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

Abstract

We construct the basic representation of the double affine Hecke algebra at critical level [Formula: see text] associated to an irreducible reduced affine root system [Formula: see text] with a reduced gradient root system. For [Formula: see text] of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights.