ON SPECIAL PIECES IN THE UNIPOTENT VARIETY

MEINOLF GECK AND GUNTER MALLE

Abstract. This article is the result of experiments performed using computer programs written in the GAP language. We describe an algorithm which computes a set of rational functions attached to a finite Coxeter group W. Conjecturally, these rational functions should be polynomials, and in the case where W is the Weyl group of a Chevalley group G defined over 𝔽_{q}, the values of our polynomials at q should give the number of 𝔽_{q}-rational points of Lusztig’s special pieces in the unipotent variety of G. The algorithm even works for complex reflection groups. We give a number of examples which show, in particular, that our conjecture is true for all types except possibly B_{n} and D_{n}.