Preliminary remarks on topological groups -- Automorphism groups of bilinear forms -- Geometry in the representation space -- Symplectic geometry -- Weakly symmetric Riemannian spaces -- The Riemannian space of all positive matrices -- A generalization of JS(X) -- The Riemannian space -- The reduction theory of positive quadratic forms -- Grรถร{159}en-characters of quadratic forms -- The modular group of degree n -- The fundamental domain of the modular group -- Modular forms of degree n -- Report on Eisenstein series of the modular group -- Dirichlet series corresponding to modular forms -- Zeta functions attached to quadratic forms -- Selberg's zeta functions -- Non-analytic Eisenstein series -- The differential operaton M? -- Final aspects