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This page is under construction (12/3/08)... 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
For each block, and each even complex nilpotent orbit O For each block there are up to three kinds of output:
**Cells and orbits**: list of cells and orbits for G and G^{v}, including information about duality, real forms of orbits, etc.**Representations**: for each even orbit on the dual side, a list of the corresponding unipotent representations of G**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.)
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 1This is the block for G = split F _{4} and G^{v} =
split F_{4}. There are 11 special complex nilpotent orbits
which play a role in this block.
Here is a sample line in the table: A2 10,15,16,21 2000 3 2/1 B3 2,6,7,8 0022 2 1and what it means: - A2 is a nilpotent orbit O for G in Bala-Carta notation
- 10,15,16,21 are the corresponding cells (from Binegar's tables)
- 2000 is the Dynking diagram of O
- The orbit O has 3 real forms (from Collingwood-McGovern's tables)
- The group A(O) is Z/2Z, and the Lusztig's quotient A-bar(O) is trivial.
- The dual orbit is B3
- The dual cells are 2,6,7,8
- The Dynkin diagram of O
^{v}is 0022 (note the Dynkin diagrams of G and G^{v}) - The orbit O
^{v}has 2 real forms - Both A(O
^{v}) and A-bar(O^{v}) are trivial
Note: if G is disconnected (which only happens for E
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: 32Here is a typical entry (there is one for each even nilpotent on the dual side): O^v diagram(O^v) O cell Unipotent representations F4(a2) 2020 A1+A1~ 19 309* 17 267 20 334*This means: - The complex nilpotent orbit O
^{v}for G^{v}is F4(a2) - The diagram of O
^{v}is 2020 - The diagram of the corresponding (dual) nilpotent orbit for G is A1+A1~
- There are three cells 17,19,20 (on the G side)
- 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).
- 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).
- 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
lambda
_{1}+lambda_{3}, 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.)
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 E Stability
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 1,1,0We see the three unipotent representations 267,309,334 again. These irreducible representations (at singular infinitesimal character) span a two dimensional space of stable characters, with basis pi(267)+pi(334) and pi(267)+pi(309). Also the parameters and dual parameters from the block files for G and G ^{v} are listed.
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