Companion points and locally analytic socle conjecture for Steinberg case

In this paper, we will modify the Breuil-Hellmann-Schraen’s (more generally, resp., Breuil-Ding’s) local model for the trianguline variety (resp., Bernstein paraboline variety) to certain semistable (resp., potentially semistable) non-crystalline point with regular Hodge-Tate weights.Then we deduce several local-global compatibility results, including a classicality result, and the existence of expected companion points on the (definite) eigenvariety and locally analytic socle conjecture for such semistable non-crystalline Galois representations, under certain hypothesis on trianguline variety and the usual Taylor-Wiles assumptions. Moreover, we also discuss slightly the coherent sheaves obtained by patching argument and the coherent sheaves which are constructed from local models and Bezrukavnikov functor, under the route of the recently work of Hellmann-Hernandez-Schraen.