Twisted Spherical Functions on the Finite Poincaré Upper Half-Plane

In this note we compute explicit formulae for the twisted spherical functions for the finite analogue of (the double cover of) the classical Poincaré upper half-plane, in any characteristic, and we obtain a uniform description for them resembling the one given by [Curtis (1993, J. Algebra157, 517–533)] for the characters of the commuting algebra of Gelʹfand–Graev representation. We also deduce some character identities.