Decomposition and Parity of Galois Representations Attached to GL ( 4 ) Dinakar Ramakrishnan

Let F be a number field, and π an isobaric ([La]), algebraic ([Cl1]) automorphic representation of GLn(AF ). We will call π quasi-regular iff at every archimedean place v of F , the associated n-dimensional representation σv of the Weil group WFv (defined by the archimedean local correspondence) is multiplicity free. For example, when n = 2 and F = Q, a cuspidal π is quasi-regular exactly when it is generated by a holomorphic newform f of weight ≥ 1. Recall that π is regular iff at each archimedean v the restriction of σv to C ∗ is multiplicity free; hence any regular π is quasi-regular, but not conversely. When F is totally real, an algebraic automorphic representation π of GLn(AF ) is said to be totally odd iff it is odd at each archimedean place v, i.e., iff the difference in the multiplicities of 1 and −1 as eigenvalues of complex conjugation in σv is at most 1; in particular if n is even, these two eigenvalues occur with the same multiplicity. One readily sees that any quasi-regular π is totally odd, but not conversely. When n = 2, π is said to be even (or that it has even parity) at an archimedean place v (of F ) iff if it is not odd at v. Still with F totally real, let c be one of the [F : Q] complex conjugations in the absolute Galois group GF =Gal(F/F ). Recall that an n-dimensional Qp-representation ρ of GF is odd relative to c if the trace of ρ(c) lies in {1, 0,−1}. It is odd if it is so relative to every c. When n = 2, ρ is said to be even relative to c if it is not odd relative to c, which is the same as the determinant of ρ(c) being 1. One says that an isobaric, algebraic π on GL(n)/F has a fixed archimedean weight iff there is an integer w such that at every archimedean place v of F , the restriction to C of σv ⊗ | · | (n−1)/2 is a direct sum of characters