Frank Lübeck   

Tables of Green Functions for Exceptional Groups

On this page we provide some data files containing the Green functions for untwisted finite groups of Lie type coming from adjoint simple algebraic groups of exceptional type in good characteristic.

The files below can be read into both GAP3 and GAP4. The data are given in terms of a parameter q which describes the order of the finite field over which a group is defined.

If in your GAP session the variable q is bound before reading one of these data files, then q will be specialized to this value when reading the data. If q is unbound it will be assigned an indeterminate over the Cyclotomics with name "q".

The data are stored in GAP records and can be accessed as follows:

values of Green functions as polynomials in q. The indices i correspond to the Green functions and j to the unipotent classes.
labeling of Green function number i. This is a pair describing the type of a Levi subgroup and a label for the conjugacy class in the relative Weyl group. (We actually provide the generalized Green functions, but only the last functions in types G2, F4 and E8 do not correspond to ordinary Green functions.)
description of unipotent class number j. Given as a pair, a label for the geometric class and a counter for the rational classes.
size of unipotent class number j. Given as polynomial in q.
order of the group. Also as polynomial in q.

The Data

Click on the filenames to view or download the data. The right colums gives the names of the GAP records containing the data. In type E8 there are two such records whose data are relevant if q is congruent to 1 or 2 modulo 3, respectively.

Filename       record name(s)
GreenG2good.g greenG2good
GreenF4good.g greenF4good
GreenE6good.g greenE6good
GreenE7good.g greenE7good
GreenE8good.ggreenE8good1mod3, greenE8good2mod3


These data were (re-)computed using the algorithm given in [1, Section 24]. The input is the generalized Springer correspondence given in [2] and character tables of Weyl groups from [3]. In fact, the mentioned algorithm from [1] does not yield the (generalized) Green functions as class functions but as linear combinations of certain functions which have support on the rational classes corresponding to one geometric unipotent class. For the values of these functions we are implicitly using the original references where all these tables were determined for the first time. For more details see the survey [4] and the references given there.

[1] Lusztig, G., Character sheaves. V, Adv. in Math., 61 (2) (1986), 103--155.

[2] Spaltenstein, N., On the generalized Springer correspondence for exceptional groups, in Algebraic groups and related topics (Kyoto/Nagoya, 1983), North-Holland, Adv. Stud. Pure Math., 6, Amsterdam (1985), 317--338.

[3] Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G., CHEVIE---a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput., 7 (3) (1996), 175--210
(Computational methods in Lie theory (Essen, 1994)).

[4] Shoji, T., Green functions of reductive groups over a finite field, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Amer. Math. Soc., Proc. Sympos. Pure Math., 47, Providence, RI (1987), 289--301.

Last updated: Mon Apr 19 14:04:14 2010 (CET)