Affine Grassmannians for \(G_2\) (Q6632090)

For a reductive group \(G\), the Bruhat-Tits building can be viewed as a combinatorial skeleton, to which the affine Grassmannians provide a richer, geometric counterpart. Expanding techniques from [Pappas-Rapoport], who developed a theory of twisted loop groups and their associated flag varieties for unitary groups using self-dual lattices, this paper describes the affine Grassmannians for the exceptional group~\(G_2\).\N\NTaking a split octonion algebra \((C,\cdot,e)\), viewed as a unital composition algebra, the embedding \(\operatorname{Aut}(C) \hookrightarrow \operatorname{SO}(C)\) induces an embedding of the associated buildings. Unlike the prototype \(\operatorname{GL}_n\), where the vertices of the building are all equivalent, being the full lattices in a fixed vector space, the building of \(G_2\) has vertices of three types, corresponding to three types of maximal parahoric subgroups.\N\NFix a perfect field \(k\) of characteristic not \(2\) or \(3\). Let \((C_s,*)\) is the split para-octonion algebra over \(k(\!(t)\!)\) obtained from \(C\), which is a symmetric (thus nonunital) composition algebra. For a \(k\)-algebra \(R\), the vertices of the \(G_2\)-building are \(R[[t]]\)-orders in \(C_s \otimes_{k(\!(t)\!)} R(\!(t)\!)\). Namely, each \(R\) is a finite projective \(R[[t]]\)-module containing the para-unit \(e\) and closed under \(*\). In type (1), \(L\) is self-dual; in type (2), \(L \subset L^{\vee} \subset t^{-1} L\) and \(L^{\vee}*L^{\vee} \subseteq t^{-1}L\); and in type (3) \(L \subset L^{\vee} \subset t^{-1} L\) and \(M = tL^{\vee}*L^{\vee}+L\) is self-dual.\N\NThe proofs use the para-octonion algebra \(C_s\) to derive explicit computations with the spin group, combined with finding clever bases of the various types of lattices.