Gelfand-Tsetlin-type algebras and q-characters for quantum symmetric pairs

It is well known that quantum affine algebras admit three distinct presentations (Kac-Moody, new Drinfeld and RTT). Relatively recently, the same has been shown to hold for a broad family of quantum affine symmetric pairs. In particular, a Drinfeld-type presentation, due to Lu-Wang, is a new and exciting development. The focus of my talk will be the relationship between the usual Drinfeld presentation of quantum affine algebras and the Lu-Wang presentation of their coideal subalgebras. Remarkably, both presentations exhibit large commutative Gelfand-Tsetlin-type subalgebras, which are of particular interest to representation theory. More specifically, I will present several results concerning the properties of the generators of these commutative subalgebras, including their behaviour under inclusion and coproduct, as well as their spectra on finite-dimensional representations. These results will then be used to define an analogue of the q-character homomorphism for quantum symmetric pairs. I will also remark on some connections to finite W-algebras via truncation and GKLO-type representations.