On symmetric power L-invariants of Iwahori level Hilbert modular forms

We compute the arithmetic ${\cal L}$-invariants (of Greenberg-Benois) of twists of symmetric powers of $p$-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on $p$-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of ${\rm GL}(2)$, we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the ${\cal L}$-invariants in terms of logarithmic derivatives of Hecke eigenvalues.