On eigenvalues related to finite Poincaré planes (Q1266419)

Suppose that \(F\) is the finite field with \(q\) elements and \(q\) is odd. The finite (Poincaré) upper half plane in the sense of \textit{A. Terras} [Fourier analysis on finite groups and applications, Cambridge Univ. Press (1999)] can be identified with the quotient \(G/K\), where \(G=GL(2,F)\) and \(K=\{\left(\begin{smallmatrix} a&b\delta\\b&a\end{smallmatrix}\right)\in G\}\), where \(\delta\) is a fixed non-square in \(F\). In an earlier paper [\textit{J. Kuang}, Proc. Am. Math. Soc. 123, 3615-3622 (1995; Zbl 0851.11074)], the author identified the adjacency operators of the finite upper half plane graphs defined in Terras [loc. cit., p. 312] with elements of the Hecke algebra \(H(G,K)\) and thus the eigenvalues of the adjacency matrices of finite upper half plane graphs are obtained from spherical irreducible representations of \(G\) [see also Terras, loc. cit., p. 352]. Here some evidence is given that the spectra of these adjacency matrices have a limiting distribution which is semi-circular as \(q\) goes to infinity.