I want to find the Hasse–Weil zeta function on the Grassmanian G(n, k), by using the definition in http://www.math.ru.nl/~jcommelin/files/localfactorsSerre.pdf, ie $$Z(T)=\prod_{m=0}^{m=2r}P_{l,m}(T)$$ where the polynomial $P_{l,m}(T)$ is given by $$P_{l,m}(T)=det(I-T\pi_{l,m})$$
But I'm not very comfortable with cohomology, could you show me how to calculate the polynomials and thus the function?
Thanks in advance!
ps I'm aware of the method of counting the points of the space, and then using $Z(T)=exp(\sum N_{r}\frac{T^{r}}{r})$, but I'd like to see how to do it using the other method.