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Gelfand-Graev characters of the finite unitary groups. (English) Zbl 1202.20016

The authors provide the values of the Gelfand-Graev characters of the finite unitary group at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand-Graev characters, and present some consequences regarding multiplicity. These results were obtained through the use of symmetric functions and combinatorial arguments.
If \(B\) is a maximal unipotent subgroup of a finite group of Lie type \(G\), then the Gelfand-Graev character is obtained by inducing a generic linear character from \(B\) to \(G\). The degenerate Gelfand-Graev characters are obtained by inducing arbitrary linear characters. The authors build on the results of A. V. Zelevinsky [Representations of finite classical groups. A Hopf algebra approach. Lect. Notes Math. 869. Berlin-Heidelberg-New York: Springer-Verlag (1981; Zbl 0465.20009)]. The authors use the characteristic map of the finite unitary group as their main combinatorial tool. The map translates the Deligne-Lusztig theory of finite unitary groups into symmetric functions and allow for combinatorial arguments to be used to establish the results.
In Section 3, the authors study the non-degenerate Gelfand-Graev character and use the formula for the character values of the Gelfand-Graev character of \(\text{GL}(n,\mathbb{F}_q)\) to obtain a corresponding formula for \(\text{U}(n,\mathbb{F}_{q^2})\). Section 4 uses tableau combinatorics to compute the decomposition of degenerate Gelfand-Graev characters. The degenerate character decomposes as the sum over \(\lambda\) of nonnegative integer multiples of \(\chi^\lambda\), where \(\lambda\) is a multipartition. In Theorem 5.2, the authors give combinatorial conditions on the multipartition \(\lambda\) to guarantee that the irreducible character \(\chi^\lambda\) appears with multiplicity one in some degenerate Gelfand-Graev character.

MSC:

20C33 Representations of finite groups of Lie type
05E05 Symmetric functions and generalizations
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields

Citations:

Zbl 0465.20009
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