Pursuing Coxeter theory for Kac-Moody affine Hecke algebras

The Kac-Moody affine Hecke algebra $\mathcal{H}$ was first constructed as the Iwahori-Hecke algebra of a $p$-adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since $\mathcal{H}$ has a Bernstein presentation, for affine types it is a positive-level variation of Cherednik's double affine Hecke algebra. Moreover, as $\mathcal{H}$ is realized as a convolution algebra, it has an additional "$T$-basis" corresponding to indicator functions of double cosets. For classical affine Hecke algebras, this $T$-basis reflects the Coxeter group structure of the affine Weyl group. In the Kac-Moody affine context, the indexing set $W_{\mathcal{T}}$ for the $T$-basis is no longer a Coxeter group. Nonetheless, $W_{\mathcal{T}}$ carries some Coxeter-like structures: a Bruhat order, a length function, and a notion of inversion sets. This paper contains the first steps toward a Coxeter theory for Kac-Moody affine Hecke algebras. We prove three results. The first is a construction of the length function via a representation of $\mathcal{H}$. The second concerns the support of products in classical affine Hecke algebras. The third is a characterization of length deficits in the Kac-Moody affine setting via inversion sets. Using this characterization, we phrase our support theorem as a precise conjecture for Kac-Moody affine Hecke algebras. Lastly, we give a conjectural definition of a Kac-Moody affine Demazure product via the $q=0$ specialization of $\mathcal{H}$.