SUZUKI FUNCTOR AT THE CRITICAL LEVEL
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SUZUKI FUNCTOR AT THE CRITICAL LEVEL ´ T. PRZEZDZIECKI School of Mathematics University of Edinburgh Edinburgh EH9 3FD United Kingdom [email protected]
Abstract. In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t = 0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero–Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding.
1. Introduction Arakawa and Suzuki [3] introduced a family of functors from the category O for sln to the category of finite-dimensional representations of the degenerate affine Hecke algebra associated to the symmetric group Sm . These functors have been generalized in many different ways, connecting the representation theory of various Lie algebras with the representation theory of various degenerations of affine and double affine Hecke algebras. Lie algebra
“Hecke” algebra
sln bn sl bn gl bn gl
degenerate affine Hecke algebra trigonometric DAHA rational DAHA (t 6= 0) cyclotomic rat. DAHA (t 6= 0)
Arakawa–Suzuki [3] Arakawa–Suzuki–Tsuchiya [4] Suzuki [52] Varagnolo–Vasserot [54]
Figure 1. Functors relating Lie algebras and “Hecke” algebras in type A
DOI: 10.1007/S00031-020-09620-1 Received July 16, 2019. Accepted July 4, 2020. Corresponding Author: T. Prze´zdziecki, e-mail: [email protected]
´ T. PRZEZDZIECKI
Other generalizations of the Arakawa–Suzuki functor may be found in, e.g., [16], [17], [23], [24], [36], [37], [46]. Here we are concerned with the third functor in the table above, introduced by Suzuki, and later studied by Varagnolo and Vasserot [54], under the assumption that t 6= 0, and the level κ is not critical. It is a functor Fκ : Cκ → Hκ+n -mod
(1)
b n -modules of level κ to the category of modules from the category Cκ of smooth gl over the rational Cherednik algebra Hκ+n (also known as the rational DAHA) b n -module associated to Sm and parameters t = κ + n, c = 1. It assigns to each gl a certain space of coinvariants: M 7→ H0 (gln [z], C[x1 , . . . , xm ] ⊗ (V∗ )⊗m ⊗ M ). In this paper we study the limit of the functor Fκ as κ → c = −n,
t → 0.
The representation theory of the rational Cherednik algebra at t = 0 differs radically from its representation theory at t 6= 0, mainly due to the fact that H0 has a large centre Z, whose spectrum can be identified with the classical Calogero
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