Unipotent Degrees of Imprimitive Complex Reflection Groups

In the representation theory of finite groups of Lie type G(q) the unipotent characters play a fundamental role. Their degrees, seen as polynomials in q, are only dependent on the Weyl group of G(q). G. Lusztig (Astérisque 212 (1993) 191–203) has shown that one can define unipotent degrees for a general finite Coxeter group. In this article we construct, for the two infinite series of n-dimensional complex reflection groups that are generated by n reflections, a set of unipotent degrees, with the same combinatorial properties as the unipotent character degrees of a finite Weyl group. In particular they are related by a Fourier transform matrix to the fake degrees, and together with the appropriate eigenvalues of Frobenius they provide a representation of SL2(Z).