Eisenstein cohomology and automorphic L-functions

Günter Harder has pioneered the theory of Eisenstein cohomology over the last few decades. This involves my own work with Harder on rank one Eisenstein cohomology for $GL(N)$ over a totally real field and the arithmetic of Rankin-Selberg $L$-functions for $GL(n) \times GL(m)$. Since then I have been involved in several projects which have the common theme of Eisenstein cohomology of some ambient reductive group and the special values of certain automorphic $L$-functions. Interesting new cases involve (i) Rankin-Selberg L-functions over a CM field, (ii) the degree $2n$ $L$-functions for $SO(n,n)$, in joint work with Bhagwat, and (iii) the degree $n^2$ Asai $L$-functions for $GL(n)$ for a quadratic extension of totally real fields, in joint work with Krishnamurthy. In this talk, which is a celebration of Harder's ideas on cohomology and $L$-functions, I will review the general principles of Eisenstein cohomology that apply to all these different contexts and how they give arithmetic information about $L$-values.