ℤ/𝕞ℤ-graded Lie algebras and perverse sheaves, III: Graded double affine Hecke algebra

Abstract

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra g \mathfrak {g} together with a Z / m Z \mathbb {Z}/m\mathbb {Z} -grading ⨁ i ∈ Z / m Z g i \bigoplus _{i\in \mathbb {Z}/m\mathbb {Z}}\mathfrak {g}_{i} and a block of D G 0 _ ( g i ) \mathcal {D}_{G_{\underline 0}}(\mathfrak {g}_{i}) as introduced in [J. Represent. Theory 21 (2017), pp. 277-321], we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot [Duke Math J. 126 (2005), pp. 251-323] and Oblomkov-Yun [Adv. Math 292 (2016), pp. 601-706] which correspond to the case of the principal block.