Local Birkhoff decompositions for loop groups and a finiteness result

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐆</m:mi> </m:math> {\mathbf{G}} denote an affine Kac–Moody group, and G its points over the local field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐆</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbf{G}(\mathbb{F}_{q}[[s]])} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐆</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>𝔽</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msup> <m:mi>s</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\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.