__abstract__
Starting from a conjugacy class in the loop Lie algebra of a reductive
group, we construct a quasi-coherent sheaf on a partial resolution of
the trigonometric commuting variety of the Langlands dual group. The
construction uses 1) Coulomb branch technology, as mathematically
developed by Braverman-Finkelberg-Nakajima and many others 2) a version
of affine Springer theory generalizing that developed by Garner-Kivinen
and Hilburn-Kamnitzer-Weekes, as well as 3) a Z-algebra construction
similar to the one used by Gordon-Stafford for rational Cherednik
algebras. When G=GL_n the partial resolution we construct is the Hilbert
scheme of n points on T*C*. For other groups, the partial resolution is
an analog of the Hilbert scheme and there are many open questions
regarding its geometry. We will explain some examples of the
construction, recovering known sheaves such as the Procesi bundle, and
conjecture the sheaves we get are coherent in general. Based on joint
work with Gorsky and Oblomkov.