Unipotent Packets for big block of Sp(6,R) special special dual orbit cells dual orbit cells #O_R lambda 6 0,3 1^7 14,15 1 (0,0,0) 42 1,4,6,7 31111 9,11,12,13 2 (1,0,0) 222 2,5,9,10,11 331 4,5,6,8,10 2 (1,1,0) 2211 12,13,14 511 1,2,3 2 (2,1,0) 1^6 15 7 0 1 (3,2,1) 33 8 322 7 not even Duality of cells: sp(6) SO(4,3) A_q for SO(4,3) 0 15 49 1 11 15 2 10 3 14 48 4 9 14 5 8 6 12 19 7 13 20 8 7 11 9 6 10 10 5 9 11 4 2,4 12 3 6 13 2 5 14 1 1,3,7,8 15 0 0 All AV for SO(4,3) are irreducible except possibly cells 8,10 These are reducible: see orbit 222 below =================================================================== special special dual orbit cells dual orbit cells #O_R lambda 6 0,3 1^7 14,15 1 (0,0,0) %stable -d -S 1,2,3 -c 14,15 lambda is singular at simple roots: 1,2,3 cells:14,15 Parameters (living at lambda): 3,4 3( 3,24): 0 0 [i1,i1,i1] 6 1 5 ( 8, *) (11, *) (12, *) 4( 4,24): 0 0 [i1,i1,i1] 5 7 6 ( 9, *) (10, *) (14, *) Dual parameters (to those living at lambda): 48,49 48(24, 3): 6 5 [r2,r2,r2] 51 46 50 (43, *) (42, *) (37, *) 3,2,3,2,1,2,3,2,1 49(24, 4): 6 5 [r2,r2,r2] 50 52 51 (44, *) (41, *) (39, *) 3,2,3,2,1,2,3,2,1 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 3,4: 1 1 -------------------------------------------------------------------- special special dual orbit cells dual orbit cells #O_R lambda 42 1,4,6,7 31111 9,11,12,13 2 (1,0,0) stable: cells stable sums 1,4 1+7+38+40 6,7 8+9+30+31 1,4,6,7 one extra, for example - 1 - 7 + 8 + 9 AV(cell 9) = AV(cell 11) = real form #1 of 31111 AV(cell 12) = AV(cell 13) = real form #2 of 31111 sophus-t43:sp6-d-new% stable -d -S 2,3 -c 9,11,12,13 %stable -d -S 2,3 -c 9,11,12,13 lambda is singular at simple roots: 2,3 cells:9,11,12,13 Parameters (living at lambda): 1,7,8,9,30,31,38,40 1( 1,24): 0 0 [ic,i1,i1] 1 3 0 ( *, *) (11, *) (15, *) 7( 7,24): 0 0 [ic,i1,i1] 7 4 2 ( *, *) (10, *) (13, *) 8( 8,23): 1 2 [r1,C+,i1] 8 18 9 ( 3, 6) ( *, *) (16, *) 1 9( 9,23): 1 2 [r1,C+,i1] 9 17 8 ( 4, 5) ( *, *) (16, *) 1 30(28,13): 3 1 [C-,i1,i1] 19 31 32 ( *, *) (36, *) (39, *) 1,2,3,2,1 31(29,13): 3 1 [C-,i1,i1] 20 30 33 ( *, *) (36, *) (37, *) 1,2,3,2,1 38(35,10): 4 3 [C-,C+,rn] 28 47 38 ( *, *) ( *, *) ( *, *) 3,1,2,3,2,1 40(36,10): 4 3 [C-,C+,rn] 26 45 40 ( *, *) ( *, *) ( *, *) 3,1,2,3,2,1 Dual parameters (to those living at lambda): 46,52,43,44,19,20,14,15 46(24, 1): 6 5 [rn,r2,r2] 46 48 45 ( *, *) (42, *) (40, *) 3,2,3,2,1,2,3,2,1 52(24, 7): 6 5 [rn,r2,r2] 52 49 47 ( *, *) (41, *) (38, *) 3,2,3,2,1,2,3,2,1 43(23, 8): 5 3 [i2,C-,r2] 43 35 44 (48,51) ( *, *) (36, *) 3,2,3,2,1,2,3,2 44(23, 9): 5 3 [i2,C-,r2] 44 34 43 (49,50) ( *, *) (36, *) 3,2,3,2,1,2,3,2 19(13,28): 3 4 [C+,r2,r2] 30 20 21 ( *, *) (16, *) (13, *) 3,2,3,2 20(13,29): 3 4 [C+,r2,r2] 31 19 22 ( *, *) (16, *) (12, *) 3,2,3,2 14(10,35): 2 2 [C+,C-,ic] 24 8 14 ( *, *) ( *, *) ( *, *) 2,3,2 15(10,36): 2 2 [C+,C-,ic] 23 7 15 ( *, *) ( *, *) ( *, *) 2,3,2 Dimension of space of stable characters: 3 Basis of stable characters expressed as sums of irreducibles 1,7,8,9,30,31,38,40: 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 -1 -1 1 1 0 0 0 0 ---------------------------------------------------------------------------------- special special dual orbit cells dual orbit cells #O_R lambda 222 2,5,9,10,11 331 4,5,6,8,10 2 (1,1,0) All cells have A(lambda) except 8,10 Cells Dimension of space of stable sums 4,8,10 1 5,6,8,10 1 4,5,6 1 (probably not basic) (not basic:) 4,5,6,8 1 4,6,8,10 1 4,5,6,10 1 4,5,8,10 1 4,5,6,8,10 2 Guess: the only AVs consistent with this are: AV(cell 4) = real form #1 of 331 AV(cell 5) = real form #2 of 331 AV(cell 6) = real form #2 of 331 AV(cell 8) = both real forms of 331 AV(cell 10) = both real forms of 331 cells stable sums 4,8,10 12+22+52 5,6,8,10 21+22+42+43: %stable -c 4,5,6,8,10 -S 1,3 -d lambda is singular at simple roots: 1,3 cells:4,5,6,8,10 Parameters (living at lambda): 21,22,42,43,52 21(21,18): 2 1 [C+,C-,i1] 32 13 19 ( *, *) ( *, *) (27, *) 2,3,2 22(22,18): 2 1 [C+,C-,i1] 33 15 20 ( *, *) ( *, *) (29, *) 2,3,2 42(38, 7): 4 2 [i2,C-,i1] 42 34 43 (44,45) ( *, *) (49, *) 2,3,2,1,2,3,2 43(39, 7): 4 2 [i2,C-,i1] 43 35 42 (46,47) ( *, *) (49, *) 2,3,2,1,2,3,2 52(44, 2): 6 5 [rn,r2,rn] 52 50 52 ( *, *) (48, *) ( *, *) 3,2,3,2,1,2,3,2,1 Dual parameters (to those living at lambda): 32,33,9,10,2 32(18,21): 4 4 [C-,C+,r2] 21 38 30 ( *, *) ( *, *) (25, *) 3,1,2,3,2,1 33(18,22): 4 4 [C-,C+,r2] 22 40 31 ( *, *) ( *, *) (26, *) 3,1,2,3,2,1 9( 7,38): 2 3 [r1,C+,r2] 9 17 10 ( 5, 7) ( *, *) ( 3, *) 3,1 10( 7,39): 2 3 [r1,C+,r2] 10 18 9 ( 6, 8) ( *, *) ( 3, *) 3,1 2( 2,44): 0 0 [ic,i1,ic] 2 0 2 ( *, *) ( 4, *) ( *, *) Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 21,22,42,43,52: 1 1 0 0 1 1 1 1 1 0 ------------------------------------------------------------------- special special dual orbit cells dual orbit cells #O_R lambda 2211 12,13,14 511 1,2,3 2 (2,1,0) Cells Dimension of space of stable sums 1 1 2,3 1 not basic: 1,2 1 1,3 1 1,2,3 2 cells stable sums 1 45+47 2,3 32,33,44,46 AV(cell 1) = real form #1 of 511 AV(cell 2) = real form #2 of 511 AV(cell 3) = real form #2 of 511 %stable -d -c 1,2,3 -S 3 lambda is singular at simple roots: 3 cells:1,2,3 Parameters (living at lambda): 32,33,44,45,46,47 32(30,13): 3 1 [C-,ic,i1] 21 32 30 ( *, *) ( *, *) (39, *) 1,2,3,2,1 33(31,13): 3 1 [C-,ic,i1] 22 33 31 ( *, *) ( *, *) (37, *) 1,2,3,2,1 44(40, 5): 5 3 [r2,C-,i1] 45 39 46 (42, *) ( *, *) (50, *) 2,3,2,1,2,3,2,1 45(40, 6): 5 3 [r2,C-,i1] 44 40 47 (42, *) ( *, *) (51, *) 2,3,2,1,2,3,2,1 46(41, 5): 5 3 [r2,C-,i1] 47 37 44 (43, *) ( *, *) (50, *) 2,3,2,1,2,3,2,1 47(41, 6): 5 3 [r2,C-,i1] 46 38 45 (43, *) ( *, *) (51, *) 2,3,2,1,2,3,2,1 Dual parameters (to those living at lambda): 21,22,5,7,6,8 21(13,30): 3 4 [C+,rn,r2] 32 21 19 ( *, *) ( *, *) (13, *) 3,2,3,2 22(13,31): 3 4 [C+,rn,r2] 33 22 20 ( *, *) ( *, *) (12, *) 3,2,3,2 5( 5,40): 1 2 [i1,C+,r2] 7 13 6 ( 9, *) ( *, *) ( 0, *) 3 7( 6,40): 1 2 [i1,C+,r2] 5 15 8 ( 9, *) ( *, *) ( 1, *) 3 6( 5,41): 1 2 [i1,C+,r2] 8 12 5 (10, *) ( *, *) ( 0, *) 3 8( 6,41): 1 2 [i1,C+,r2] 6 14 7 (10, *) ( *, *) ( 1, *) 3 Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 32,33,44,45,46,47: 0 0 0 1 0 1 1 1 1 0 1 0 ------------------------------------------------------------------- special special dual orbit cells dual orbit cells #O_R lambda 1^6 15 7 0 1 (3,2,1) trivial representation #50 at rho %stable -d -c 0 Parameters (living at lambda): 50 50(44, 0): 6 5 [r2,r2,r1] 51 52 50 (49, *) (48, *) (44,46) 3,2,3,2,1,2,3,2,1 Dual parameters (to those living at lambda): 0 0( 0,44): 0 0 [i1,i1,i2] 1 2 0 ( 3, *) ( 4, *) ( 5, 6) Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 50: 1