Let \(H_n=H_n(\mathbb R)\) be the Heisenberg group which can be identified with \(\mathbb R^{n}\times\mathbb R^{n} \times\mathbb R=\mathbb C^n \times\mathbb R\). The action of the unitary group \(U(n)=U(n,\mathbb C)\) on the Heisenberg group \(H_n\) yields a Gelfand pair \((U(n) \ltimes H_n,U(n))\) whose bounded spherical functions are well known where \(U(n) \ltimes H_n\) is the semidirect product of \(U(n)\) and \(H_n\). In this paper the authors derive a discrete counterpart by replacing the real numbers by a finite field of odd characteristic.
Let \(\mathbb F=\mathbb F_{q}\) denote the field with \(q\) elements where \(q=p^m\) for some odd prime \(p\) and positive integer \(m\). Choose any non-square \(\varepsilon \in \mathbb F^{\times} \setminus\mathbb F^{\times})^2\) in \(\mathbb F\) and form the quadratic extension field \(\widetilde{\mathbb F}=\mathbb F(\sqrt{\varepsilon})\).
We adapt complex notation to \(\widetilde{\mathbb F}\), writing, for \(z=x+y\sqrt{\varepsilon} \in \widetilde{\mathbb F}\), \(\bar{z}=x-y\sqrt{\varepsilon}\).
\[ \langle {\mathbf z},{\mathbf z}' \rangle =z_{1}\bar z'_{1}+\cdots +z_{n}\bar z'_{n} \]
is a Hermitian inner product on \(\widetilde{\mathbb F}^n\). The unitary group \(U(n,\widetilde{\mathbb F})\) is the set of \(\widetilde{\mathbb F}\)-linear transformations on \(\widetilde{\mathbb F}^n\) preserving \(\langle \cdot , \cdot \rangle\).
Let \(H_{n}(\mathbb F)=\mathbb F^{n}\times\mathbb F^{n}\times\mathbb F =\widetilde{\mathbb F}^n \times \mathbb F\) denote the associated Heisenberg group. The action of the unitary group \(U(n,\widetilde{\mathbb F})\) on \(H_{n}(\mathbb F)\) yields a finite Gelfand pair \((U(n,\widetilde{\mathbb F}) \ltimes H_n(\mathbb F), U(n,\widetilde{\mathbb F}))\).
Spherical functions for this finite Gelfand pair are computed explicitly. These formulae resemble the classical Gauss sums.
For the entire collection see [Zbl 1170.22002].