Spherical functions on a finite affine space with a series of Zelevinsky subgroups
In the complete linear group \(G_n= GL(n,F)\) over a finite field \(F\) the Zelevinsky series [cf. \textit{A. V. Zelevinsky}, Representations of finite classical groups. A Hopf algebra approach. Lect. Notes Math. 869 (Berlin etc. 1981; Zbl 0465.20009)] \(G_{n-1/2} \supset G_{n-1} \supset\cdots \supset G_{3/2} \supset G_1\) of subgroups \(G_{n-i +1/2}\), \(G_{n-i}\), \(i=1,2, \dots, n-1\), is given. The series generates a simple branching quasiregular representation of the group \(G_n\) associated with the homogeneous space \(G_n/G_{n-1/2}\). The spherical subgroup \(G_{n-1/2}\) separates the subspace of spherical functions, and the series introduces a Gel'fand-Tsetlin basis of basic spherical functions. In the paper the explicit form of these spherical functions is found.
scientific article; zbMATH DE number 846503