The completed Kirillov model and local-global compatibility for functions on Igusa varieties

We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tilde{\mu}_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms.