We discuss cuspidal character sheaves in the setting of cyclically graded Lie algebras. Via a nearby cycle construction representations of Hecke algebras of complex reflection groups at roots of unity enter their description. We explain a conjectural level-rank (Koszul) duality arising from connections with Lusztig-Yun’s work, where Fourier transforms of character sheaves are related to representations of trigonometric double affine Hecke algebras. We will also discuss connections with d-Harish-Chandra series introduced by Broué-Malle-Michel and Oblomkov-Yun’s construction of rational Cherednik algebra modules using affine Springer fibres.

This is based on joint work with various co-authors, Grinberg, Liu, Trinh, Tsai, and Vilonen.

Place: Room 206, RIMS