(1987).

a₁ affine space arbitrary C₁ Cartan commutative complex numbers consider constructed corresponding cosets D₁ decomposed decomposition defined denote Diff differential operator dimension direct sum Dokl e₁ eigenvalues elements equal equation equivalent exists finite finite-dimensional follows formula function g G₁ Gaussian measure Gelfand given group G h₁ h₂ Harish-Chandra Hence Hilbert space homogeneous space horosphere indecomposable integral invariant irreducible representations irreducible unitary representations isomorphism k₁ Lemma Lie algebra linear Lorentz group m₁ manifold mapping Math matrices module monomial multiplication n₁ norm obtain P₁ P₂ pair polynomial positive definite principal series proof Proposition prove quasi-invariant measure relation representation of G respect ring satisfying the condition sequence space H spherical functions subset subspace symmetric Theorem tion transformation unimodular uniquely unitary representation V₁ vector w₁ w₂ Weyl Weyl group X₁ z₁ zero