Atlas of Lie Groups and Representations 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 Gv. Not every such pair arises. For each block, and each even complex nilpotent orbit Ov for Gv(C), there is a collection of unipotent representations, occuring at a certain (usually singular) infinitesimal character determined by Ov. For each block there are up to three kinds of output: Cells and orbits: list of cells and orbits for G and Gv, 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 are examples of each kind of output, and how to interpret them. 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 ``` This is the block for G = split F4 and Gv = split F4. 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 1 ``` and 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 Ov is 0022 (note the Dynkin diagrams of G and Gv) The orbit Ov has 2 real forms Both A(Ov) and A-bar(Ov) 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 ``` Here 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 Ovfor Gv is F4(a2) The diagram of Ov 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 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 Gv. This means that the infinitesimal character is singular on the short roots for G, i.e. roots 1 and 2 in atlas numbering.) There is further information at the end of the file. 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. 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 1,1,0 ``` We 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 Gv are listed. Back to the tables.