Harish-Chandra isomorphism for cyclotomic double affine Hecke algebras

TL;DR

This paper proves a conjecture linking the spherical subalgebra of cyclotomic DAHA to quantized multiplicative quiver varieties, establishing new isomorphisms in the structure of these algebraic objects.

Contribution

It confirms a conjecture by Braverman, Etingof, and Finkelberg, and constructs an explicit isomorphism as a q-analogue of known maps, advancing understanding of cyclotomic DAHA.

Findings

Spherical subalgebra of cyclotomic DAHA is isomorphic to a quantized multiplicative quiver variety.

Constructs an explicit q-analogue of Oblomkov's radial parts map.

Proves isomorphism with the image of a shifted quantum toroidal algebra under GKLO homomorphism.

Abstract

We confirm a conjecture of Braverman--Etingof--Finkelberg that the spherical subalgebra of their cyclotomic double affine Hecke algebra (DAHA) is isomorphic to a quantized multiplicative quiver variety for the cyclic quiver, as defined by Jordan. The isomorphism is constructed as a q-analogue of Oblomkov's cyclotomic radial parts map for the rational case. In the appendix, we also prove that the spherical cyclotomic DAHA is isomorphic to the image of a shifted quantum toroidal algebra under Tsymbaliuk's GKLO homomorphism.