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Yangians, Mirabolic Subalgebras, and Whittaker Vectors

Artem Kalmykov

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.

Yangians, Mirabolic Subalgebras, and Whittaker Vectors

Abstract

We construct an element in a completion of the universal enveloping algebra of , 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 , 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 -matrix. As application, we construct a universal vertex-IRF transformation from the standard dynamical -matrix to this constant one in categorical terms.
Paper Structure (25 sections, 58 theorems, 135 equations)

This paper contains 25 sections, 58 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Background: dynamical quantum groups
  3. General setup
  4. Harish-Chandra bimodules
  5. Constant and dynamical $\boldsymbol{R}$-matrices
  6. Extremal projector
  7. Parabolic restriction functor
  8. Mirabolic subgroup
  9. Yangian
  10. General definition and properties
  11. Technical lemmas
  12. Kirillov projector
  13. Kirillov projector: right $\boldsymbol{\mathfrak{n}_-^{\psi}}$-invariance
  14. Global induction
  15. Local induction
  16. ...and 10 more sections

Key Result

Theorem A

For any right Whittaker module over $\mathfrak{m}_N$, the Kirillov projector defines a unique linear operator satisfying and acting by identity on the space of Whittaker vectors.

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: BezrukavnikovFinkelberg
  • Remark 2.4
  • Proposition 2.5: KalmykovSafronov
  • Definition 2.6
  • ...and 79 more