Abstract
In this chapter we study an example of a multiplicity-free triple where the representation that we induce has dimension greater than one.
Let q = p h with p an odd prime and h ≥ 1. Set
Moreover (cf. Sect. 3.5), we denote by B j (resp. U j, resp. C j) the Borel (resp. the unipotent, resp. the Cartan) subgroup of G j, for j = 1, 2. Throughout this chapter, with the notation as in Sect. 5.3, we let \(\nu \in \widehat {\mathbb F_{q^2}^*}\) be a fixed indecomposable character. We assume that \(\nu ^\sharp = \mathrm {Res}^{\mathbb F_{q^2}^*}_{\mathbb F_{q}^*} \nu \) is not a square: this slightly simplifies the decomposition into irreducibles. Finally, ρ ν denotes the cuspidal representation of G 1 associated with ν.
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References
T. Ceccherini-Silberstein, F.Scarabotti, F.Tolli, Discrete Harmonic Analysis: Representations, Number Theory, Expanders, and the Fourier Transform. Cambridge Studies in Advanced Mathematics, vol. 172 (Cambridge University Press, Cambridge, 2018)
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Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F. (2020). Harmonic Analysis of the Multiplicity-Free Triple \((\mathrm {GL}(2,\mathbb F_{q^2}), \mathrm {GL}(2,\mathbb F_{q}), \rho _\nu )\) . In: Gelfand Triples and Their Hecke Algebras. Lecture Notes in Mathematics, vol 2267. Springer, Cham. https://doi.org/10.1007/978-3-030-51607-9_6
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DOI: https://doi.org/10.1007/978-3-030-51607-9_6
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