An operator on a one-sheeted hyperboloid over a finite field (Q1804160)

Let \(k\) be a finite field, \(\text{char }k \neq 2\). The hyperboloid \(X\): \(x^2 + y^2 - z^2 = 1\) in \(k^3\) can be realized as the set of matrices \(m \in \text{Mat} (2,k)\) with \(\text{tr }m = 0\), \(\text{det }m = -1\). The group \(G = \text{PSL} (2,k)\) acts on \(X\) transitively by the adjoint representation. One can embed \(X\) in the direct product \(P^1 \times P^1\) where \(P^1\) is the projective line over \(k\). This admits the construction of an operator \(I_\pi\) which is an analogue of the Berezin operator in quantization on symplectic manifolds. Here \(\pi\) is a multiplicative character of \(k\) such that \(\pi^2 \neq 1\). The spectrum of \(I_\pi\) is calculated explicitly.