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 \(\mathcal{H}\).