Abstract
In this paper, we compute the conjugacy classes and the list of irreducible characters of \(\mathrm{GSp}(4,q)\), where \(q\) is odd. We also determine precisely which irreducible characters are non-cuspidal and which are generic. These characters are then used to compute dimensions of certain subspaces of fixed vectors of smooth admissible non-supercuspidal representations of \(\mathrm{GSp}(4,F)\), where \(F\) is a non-archimedean local field of characteristic zero with residue field of order \(q\).
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Notes
The referee has informed us that Shinoda had independently obtained the conjugacy classes [17] and the complete character table [16] of \(\mathrm{GSp}(4,q)\), which he calls \(\mathrm{CSp}(4,q),\) using the modern approach of Deligne-Lusztig theory. Shinoda remarks at the end of the introduction in [16] that Reid had also obtained this character table in [12]. We were not aware of these results when doing the computations in this paper. We give a correspondence between our notations and Shinoda’s notations for the conjugacy classes and the irreducible characters of \(\mathrm{GSp}(4,q)\) in Sects. 3 and 5, respectively.
There is a misprint in [18], page 523. Note that \(\dim (\theta _1)\) is indeed \(\frac{1}{2}q^2(q^2+1).\)
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Acknowledgments
The author would like to thank Ralf Schmidt and Alan Roche for their valuable notes and comments on this paper. The author would especially like to thank the referee for the detailed comments and careful reading of this paper.
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Breeding, J. Irreducible characters of \(\mathrm{GSp}(4, q)\) and dimensions of spaces of fixed vectors. Ramanujan J 36, 305–354 (2015). https://doi.org/10.1007/s11139-014-9622-3
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DOI: https://doi.org/10.1007/s11139-014-9622-3