  Let F be a number field with adele ring A_F, and \pi an isobaric, algebraic
automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is
quasi-regular, meaning that at every archimedean place v of F, the
4-dimensional representation \sigma_v of the Weil group W_{F_v} attached to
\pi_v is multiplicity free. Suppose there is an associated 4-dimensional,
Hodge-Tate p-adic representation \rho of the absolute Galois group G_F, whose
local L-factors agree with those of \pi (up to a shift) at almost all primes P
of F. Then our first result is that the semisimplification of \rho does not
contain any irreducible 2-dimensional Galois representation which is even. The
second result is that if \pi is regular and \rho crystalline, then for
sufficiently large p (see the article for a precise statement), the
decomposition type of \rho is the same as the isobaric type of \pi. A
consequence is that \rho is irreducible when \pi is 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 (\sigma, \sigma') 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
G_F-invariants of the tensor product \eta of the dual of \sigma with \sigma'
equals, for large p, the order of pole at s=1 of the L-function of \eta (with
the bad factors removed). This is as predicted by the Tate conjecture when
\sigma, \sigma' 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.
