Cocenter of Hecke algebras of Kac-Moody groups
Let $H$ be the generic Hecke algebra over $\mathbb{Z}[\mathbf{q}^{\pm 1}]$ associated to a split Kac--Moody group $G$, arising as the deformation of the group algebra of its Weyl group $W$. The cocenter $\overline{H} = H/[H, H]$ encodes the trace and character theory of $H$, playing a fundamental role in representation theory and harmonic analysis. Through deep combinatorial results on cyclic reductions in Coxeter groups, each conjugacy class $\mathcal{O}$ of $W$ determines a canonical element $T_{\mathcal{O}}$ in $\overline{H}$, and these elements are known to span the cocenter. However, establishing their linear independence has remained an open problem outside of finite and affine types. In this paper, we solve this problem: the canonical elements form a $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-basis of the cocenter $\overline{H}$. Our approach differs from earlier representation-theoretic methods in finite and affine types. To construct explicit functionals that separate all conjugacy classes, we develop a new framework based on re-normalized orbital integrals. This framework synthesizes parabolic induction, traces of infinite-dimensional bimodules, and Kac--Moody harmonic analysis into an almost-dual basis for the cocenter. As a key local ingredient, we establish a generic duality theorem for finite groups of Lie type relating the cocenter to regular semisimple conjugacy classes, and determine precisely when this pairing is non-degenerate. Finally, we deduce the existence and uniqueness of generic class polynomials for $W$, and prove a uniform ``dimension=degree'' theorem for basic Deligne--Lusztig varieties of the split Kac--Moody group $G$.