Harmonic analysis on the finite symmetric space \(\mathrm{GL}(n,K)/\mathrm{GL}(1,K)\times \mathrm{GL}(n-1,K)\)
This paper gives a classification of all irreducible spherical representations of the group \(G = \mathrm{GL}(n,K)\) with respect to the subgroup \(H = \mathrm{GL}(1,K) \times \mathrm{GL}(n - 1,K)\). Here \(K\) denotes a finite field of odd characteristic. The spherical functions associated to these representations are also computed. Among other results, the author shows that \((G,H)\) is not a Gelfand pair and there are no irreducible spherical cuspidal representations for \(n > 2\). Cusp forms and cuspidal representations play an important role in proofs following some ideas of \textit{Harish-Chandra} [Collected Papers. Vol. IV, Springer, Berlin, 8--20 (1984)]. As the author mentions, this work may be considered as a particular case of some results of \textit{G. Lusztig} [Prog. Math. 88, 57--81 (1990; Zbl 0773.20012)]. These ideas may be used in related matters of harmonic analysis on finite symmetric spaces.
irreducible spherical cuspidal representations
scientific article; zbMATH DE number 792822