Edifices: Building-like spaces associated to linear algebraic groups

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