Quantum Sugawara operators in type A

Abstract

The quantum Sugawara operators associated with a simple Lie algebra $g$ are elements of the center of a completion of the quantum affine algebra $U_q\left(\hat{g}\right)$ at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series ${\ell }_V\left(z\right)$ associated with every finite-dimensional representation V of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over $U_q\left(\hat{g}\right)$ at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the q-characters and q-deformed $W$-algebras via the Harish-Chandra homomorphism.

We produce explicit quantum Sugawara operators for the quantum affine algebra of type A which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the q-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.