On defining characteristic representations of finite reductive groups.

*(English)*Zbl 1290.20011From the introduction: We consider series of finite groups of Lie type which are specified by a root datum and a finite order automorphism of that root datum. For each power \(q\) of a prime \(p\) this determines a connected reductive algebraic group \(\mathbf G\) over \(\overline{\mathbb F}_p\) (an algebraic closure of the field with \(p\) elements) and a group of fixed points \(\mathbf G^F\) of a Frobenius morphism \(F\colon\mathbf G\to\mathbf G\), up to isomorphism.

We are interested in a parameterization of the irreducible modules of \(\mathbf G^F\) over \(\overline{\mathbb F}_p\).

In the literature on representations of connected reductive algebraic groups and finite groups of Lie type in their defining characteristic most authors restrict their descriptions to the case of simply-connected algebraic groups and the finite groups of Lie type arising from these.

In this paper we give a parameterization of the irreducible representations in defining characteristic for arbitrary finite groups of Lie type. It is very concrete and computable starting from the given root datum for the algebraic group and Frobenius action on the root datum. The description will not become more complicated for twisted Frobenius actions.

Here is an overview of the content of the other sections of this paper. Section 2 contains a description of our setup. We describe how root data and Frobenius actions on root data can be represented and how to compute certain related data. Some of the results in this section may be of independent interest. For example, we describe a construction of a certain covering group of an arbitrary connected reductive group, which generalizes the well-known simply-connected coverings of semisimple groups (see Proposition 2.5).

In Section 3 we first recall the results about defining characteristic representations of the algebraic groups and the finite groups of Lie type arising from simply-connected semisimple groups which we have mentioned above. Then we state our main result in Theorem 3.5 where we consider arbitrary finite groups of Lie type. In the end of that section we work out an example in some detail (certain centralizers of semisimple elements in exceptional groups of type \(E_8\)).

In Section 4 we give a more detailed description of the parameter sets in our main theorem for finite groups of Lie type arising from any simple connected reductive group. As an application of these results we work out the number of semisimple conjugacy classes for all of these finite groups. The results of this application were obtained before by the first named author with a completely different proof. The new proof given here is more elementary.

We are interested in a parameterization of the irreducible modules of \(\mathbf G^F\) over \(\overline{\mathbb F}_p\).

In the literature on representations of connected reductive algebraic groups and finite groups of Lie type in their defining characteristic most authors restrict their descriptions to the case of simply-connected algebraic groups and the finite groups of Lie type arising from these.

In this paper we give a parameterization of the irreducible representations in defining characteristic for arbitrary finite groups of Lie type. It is very concrete and computable starting from the given root datum for the algebraic group and Frobenius action on the root datum. The description will not become more complicated for twisted Frobenius actions.

Here is an overview of the content of the other sections of this paper. Section 2 contains a description of our setup. We describe how root data and Frobenius actions on root data can be represented and how to compute certain related data. Some of the results in this section may be of independent interest. For example, we describe a construction of a certain covering group of an arbitrary connected reductive group, which generalizes the well-known simply-connected coverings of semisimple groups (see Proposition 2.5).

In Section 3 we first recall the results about defining characteristic representations of the algebraic groups and the finite groups of Lie type arising from simply-connected semisimple groups which we have mentioned above. Then we state our main result in Theorem 3.5 where we consider arbitrary finite groups of Lie type. In the end of that section we work out an example in some detail (certain centralizers of semisimple elements in exceptional groups of type \(E_8\)).

In Section 4 we give a more detailed description of the parameter sets in our main theorem for finite groups of Lie type arising from any simple connected reductive group. As an application of these results we work out the number of semisimple conjugacy classes for all of these finite groups. The results of this application were obtained before by the first named author with a completely different proof. The new proof given here is more elementary.

##### MSC:

20C33 | Representations of finite groups of Lie type |

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

##### Keywords:

finite groups of Lie type; finite reductive groups; parameterizations of irreducible representations; defining characteristic representations; numbers of semisimple classes; root data; groups of fixed points; Frobenius morphisms; irreducible modules; Frobenius actions; covering groups##### References:

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