Invariant Integral on Classical Groups and Algebraic Harmonic Analysis

Let G = SpecA be a linearly reductive group and let wG ∈ A ∗ be the invariant integral on G. We establish the algebraic harmonic analysis on G and we compute wG when G = Sln, Gln, On, Sp2n by geometric arguments and by means of the Fourier transform. Introduction An affine k-group G = SpecA is linearly semisimple (that is, linearly reductive) if and only if A splits into the form A = k × B as k-algebras, where the first projection π1 : A ∗ → k is the morphism π1(w) := w(1) ([A2, 2.6]). The linear form wG := (1, 0) ∈ k ×B ∗ = A will be referred to as the invariant integral on G. The aim of this article is to establish the algebraic harmonic analysis onG and the explicit calculation of wG when G = Sln, Gln, On, Sp2n (char k = 0) by geometric arguments and by means of the Fourier transform, which is defined below. An affine k-group G = SpecA is linearly semisimple if and only if A = ∏ i A ∗ i , where Ai are finite simple k-algebras ([A1, 6.8]). On à := ⊕iA ∗ i ⊆ A ∗ one has the non-singular trace metric and its associated polarity φ : à ≃ A. This morphism is essentially characterized by being a morphism of left and right A-modules or equivalently of left and right G-modules. Let ∗ : A → A, a 7→ a be the morphism induced by the morphism G → G, g 7→ g. We prove that the Fourier Transform F : A → A, F (a) = wG(a ∗ · −), where wG(a ∗ · −)(b) := wG(a ∗ · b), is the inverse morphism of φ, because is a morphism of left and rightG-modules. The product operation on A defines, via the Fourier Transform, a product on A, which is the convolution product in the classical examples. We prove algebraically Parseval’s Identity (Theorem 2.3), the Peter-Weyl Theorem (Eq. 6), the Inversion Formula (Eq. 12), etc. Harmonic Analysis has been developed from an algebraic point of view in finite groups and in the Cartier Duality on multiplicative groups (see [Se], [DG]). A. Van Daele, in the more elaborate algebraic framework of regular multiplier Hopf algebras with invariant functionals, defined the Fourier Transform and proved Plancherel’s formula (see [V]). In order to compute wG, when G = Sln, Gln, On, Sp2n, we consider a system of coordinates in G, that is, we consider G = SpecA as a closed subgroup of a semigroup of matrices Mn = SpecB. Then A is the quotient of B by the ideal I of the functions of Mn vanishing on G. Hence, A ∗ is a subalgebra of B and one has that k · wG = A ∗G = {w ∈ B : w(I) = 0}. Moreover, B (which is the ring of functions of Mn/G), coincides essentially with B , via the Fourier transform. 2000 Mathematics Subject Classification. Primary 14L24. Secondary 14L17.