Derived Hecke algebra and automorphic L-invariants

TL;DR

This paper studies automorphic L-invariants associated with cohomological automorphic representations of PGL(2) over number fields, showing their equivalence under certain conditions involving derived Hecke algebras.

Contribution

It establishes a connection between automorphic L-invariants and the structure of cohomology as a module over Venkatesh's derived Hecke algebra, revealing their equivalence in specific cases.

Findings

Automorphic L-invariants are essentially the same across cohomological degrees.

The invariants coincide when the cohomology is generated by minimal degree cohomology.

The work applies to representations with Steinberg local components at primes.

Abstract

Let $\pi$ be a cohomological automorphic representation of $PGL(2)$ over a number field of arbitrary signature and assume that the local component of $\pi$ at a prime $\mathfrak{p}$ is the Steinberg representation. In this situation one can define an automorphic $\mathcal{L}$-invariant for each cohomological degree in which the system of Hecke eigenvalues associated to $\pi$ occurs. We show that these $\mathcal{L}$-invariants are (essentially) the same if the $\pi$-isotypic component of the cohomology is generated by the minimal degree cohomology as a module over Venkatesh's derived Hecke algebra.