## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 275 - 302

### On the functor of Arakawa, Suzuki and Tsuchiya

Sergey Khoroshkin and Maxim Nazarov

#### Abstract

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{C}_N$ depending on a parameter $\kappa\in\mathbb{C}\,$, and certain modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_m$ of level $\kappa-m\,$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{\mathfrak{gl}}_m$ alongside of $\,\widehat{\mathfrak{sl}}_m\,$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{H}_N$ and all modules of $\mathfrak{sl}_m$ or $\mathfrak{gl}_m\,$. The latter correspondence was constructed earlier by Cherednik.

#### Article information

**Dates**

Received: 4 December 2015

Revised: 21 April 2016

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499429

**Digital Object Identifier**

doi:10.2969/aspm/07610275

**Mathematical Reviews number (MathSciNet)**

MR3837925

**Zentralblatt MATH identifier**

07039306

**Subjects**

Primary: 17B35: Universal enveloping (super)algebras [See also 16S30]

**Keywords**

Affine Lie algebras Cherednik algebras

#### Citation

Khoroshkin, Sergey; Nazarov, Maxim. On the functor of Arakawa, Suzuki and Tsuchiya. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 275--302, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610275. https://projecteuclid.org/euclid.aspm/1537499429