  We introduce a refined version of group cohomology and relate it to the space
of polynomials on the group in question. We show that the polynomial cohomology
with trivial coefficients admits a description in terms of ordinary cohomology
with polynomial coefficients, and that the degree one polynomial cohomology
with trivial coefficients admits a description directly in terms of
polynomials. Lastly, we give a complete description of the polynomials on a
connected, simply connected nilpotent Lie group by showing that these are
exactly the maps that pull back to classical polynomials via the exponential
map.
