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

We describe the cuspidal functions ${\mathbb{V}}_b^{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}$ on the ordinary Caraiani-Scholze Igusa variety for ${\mathrm{G}\mathrm{L}}_2$ 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 ${\tilde{\mu }}_{p^{\mathrm{\infty }}}$ on ${\mathbb{V}}_b^{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in ${\mathbb{V}}_b^{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}$ 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.