On the basic representation of the double affine Hecke algebra at critical level

TL;DR

This paper constructs the fundamental representation of the double affine Hecke algebra at critical level for certain root systems, extending prior work by Oblomkov and Gehles to a broader class.

Contribution

It provides a new construction of the basic representation at critical level for irreducible reduced affine root systems with reduced gradient roots.

Findings

Constructed the basic representation at critical level q=1.

Extended previous results to a broader class of root systems.

Provides a foundation for further study of representations at critical level.

Abstract

We construct the basic representation of the double affine Hecke algebra at critical level $q=1$ associated to an irreducible reduced affine root system $R$ with a reduced gradient root system. For $R$ of untwisted type such a representation was studied by Oblomkov [O04] and further detailed by Gehles [G06] in the presence of minuscule weights.