Geometric Calculation of the Invariant Integral of Classic Groups

Let G = SpecA be a linearly reductive group, and wG ∈ A ∗ be G-invariant and wG(1) = 1. We establish the 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 semisimple if and only if A splits in the form A = k ×B∗ as k-algebras, where the first projection π1 : A → k is the morphism π1(w) := w(1) ([A2] Theorem 2.6). The linear form wG := (1, 0) ∈ k × B = A will be referred of as the invariant integral on G. In the theory of invariants the calculation of the invariant integral wG is of great interest, because it yields the calculation of the invariants of any representation. The aim of this article is the explicit calculation of wG when G = Sln, Gln, On, Sp2n (char k = 0), by geometric arguments, and by means of the Fourier transform, defined below. Although G is not a compact group it is possible to define the invariant integral of G, the Fourier transform, the convolution product (2.6) and prove the Parseval identity (2.3), inversion formula, etc. Let Ai be simple (and finite) k-algebras and A ∗ = ∏ iA ∗ i . On every A ∗ i , one has the non singular trace metric and its associated polarity. Hence, one obtains a morphism of A-modules φ : A = ⊕iAi →֒ ∏ i A ∗ i = A . If G = SpecA is a semisimple affine k-group and ∗ : A → A, a 7→ a is the morphism induced by the morphism G → G, g 7→ g, we prove that φ is the morphism A → A, a 7→ wG(a · −) where wG(a ∗ · −)(b) := wG(a · b). We shall call φ the Fourier transform. The product operation in A defines, via the Fourier transform, a product on A, which is the convolution product in the classical examples. Let us consider a system of coordinates in G, that is, let us consider G = SpecA as a closed subgroup of a group 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 = {w ∈ B : w(I) = 0}. Moreover, B (which is the ring of functions of Mn/G), coincides essentially with B , via the Fourier transform. Finally, we prove that given w ∈ B, the condition w(I) = 0 is equivalent to w(I) = 0, which is a finite system of equations “in each degree”. 2000 Mathematics Subject Classification. Primary 14L24. Secondary 14L17.