Abstract
In this chapter we study a family of multiplicity-free triples on \(\mathrm {GL}(2, \mathbb F_q)\) that generalize the well known Gelfand pair associated with the finite hyperbolic plane (see [Terras, Fourier analysis on finite groups and applications. London mathematical society student texts, vol 43. Cambridge University Press, Cambridge, 1999, Chapters 19, 20, 21, and 23]). We suppose that q is an odd prime power (cf. Sect. 3.5) and we denote by \(\widehat {\mathbb F_q^*}\) (respectively \(\widehat {\mathbb F_{q^2}^*}\)) the multiplicative characters of \(\mathbb F_q\) (respectively \(\mathbb F_{q^2}\)).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. Ceccherini-Silberstein, A. Machì, F. Scarabotti, F. Tolli, Induced representations and Mackey theory. J. Math. Sci. (N.Y.) 156(1), 11–28 (2009)
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains. Cambridge Studies in Advanced Mathematics, vol. 108 (Cambridge University Press, Cambridge, 2008)
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation Theory of the Symmetric Groups: The Okounkov–Vershik Approach, Character Formulas, and Partition Algebras. Cambridge Studies in Advanced Mathematics, vol. 121 (Cambridge University Press, Cambridge, 2010)
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Mackey’s criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups. Tohoku Math. J. 67(4), 553–571 (2015)
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)
I. Piatetski-Shapiro, Complex Representations of GL(2,K) for Finite Fields K. Contemporary Mathematics, vol. 16 (American Mathematical Society, Providence, 1983)
F. Scarabotti, F. Tolli, Fourier analysis of subgroup-conjugacy invariant functions on finite groups. Monatsh. Math. 170, 465–479 (2013)
A. Terras, in Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, vol. 43 (Cambridge University Press, Cambridge, 1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F. (2020). Harmonic Analysis of the Multiplicity-Free Triple \((\mathrm {GL}(2,\mathbb F_q), C, \nu )\) . 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_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-51607-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-51606-2
Online ISBN: 978-3-030-51607-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)