Supercharacters and generalized Gelfand-Graev characters for orthogonal groups

Abstract

The generalized Gelfand-Graev characters defined by N. Kawanaka are an important tool for the analysis of unipotent characters of finite groups of Lie type. But their construction is notoriously difficult, as among other things it relies on a not uniquely defined unipotent subgroup of the underlying algebraic group. Therefore, it was of great benefit that S. Andrews and N. Thiem were able to construct generalized Gelfand-Graev characters for the finite special linear group from supercharacters of the finite group of unitriangular matrices. An immediate idea is to expand this connection between generalized Gelfand-Graev characters and supercharacter theory to other finite groups of Lie type. We will investigate the case of the finite special orthogonal group of even dimension in good characteristic. This supercharacter theory of the finite group of unitriangular matrices was originally introduced by N. Yan for the purpose of approximating the classification of the irreducible characters of this group, which in itself is a wild problem. It centers around a 1-cocycle from the group of unitriangular matrices onto its algebra. Expanding their utilization to other finite groups of Lie type, C. A. M. André and A. M. Neto defined such supercharacter theories for the maximal unipotent subgroups of the finite groups of type Bn, Cn and Dn by using so-called elementary characters which are induced from linear characters of root subgroups. Reintroducing the use of a 1-cocycle M. Jedlitschky decomposed the supercharacters of C. A. M. André and A. M. Neto for the finite special orthogonal group of even dimension in good characteristic, which no longer form a supercharacter theory owing to not having an accompanying set of superclasses but still retain the other properties of a supercharacter theory. For the classification of these characters, we can define a derivation of a Gram matrix for each such character, which not only allows us to identify identical characters but also contains the information about the irreducibility of such characters. While S. Andrews and N. Thiem were able to establish the generalized Gelfand-Graev characters of the finite special linear group directly from supercharacters of its maximal unipotent subgroup, avoiding the laborious construction defined by N. Kawanaka, the extension of the link between supercharacter theories and generalized Gelfand-Graev characters to the case of the special orthogonal group of even dimension is not immediately possible, as the supercharacters defined by C. A. M. André and A. M. Neto in general do not fit this purpose. Yet, with the aid of the aforementioned Gram matrix, we can use the constituents of these supercharacters defined by M. Jedlitschky to produce the generalized Gelfand-Graev characters of the finite special orthogonal group.