On Residual Cohomology Classes Attached to Relative Rank One Eisenstein Series for the Symplectic Group

Abstract

The cohomology of an arithmetically defined subgroup of the symplectic Q-group Spn is closely related to the theory of automorphic forms. This paper gives a structural account of that part of the cohomology which is generated by residues or derivatives of Eisenstein series of relative rank one. In particular we determine a set of conditions subject to which residues of Eisenstein series may give rise to non-vanishing cohomology classes. A non-vanishing condition on the value at s = 1/2 of certain automorphic L-functions which naturally appear in the constant terms of the Eisenstein series plays a major role.

Eisenstein series, cohomology of arithmetic groups, and automorphic L-functions at half integral arguments

“…The action of minimal coset representatives (and the whole Weyl group) on L a P 0 , written in that basis, is directly related to the description of the Weyl group in terms of permutations and sign changes. This point of view provides the explicit formulas of Section 3, which are quite useful not only for obtaining divisibility properties, but also for obtaining very precise description of possible residual Eisenstein cohomology classes as for example in [9]. On the other hand, in the case of split classical groups, the evaluation point written in this basis corresponds to the character of L P given in terms of powers of absolute values of the determinant as in Section 4.1.…”

“…We consider separately the case of the general linear group GL n , the even special orthogonal group SO 2n , and the remaining two cases. The description in the case of a maximal proper parabolic Q-subgroup of a symplectic group was already used in [9] to study the corresponding Eisenstein cohomology space.…”

“…Our main theorem is a generalization of this result to any proper parabolic Q-subgroup of any classical group. In turn, the results in [9] may be viewed as an application of the general approach taken here.…”

“…In the previous work [9] a quite precise description of possible residual Eisenstein cohomology classes attached to cuspidal automorphic representations of the Levi factor of a maximal proper parabolic Q-subgroup in the Q-split symplectic group Sp n of Q-rank n is given. That paper already reveals that in the case of a maximal proper parabolic subgroup, regardless to the analytic properties of the Eisenstein series, the necessary non-vanishing conditions themselves imply that the evaluation point s 2 R is necessarily a half-integer (where s corresponds to a character det s of the Levi factor).…”

The Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup

“…Some of these facts were already used, without proof and in a less explicit form, in several papers dealing with the case of a maximal proper parabolic Q-subgroup of a particular classical group G. The case G = GL n is considered in [8,Section 5], the case G = Sp 2n in [12], and the case G = GL 2 over a division algebra in [13]. In all these papers, the motivation for studying the Franke filtration is to determine the contribution of residues of Eisenstein series supported in a maximal proper parabolic Q-subgroup to the Eisenstein cohomology of the considered group G (with coefficients given by a finitedimensional algebraic representation of G).…”