Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 1, pp. 115-140 (2000)

Green Functions and a Conjecture of Geck and Malle

Toshiaki Shoji

Department of Mathematics, Science University of Tokyo, Noda, Chiba 278-8510, Japan

Abstract: For any special unipotent class $C$ of a split reductive group $G$ over a finite field $\bf{F}_q$, a special piece $\widehat C$ is defined. It is known that the cardinality of $\widehat C(\bf{F}_q)$ is a polynomial in $q$. Geck and Malle proposed a conjectural algorithm for computing these polynomials, and verified it in the case of exceptional groups. In this paper we show, in the case of classical groups, that their conjecture is reduced to another conjecture concerning Springer representations of $W$. We verify this conjecture in the case where $G$ is of type $B_n,C_n$ or $D_n$ with $n\le 6$.

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