Partial flag varieties, stable envelopes and weight functions

Abstract: We consider the cotangent bundle T^*F_\lambda of a GL_n partial flag variety, \lambda = (\lambda_1,...,\lambda_N), |\lambda|=\sum_i\lambda_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*T(T^*F\lambda). In [MO], a Yangian module structure was introduced on \oplus_{|\lambda|=n} H^*T(T^*F\lambda). We identify this Yangian module structure with the Yangian module structure introduced in [GRTV]. This identifies the operators of quantum multiplication by divisors on H^*T(T^*F\lambda), described in [MO], with the action of the dynamical Hamiltonians from [TV2, MTV1, GRTV]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [MO]. The formula is in terms of the Yangian weight functions introduced in [TV1], c.f. [TV3, TV4], in order to construct q-hypergeometric solutions of qKZ equations.