We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor power (C^N)^{otimes n}.
We propose a formula expressing Perron - Frobenius eigenvectors of Cartan matrices in terms of products of values of the Gamma function.
We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $frak{gl}(m)$ is proposed.
Using a classical result of Avramov-Golod we strengthen a recent result of Gorodentsev, Khoroshkin and Rudakov on syzygies of highest weight orbit closure.
