For GL2, cuspidal functions on the ordinary Caraiani-Scholze Igusa variety form a completed Kirillov model of classical cusp forms, and their coinvariants recover Hida-ordinary forms and yield a weak local-global compatibility theorem.
The conjectural connections between automorphic representa- tions and Galois representations
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The completed Kirillov model and local-global compatibility for functions on Igusa varieties
For GL2, cuspidal functions on the ordinary Caraiani-Scholze Igusa variety form a completed Kirillov model of classical cusp forms, and their coinvariants recover Hida-ordinary forms and yield a weak local-global compatibility theorem.