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Supercharacters and generalized Gelfand-Graev characters for orthogonal groups
Get PDFThe 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
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Representations of Finite Groups
Get PDFThe workshop Representations of Finite Groups was organised by Joseph Chuang (London), Meinolf Geck (Stuttgart), Radha Kessar (London) and Gabriel Navarro (Valencia). It covered a wide variety of aspects of representation theory of finite groups and its relations to other areas of mathematics
Representations of Finite Groups
No full textThe workshop Representations of Finite Groups was organised by Joseph Chuang (London), Meinolf Geck (Stuttgart), Radha Kessar (London) and Gabriel Navarro (Valencia). It covered a wide variety of aspects of representation theory of finite groups and its relations to other areas of mathematics
Fourier transforms and Frobenius eigenvalues for finite Coxeter groups
No full textAbstractLusztig's classification of the unipotent characters of a finite Chevalley or Steinberg group involves a certain non-abelian Fourier transformation. We construct analogous transformations for the Suzuki and Ree groups, based on a set of axioms derived from Lusztig's theory of character sheaves. We also determine Fourier matrices for the “spetses” (in the sense of Broué, Michel, and the second author) associated with twisted dihedral groups. This completes the determination of Fourier matrices for all “spetses” associated with finite Coxeter groups. We end by collecting common properties of these Fourier matrices and the eigenvalues of Frobenius of character sheaves and unipotent characters
On cuspidal and supercuspidal representations of finite reductive groups
No full textThis thesis is about the modular representation theory of finite reductive groups in non-defining characteristic and, in particular, about the development of methods which lead to a better understanding of cuspidal and especially supercuspidal representations of these groups. By a conjecture by Gerhard Hiss, every irreducible supercuspidal representation lifts to a representation in characteristic 0. An even stronger conjecture by Meinolf Geck states that the reductions modulo l of certaincuspidal irreducible representations are also irreducible. An important part of this thesis is the reduction of these two conjectures to the case where the algebraic group to be considered is simple
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Representations of Finite Groups
Get PDFThe workshop Representations of Finite Groups was organised by Joseph Chuang (London), Meinolf Geck (Stuttgart), Markus Linckelmann (London), and Gabriel Navarro (Valencia). It covered a wide variety of aspects of the representation theory of finite groups and related objects, such as algebraic groups
On the decomposition numbers of the finite unitary groups in non-defining characteristic.
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