For degenerate principal series of GL_n satisfying a balanced condition, the Eisenstein map on the bottom-degree cohomology is Aut(C)-equivariant, yielding a rational structure on this Eisenstein cohomology.
Eisenstein Cohomology for GL(N) and the special values of Rankin-Selberg L-functions over a totally imaginary number field
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abstract
Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.
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On rationality of certain Eisenstein cohomology
For degenerate principal series of GL_n satisfying a balanced condition, the Eisenstein map on the bottom-degree cohomology is Aut(C)-equivariant, yielding a rational structure on this Eisenstein cohomology.