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:

`.table[i][j]`

- values of Green functions as polynomials in q. The indices i correspond to the Green functions and j to the unipotent classes.
`.charInfo[i]`

- 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.)
`.classInfo[j]`

- description of unipotent class number j. Given as a pair, a label for the geometric class and a counter for the rational classes.
`.classLengths[j]`

- size of unipotent class number j. Given as polynomial in
`q`

. `.GFOrder`

- 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.g `greenE8good1mod3`

,
`greenE8good2mod3`

### References

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)