Relation between the Riemann zeta-function and the Casimir operator

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The principal purpose of our research was to develop a theory to grasp the Riemann zeta-function ζ(s) in the geometric perspective of Lie groups, dispensing with the theory of Kloosterman sums. We, under a joint research activity with R.W. Bruggeman, achieved this aim. We could embed the fourth power moment of ζ(s) in the space L^2(PSL_2(Z)\PSL_2R) as a special value of a certain Poincare series, and retrieved its known spectral decomposition in a new way, i.e., completely without recourse to the theory of Kloosterman sums. This work of ours revealed also a deep relation between ζ(s) and the Jacquet-Langlands local functional equation. This fact will possibly make it feasible to extend our theory to higher dimensional situation, which otherwise should have been almost impossible. Our result could be assessed as to be a basic achievement in the theory of zeta-functions.