The elliptic Hall algebra EHA is a widely studied object in geometric representation theory and combinatorics. This algebra arises as the Hall algebra of the category of coherent sheaves of a smooth elliptic curve. The Bqt algebra arose in the work of Carlsson-Gorsky-Mellit as a certain algebra of operators on the K-theory of parabolic flag Hilbert schemes. In recent work, alongside Gorsky and Simental, we showed that Bqt contains EHA as its spherical subalgebra. In particular, EHA is but one in a family of so-called K-theoretic Hall algebras, which depend on a quiver Q.
In this talk I will discuss work Gorsky and Simental where we define new quiver Bqt algebras whose (conjecturally) spherical subalgebras are precisely the associated K-theoretic Hall algebras. To do, we introduce split parabolic quiver varieties, new spaces which generalize Nakajima quiver varieties, on which we define a family of operators that yield our desired algebra.