Unipotent Representatons of E6(quat) in block dual to E6(F4) G= E6(quaternionic) = compact inner class, quasisplit (adjoint) G^vee= E6(F4) = split inner class, not split (simply connected) cpt herm quat F4 0 0 45 split 1 513 1881 quaternionic E6(F4) special special orbit cells dualorbit cells diagram #O_R E6 0 0 2 000000 1 E6(a1) 1 2A1 1 100001 (not even) D5 2 2A2 0 200002 1 #O_R is the number of real orbits of real points of (simply connected) E6(F4). This group is connected, so this can be read off from the tables in Collingwood-McGovern. b | abcdef means a-c-d-e-f 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) => ** Stability is empty for this block of E6(quaterninic) ** E6(F4) cell aq 0: 0,1,2,3,7,9 1: 22 2: 44 Summary: each cell for E6(F4) contains a distinguished A(lambda) (7,22,44) Orbit 2A1 on dual side isn't even Each (even) unipotent packet for E6(quaternionic) in the block is a singleton, dual to the distinguished A(lambda) E6(F4) E6(F4) E6(quat) E6(quat) dual orbit dual A(lambda) cell unipotent 0 44 0 0 (large fundamental series at 0) 2A2 7 2 37 ==================================================================== quaternionic E6(F4) special special orbit cells dualorbit cells diagram #O_R E6 0 0 2 000000 1 cells stable sums 0 0 %stable -d -S 1,2,3,4,5,6 -c 2 lambda is singular at simple roots: 1,2,3,4,5,6 cells:2 Parameters (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 Dual parameters (to those 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 Dimension of space of stable characters: 1 Everything is stable ------------------------------------------------------------------- quaternionic E6(F4) special special orbit cells dualorbit cells diagram #O_R D5 2 2A2 0 200002 1 %stable -d -c 0 -S 2,3,4,5 lambda is singular at simple roots: 2,3,4,5 Parameters (living at lambda): 37 37(1762, 7): 17 4 [C-,C+,rn,rn,rn,C-] 30 41 37 37 37 32 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 1,3,1,4,2,3,1,5,4,2,3,1,4,3,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 Dual parameters (to those living at lambda): 7 7( 7,1762): 3 0 [C+,C-,ic,ic,ic,C+] 14 3 7 7 7 12 ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) ( *, *) 2,4,3,5,4,2 Dimension of space of stable characters: 1 Everything is stable