(u)/~G(u)O have been determined by Alexeevski [1]. If G has type Al then ~G(u)/~G(u)O = 1. If G has type BI, CI, DI then ~du)/~G(u)O is isomorphic to 'lL2 x 'lL2 X ... X 'lL2 (e factors) for some e depending on u. If G has type G2, F4, E6, E7, E8 then ~G(u)/~G(u)O is isomorphic to the symmetric group Sn for some n ::;; 5 depending on u. If the Frobenius map acts trivially on the Dynkin diagram of G then every unipotent class of G is F-stable, although this is not always true when the Frobenius map acts non-trivially on the diagram.

The details can be found in Steinberg [3]. Thus all irreducible components of the Gelfand-Graev character r occur with multiplicity 1. In order to determine how many such components there are we consider the dual E = r*. E has a very simple form. It is the generalized character of GF which takes the value IZFlql on all the regular unipotent elements of GF and value 0 on all other elements of GF • (An element is regular if and only if its centralizer has dimension equal to the rank of G). Now the number of regular unipotent elements of GF can be shown to be IGFI/IZFlql.

In the first place we have 1* = St. Thus the dual of the principal character is the Steinberg character. It follows of course that St* = 1. Secondly we take a Deligne-Lusztig generalized character R T ,6 of GF • Then we have R},6 = BGBT R T ,6' This result was proved by Deligne and Lusztig in [2]. The dual of an irreducible character ~ of GF need not be irreducible. 6 The Gelfand-Graev Character of GF We shall continue to assume that the centre Z of G is connected. Let G* be the dual group of G and (G*)' be the semisimple part of G*.