Automorphic ℒ‐invariants for reductive groups

Abstract

Abstract Let G be a reductive group over a number field F, which is split at a finite place 𝔭 {\mathfrak{p}} of F, and let π be a cuspidal automorphic representation of G, which is cohomological with respect to the trivial coefficient system and Steinberg at 𝔭 {\mathfrak{p}} . We use the cohomology of 𝔭 {\mathfrak{p}} -arithmetic subgroups of G to attach automorphic ℒ {\mathcal{L}} -invariants to π. This generalizes a construction of Darmon (respectively Spieß), who considered the case G = GL 2 {G={\mathrm{GL}}_{2}} over the rationals (respectively over a totally real number field). These ℒ {\mathcal{L}} -invariants depend a priori on a choice of degree of cohomology, in which the representation π occurs. We show that they are independent of this choice provided that the π-isotypic part of cohomology is cyclic over Venkatesh’s derived Hecke algebra. Further, we show that automorphic ℒ {\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 ℒ {\mathcal{L}} -invariants are equal to the Fontaine–Mazur ℒ {\mathcal{L}} -invariants of the associated Galois representation.