Automorphic L-invariants for reductive groups

TL;DR

This paper generalizes the construction of automorphic L-invariants to reductive groups over number fields, linking them to Galois representations and showing their independence from cohomological choices under certain conditions.

Contribution

It extends the concept of automorphic L-invariants beyond GL(2), demonstrating their independence from cohomological degrees and their detection via completed cohomology, connecting automorphic and Galois invariants.

Findings

Automorphic L-invariants are independent of cohomological degree choices under cyclicity conditions.

They can be detected through completed cohomology.

For certain unitary groups, automorphic L-invariants match Fontaine-Mazur invariants.

Abstract

Let $G$ be a reductive group over a number field $F$, which is split at a finite place $\mathfrak{p}$ of $F$, and let $\pi$ be a cuspidal automorphic representation of $G$, which is cohomological with respect to the trivial coefficient system and Steinberg at $\mathfrak{p}$. We use the cohomology of $\mathfrak{p}$-arithmetic subgroups of $G$ to attach automorphic $\mathcal{L}$-invariants to $\pi$. This generalizes a construction of Darmon (respectively Spie\ss), who considered the case $G=GL_2$ over the rationals (respectively over a totally real number field). These $\mathcal{L}$-invariants depend a priori on a choice of degree of cohomology, in which the representation $\pi$ occurs. We show that they are independent of this choice provided that the $\pi$-isotypical part of cohomology is cyclic over Venkatesh's derived Hecke algebra. Further, we show that automorphic $\mathcal{L}$-invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic $\mathcal{L}$-invariants are equal to the Fontaine-Mazur $\mathcal{L}$-invariants of the associated Galois representation.