Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 025, 54 pages      arXiv:2310.06669      https://doi.org/10.3842/SIGMA.2025.025

Yangians, Mirabolic Subalgebras, and Whittaker Vectors

Artem Kalmykov ab
a) Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA, 02139, USA
b) Saint Petersburg University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

Received March 26, 2024, in final form April 04, 2025; Published online April 18, 2025

Abstract
We construct an element in a completion of the universal enveloping algebra of $\mathfrak{gl}_N$, 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 $\mathfrak{gl}_N$, 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 application, we construct a universal vertex-IRF transformation from the standard dynamical $R$-matrix to this constant one in categorical terms.

Key words: Whittaker modules; Yangian; extremal projector; vertex-IRF transformation.

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