Double-affine Kazhdan-Lusztig polynomials via masures

Abstract

Masures (previously also known as hovels) are a generalization of the theory of affine buildings for arbitrary $p$-adic Kac-Moody groups. Gaussent and Rousseau invented masures to compute the Satake transform for $p$-adic Kac-Moody groups. Their answer is given as a sum over Hecke paths, which are certain piecewise linear paths. Guided by their method we give a definition of double affine Kazhdan-Lusztig $R$-polynomials as a sum over piecewise linear paths that we call $I_\infty$-Hecke paths. Remarkably, the notion of $I_\infty$-Hecke path, which arises from masure theoretic considerations, is closely related to chains in the double affine Bruhat order. Our main result is that there are finitely many $I_\infty$-Hecke paths in untwisted affine ADE type. This implies that $R$-polynomials are well-defined in this case. This finiteness result follows from earlier known finiteness results for the double affine Bruhat order. Combined with other results on the double affine Bruhat order, we now have all the ingredients to define double affine Kazhdan-Lusztig $P$-polynomials.