Abstract

Given a semisimple linear algebraic $k$-group $G$, one has a spherical building ${\mathrm{\Delta }}_G$, and one can interpret the geometric realisation ${\mathrm{\Delta }}_G(ℝ)$ of ${\mathrm{\Delta }}_G$ in terms of cocharacters of $G$. The aim of this paper is to extend this construction to the case when $G$ is an arbitrary connected linear algebraic group; we call the resulting object ${\mathrm{\Delta }}_G(ℝ)$ the spherical edifice of $G$. We also define an object $V_G(ℝ)$ which is an analogue of the vector building for a semisimple group; we call $V_G(ℝ)$ the vector edifice. The notions of linear map and of isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on $V_G(ℝ)$ and show they are all bi-Lipschitz equivalent to each other; with this extra structure, $V_G(ℝ)$ becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits centre conjecture.

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