Unipotent Representatons of E6(F4) only one block, dual to E6(quaternionic) cpt herm quat F4 0 0 45 split 1 513 1881 E6(F4) quaternionic special special orbit cells dualorbit cells diagram #O_R 0 2 E6 0 222222 1 2A1 1 E6(a1) 1 222022 1 2A2 0 D5 2 220202 2 #O_R is the number of real orbits of real points of E6_ad_quaternionic. This group is connected, so this can be read off from the tables in Collingwood-McGovern. b | abcdef means a-c-d-e-f the diagram gives the simple roots at which lambda is singular example: 220202 means lambda is singular at roots 3,5 Note: adjoint/simply connected don't matter for E6 since Z=Z/3. Note: E6(F4) has only one Cartan (fundamental, (C^x)^2 x (S^1)^2, connected) Dual Cartan for E6(quat) is the most split one, (R^x)^2 x (C^x)^2 Even though dual Cartan is not connected: ** Stability is empty for E6(F4) ** E6(quat) has no A(lambda)s in this block L-packets are singletons W_i=W_{i,c}=W(D_4) Recall W(G,H)=W(F_4)=W(D_4)\rtimes W(A_2) Summary: E6(quat) E6(F4) E6(F4) E6(quat) dual orbit orbit cell unipotents E6 0 2 44 (trivial) E6(a1) 2A1 1 28,32,33,37,38 1 packet D5 2A2 0 0,2,13,19 2 packets ==================================================================== E6(F4) quaternionic special special orbit cells dualorbit cells diagram #O_R 0 2 E6 0 222222 1 %stable -d -c 0 cell stable sums 0 44 (trivial representation) Parameters (living at lambda): 44 44(44, 851): 12 0 [C-,ic,ic,ic,ic,C-] 42 44 44 44 44 43 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,4,2,5,4,3,1,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 0 0( 851,44): 8 4 [C+,rn,rn,rn,rn,C+] 2 0 0 0 0 1 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,2,3,4,5,4,2,3,4,5 Dimension of space of stable characters: 1 Everything is stable ------------------------------------------------------------------- E6(F4) quaternionic special special orbit cells dualorbit cells diagram #O_R 0 2 E6 0 222222 1 2A1 1 E6(a1) 1 222022 1 2A2 0 D5 2 220202 2 cell stable sums 1 28,32,33,37,38 (everything stable) %stable -d -c 1 -S 4 lambda is singular at simple roots: 4 cells:1 Parameters (living at lambda): 28,32,33,37,38 28(28,1537): 7 0 [ic,C-,C-,C+,C-,ic] 28 22 24 31 23 28 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,1,4,3,5,4,3,1,6,5,4,2,3 32(32,1432): 8 0 [ic,C-,ic,C+,C-,C+] 32 25 32 35 27 37 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,5,4,2,3,1,4,3,5,4,2,6,5,4,2,3 33(33,1431): 8 0 [C+,C-,C-,C+,ic,ic] 38 26 29 36 33 33 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,1,4,2,5,4,3,1,6,5,4,2,3,4,5 37(37,1305): 9 0 [ic,C-,ic,C+,C-,C-] 37 30 37 39 30 32 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,5,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 38(38,1304): 9 0 [C-,C-,C-,C+,ic,ic] 33 34 34 41 38 38 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,4,2,5,4,3,1,6,5,4,2,3,4,5,6 Dual parameters (to those living at lambda): 16,12,11,7,6 16(1537,28): 13 4 [rn,C+,C+,C-,C+,rn] 16 22 20 13 21 16 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,4,2,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4,6 12(1432,32): 12 4 [rn,C+,rn,C-,C+,C-] 12 19 12 9 17 7 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,4,2,3,1,4,3,6,5,4,2,3,1,4,3,5,4,6 11(1431,33): 12 4 [C-,C+,C+,C-,rn,rn] 6 18 15 8 11 11 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,4,2,3,4,5,4,2,3,4,5,6,5,4,2,3,1,4,5,6 7(1305,37): 11 4 [rn,C+,rn,C-,C+,C+] 7 14 7 5 14 12 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4 6(1304,38): 11 4 [C+,C+,C+,C-,rn,rn] 11 10 10 3 6 6 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 4,2,3,4,5,4,2,3,4,5,6,5,4,2,3,4,5,6 Dimension of space of stable characters: 5 Everything is stable ------------------------------------------------------------------- E6(F4) quaternionic special special orbit cells dualorbit cells diagram #O_R 2A2 0 D5 2 220202 2 cell stable sums 2 0,2,13,19 (everything is stable) Note: two real forms of orbit, can't learn anything from this %stable -d -c 2 -S 3,5 lambda is singular at simple roots: 3,5 cells:2 Parameters (living at lambda): 0,2,13,19 0( 0,1790): 0 0 [C+,ic,C+,ic,C+,C+] 2 0 1 0 1 2 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2( 2,1786): 1 0 [C-,ic,C+,ic,C+,C-] 0 2 4 2 5 0 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,6 13(13,1731): 4 0 [C-,C+,C+,C-,C+,C-] 6 19 15 9 18 8 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,4,3,1,5,6,5,4 19(19,1684): 5 0 [C-,C-,C+,ic,C+,C-] 12 13 23 19 24 14 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,4,3,1,5,6,5,4,2 Dual parameters (to those living at lambda): 44,42,31,25 44(1790, 0): 20 4 [C-,rn,C-,rn,C-,C-] 42 44 43 44 43 42 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,2,3,1,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,2,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 42(1786, 2): 19 4 [C+,rn,C-,rn,C-,C+] 44 42 40 42 39 44 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,1,4,2,3,1,4,3,5,4,2,3,1,4,3,5,4,2,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3 31(1731,13): 16 4 [C+,C-,C-,C+,C-,C+] 38 25 29 35 26 36 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,3,1,4,3,1,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,2,6,5 25(1684,19): 15 4 [C+,C+,C-,rn,C-,C+] 32 31 21 25 20 30 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 3,1,4,3,1,5,4,2,3,1,4,3,5,4,6,5,4,2,3,1,4,3,5,4,6,5 Dimension of space of stable characters: 4 Everything is stable