Yangians, Mirabolic Subalgebras, and Whittaker Vectors

Abstract

We construct an element in a completion of the universal enveloping algebra of 𝖌𝔩N, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is defined using the evaluation homomorphism from the Yangian of 𝖌𝔩N, on the other hand, it gives a canonical projection onto the space of Whittaker vectors for any Whittaker module over the mirabolic subalgebra. Using the Kirillov projector, we deduce some categorical properties of Whittaker modules; for instance, we prove a mirabolic analog of Kostant's theorem. We also show that it quantizes a rational version of the Cremmer-Gervais 𝑟-matrix. As an application, we construct a universal vertex-IRF transformation from the standard dynamical 𝑅-matrix to this constant one in categorical terms.