GL_n(R) Action on Pos_n(R)

Abstract

Let G = GL_n(R) or SL_n(R) and Γ_n = GL_n(Z). Let Posn(R) be the space of positive symmetric real n × n matrices. Recall that symmetric real n × n matrices Z have an ordering, defined by Z ≧ 0 if and only if 〈Zx, x〉 ≧ 0 for all x ∈ Rⁿ. We write Z₁ ≧ Z₂ if and only if Z₁ - Z₂ ≧ 0. If Z ≧ 0 and Z is non-singular, then Z ≫ 0, and in fact Z ≧ λI if λ is the smallest, necessarily positive, eigenvalue.