We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p–adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p–ad...

In this paper we develop a theory of class invariants associated to p-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to p-adic representations in terms of resolvends associated to torsors of finite group schemes.

Regrettably, the paper [1] had two incorrect lemmas. Nevertheless, statements of theorems in are correct with no modification required. Specifically, [1, Lemma 6.4] does not hold for spherical spaces, and also conclusion B.2] is valid generality stated there. The effect error on proof Theorem 6.1] significant. In § 1 one we provide a that relay upon 6.4]. minor explain nessecary modifications §2.

We compare the Pontryagin duals of fine Selmer groups two congruent $p$-adic Galois representations over admissible pro-$p$, Lie extensions $K_\infty$ number fields $K$. prove that in several natural settings $\pi$-primary submodules are pseudo-isomorphic Iwasawa algebra; if coranks not equal, then we can still inequalities between $\mu$-invariants. In special case a $\mathbb{Z}_p$-extension $K...