Decomposition and parity of p-adic representations attached to algebraic automorphic forms on GL(4)

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Let F be a number field with adele ring AF, and πan isobaric, algebraic automorphic representation of GL₄(AF) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation σᵥ of the Weil group WFᵥ attached to πᵥ is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation ρof the absolute Galois group GF, whose local L-factors agree with those of π(up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of ρdoes not contain any irreducible 2-dimensional Galois representation which is even. The second result is that if πis regular and ρcrystalline, then for sufficiently large p (see the article for a precise statement), the decomposition type of ρis the same as the isobaric type of π. A consequence is that ρis irreducible when πis cuspidal (and regular algebraic), which has also been proved by F. Calegari and T. Gee, in fact with no hypothesis on p. The third and final result, using Taylor's potential modularity theorem, is that given a pair (σ, σ') of odd, 2-dimensional p-adic representations of the same weight and distinct Hodge-Tate types, such that their direct sum is automorphic, the dimension of the GF-invariants of the tensor product ηof the dual of σwith σ' equals, for large p, the order of pole at s=1 of the L-function of η(with the bad factors removed). This is as predicted by the Tate conjecture when σ, σ' occur in p-adic etale cohmology of smooth projective varieties over F, and it also provides a useful link to a small piece of the work of C. Skinner and E. Urban.