Derived Hecke algebra and automorphic ℒ-invariants

Abstract

Let π \pi be a cohomological cuspidal automorphic representation of PGL 2 _2 over a number field of arbitrary signature. Under the assumption that the local component of π \pi at a prime p {\mathfrak {p}} is the Steinberg representation, the automorphic L {\mathcal {L}} -invariant of π \pi at p {\mathfrak {p}} has been defined using the lowest degree cohomology in which the system of Hecke eigenvalues associated with π \pi occurs. In this article we define automorphic L {\mathcal {L}} -invariants for each cohomological degree and show that they behave well with respect to the action of Venkatesh’s derived Hecke algebra. As a corollary, we show that these L {\mathcal {L}} -invariants are (essentially) the same if the following conjecture of Venkatesh holds: the π \pi -isotypic component of the cohomology is generated by the minimal degree cohomology as a module over the p p -adic derived Hecke algebra.