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We define a “compactification” of the representation ring of the linear group scheme over Specℤ, in the spirit of Arakelov geometry. We show that it is a λ-ring which is canonically isomorphic to a localized polynomial ring and that it plays a universal role with respect to natural operations on theK 0-theory of hermitian bundles defined by Gillet-Soulé. As a byproduct, we prove that the natural pre-λ-ring structure of theK 0-theory of hermitian bundles is a λ-ring structure. This last result plays a key role in the proof of the main results of [18] and [12].

Lambda-structure on Grothendieck groups of Hermitian vector bundles