These tables are not particularly user friendly, and require some knowledge on the part of the user. See Notes on Cells of Harish-Chandra modules and special unipotent representations. Other references under papers are also useful.

These tables are based on Birne Binegar's matching of cells and nilpotent orbits.

Each square in the tables corresponds to a block for a real form of an exceptional group. Such a block is parametrized by a pair: a real form of G and a real form of the dual group G^v. Not every such pair arises.

For each block, and each even complex nilpotent orbit O^v for G^v(C), there is a collection of unipotent representations, occuring at a certain (usually singular) infinitesimal character determined by O^v.

For each block there are up to three kinds of output:

1. Cells and orbits: list of cells and orbits for G and G^v, including information about duality, real forms of orbits, etc.
2. Representations: for each even orbit on the dual side, a list of the corresponding unipotent representations of G
3. Stability: for each even orbit on the dual side, information about stable sums of the corresponding unipotent representations. (This is not computed in all cases.)

Cells and Orbits:

Here is a sample output file:

Unipotent Packets for big block of split F4

G=F4 split
G^v=F4 split

type: F4 sc s
real form: 2
dual real form: 2

G:   1--2=>=3--4
G^v: 1--2=<=3--4

Special                            Special                           
Orbit   Cells       Diagram #R A   Dual Orbit  Cells       Diagram #R A
F4      0	    2222    1  1   0           24          0000	   1  1
F4(a1)  1,3         2202    2  2   A1~         22,23       1000	   2  2
F4(a2)  4,5,11      0202    2  2/1 A1+A1~      17,19,20    0010	   2  1		   
F4(a3)  9,13,14     0200    3  S4  F4(a3)      9,13,14     0020	   3  S4
C3      12          1012    1  1   A2~         18          2000	   1  1 
B3      2,6,7,8     2200    2  1   A2          10,15,16,21 0002	   3  2/1	
A2      10,15,16,21 2000    3  2/1 B3          2,6,7,8     0022	   2  1	
A2~     18          0002    1  1   C3          12          2101	   1  1
A1+A1~  17,19,20    0100    2  1   F4(a2)      4,5,11      2020	   2  2/1
A1~     22,23       0001    2  2   F4(a1)      1,3         2022	   2  2
0       24(trivial) 0000    1  1   F4          0           2222	   1  1

Here is a sample line in the table:

A2      10,15,16,21 2000    3  2/1 B3          2,6,7,8     0022	   2 1	

1. A2 is a nilpotent orbit O for G in Bala-Carta notation
2. 10,15,16,21 are the corresponding cells (from Binegar's tables)
3. 2000 is the Dynking diagram of O
4. The orbit O has 3 real forms (from Collingwood-McGovern's tables)
5. The group A(O) is Z/2Z, and the Lusztig's quotient A-bar(O) is trivial.
6. The dual orbit is B3
7. The dual cells are 2,6,7,8
8. The Dynkin diagram of O^v is 0022 (note the Dynkin diagrams of G and G^v)
9. The orbit O^v has 2 real forms
10. Both A(O^v) and A-bar(O^v) are trivial

Note: if G is disconnected (which only happens for E₇ adjoint) we cannot read off the number of real forms of a nilpotent orbit from the tables in Collingwood-McGovern. In cases where we can't easily determine this, a (2) or (4) indicates that there are 2 or 4 real forms in the simply connected case respectively; for the adjoint group the number is the same or this/2.

Unipotent Representations Here is the corresponding list of unipotent representations for the big block of split F₄:

Unipotent representations for F4(split)/G^v=F4(split)
Atlas version 0.3./Build date: Nov 19 2007 at 06:09:46.

O^v           diagram(O^v)    O             cell        Unipotent representations

0             0000            F4            0           7*

F4(a3)        0020            F4(a3)        14          98,161,225,285*
                                            13          35,146,191,244,328*
                                            9           81,192,193,194,295*

A2~           2000            C3            12          212*

A2            0002            B3            2           67
                                            6           207*
                                            8           251*
                                            7           324*

B3            0022            A2            15          257
                                            10          149
                                            16          293*
                                            21          325*

F4(a2)        2020            A1+A1~        19          309*
                                            17          267
                                            20          334*

F4(a1)        2022            A1~           22          290,313*
                                            23          299,332*

F4            2222            0             24          331*

Number of orbits: 11
Number of even orbits: 8
Number of cells: 19
Number of unipotent representations: 32

O^v           diagram(O^v)    O             cell        Unipotent representations
F4(a2)        2020            A1+A1~        19          309*
                                            17          267
                                            20          334*

1. The complex nilpotent orbit O^vfor G^v is F4(a2)
2. The diagram of O^v is 2020
3. The diagram of the corresponding (dual) nilpotent orbit for G is A1+A1~
4. There are three cells 17,19,20 (on the G side)
5. Cell 19 contains 1 unipotent representation #309 in the output of the block command for G. The * indicates that this representation is dual to an A(lambda).
6. Cell 17 contains a single unipotent representation 267, not dual to an A(lambda); cell 20 contains a unipotent representation 334 dual to an A(lambda).
7. The infinitesimal character of these unipotent representations is determined by the labelling of the Dynkin diagram: it is the sum of the fundamental weights for the nodes with label 2. In this case it is lambda₁+lambda₃, and is therefore singular on roots 1 and 3.
   (There may be some confusion about roots versus dual roots here. For a clearer example, orbit B3 on the dual side has diagram 0022. By the Dynkin diagram above, the zeroes correspond to long roots for G^v. This means that the infinitesimal character is singular on the short roots for G, i.e. roots 1 and 2 in atlas numbering.)

Note on infinitesimal character characters and strong real forms

If G is simply connected then it is possible to translate to any singular integral infinitesimal character.

If not, given a block and a dual complex orbit, there may not be a translation functor from the infinitesimal character of the block to the infinitesimal character defined by the orbit. Therefore certain orbits are not allowed.

On the other hand there may be more than one strong real form on the dual side, in which case there are more than one corresonding block.

If G is a real simple exceptional group, it is connected unless it is adjoint of type E₇. Consequently these subtleties arise primarily in this case. See the files for real forms of adjoint E₇.

Here is an excerpt from the Stability file for this block, corresponding to the orbit F4(a2) on the dual side:

Special                         Special                           
Orbit   Cells       Diagram A   Dual Orbit  Cells       Diagram  A^v
A1+A1~  17,19,20    0100 2  1   F4(a2)      4,5,11      2020 2   2/1

Parameters: 267,309*,334*
267(204, 70):  10  3  [C-,i1,C-,C+]  231  265  240  290   (  *,  *)  (283,  *)  (  *,  *)  (  *,  *)   1,2,3,2,1,3,2,3,4,3,2,1,3,2,4,3,2,1
309(223, 28):  12  4  [C-,i2,C-,i2]  285  309  292  309   (  *,  *)  (321,323)  (  *,  *)  (313,315)   1,2,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3
334(228, 10):  14  7  [r2,rn,r2,rn]  330  334  331  334   (311,  *)  (  *,  *)  (317,  *)  (  *,  *)   1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4

Dual Parameters: 75,28*,10*
 75( 70,204):   4  3  [C+,r2,C+,C-]  107   74  112   45   (  *,  *)  ( 63,  *)  (  *,  *)  (  *,  *)   4,3,2,3,4,2
 28( 28,223):   2  4  [C+,r1,C+,r1]   49   28   48   28   (  *,  *)  ( 23, 25)  (  *,  *)  ( 15, 17)   4,2
 10( 10,228):   0  0  [i1,ic,i1,ic]    6   10    7   10   ( 13,  *)  (  *,  *)  ( 19,  *)  (  *,  *)   e

Dimension of space of stable characters: 2

Basis of stable characters as sums of irreducibles:
267+334*
267+309*

Basis of stable characters as matrix of coefficients:
1,0,1
1,1,0

Back to the tables.