Local Birkhoff decompositions for loop groups and a finiteness result

Abstract

Let

denote an affine Kac–Moody group, and G its points over the local field

𝔽 q ⁢ ( ( s ) ) {\mathbb{F}_{q}((s))}

. We establish a local Birkhoff decomposition for a subset of G in terms of a pair of subgroups roughly of the form

𝐆 ⁢ ( 𝔽 q ⁢ [ [ s ] ] ) {\mathbf{G}(\mathbb{F}_{q}[[s]])}

and

𝐆 ⁢ ( 𝔽 q ⁢ [ s - 1 ] ) {\mathbf{G}(\mathbb{F}_{q}[s^{-1}])}

. Our techniques are global-to-local and use the reduction theory for loop groups due to H. Garland. Building on these ideas, we establish the finiteness of a set whose cardinality is related to spherical R-polynomials in D. Muthiah’s conjectural double-affine Kazhdan–Lusztig theory.