We show that local-global compatibility (at split primes) away from p holds at all points of the p-adic eigenvariety of a definite n-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GLn across the eigenvariety by considering the fibers of its defining coherent sheaf. W...

We prove a realization of the local Langlands correspondence for two-dimensional representations Weil group conductor three in cohomology Lubin–Tate curves by purely geometric method.