Character sheaves and Hecke algebras


The Springer theory relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of Coxeter groups. We discuss a Springer theory for graded Lie algebras and the character sheaves arising in this setting, concentrating on the construction of cuspidal character sheaves. Irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. We will explain the connection between our work and the recent work of Lusztig and Yun, where irreducible representations of trigonometric double affine Hecke algebras appear in the picture. This is based on joint work with Kari Vilonen and partly with Misha Grinberg.