Geometric Reductivity at Archimedean Places

Let G → GL(n,C) be a representation of a complex reductive group. A theorem of Hilbert says that the algebra C[x1, . . . , xn] of invariant polynomials is finitely generated. Let Y be the projective variety defined by this graded algebra, and then we have a rational morphism π : C · · · → Y(C). A theorem on geometric reductivity of Mumford says that a point x ∈ C is regular for the map π if and only if the closure of the orbit Gx does not contain the origin 0. Such results have been generalized to more general bases by Haboush and Seshadri, et al., and have been used in constructing moduli spaces of various geometric objects. We would like to have some analogous results in the Arakelov theory, and apply them to the arithmetic problem. In other words, we want to consider a representation for a reductive group overZ, and invariants with length induced from the standard hermitian structure of C. The first paper to appear on this aspect was Burnol’s [B], in which he proved a p-adic analogue of a result of Kempf and Ness on stability and length function. In this paper, we want to prove some analogues of Hilbert’s theorem and Mumford’s theorem in Arakelov theory. More precisely, we will formulate and prove the geometric reductivity of a reductive group over archimedean places, and give a HilbertSamuel formula for the volumes of integral invariants. More details are explained in what follows. In §1, following a paper of Burnol [B], we will analyze the geometric reductivity of Mumford-Seshadri over a discrete valuation ring in terms of valuations. This will lead to a notion of geometric reductivity at archimedean places. Then we will explain that