Unipotent Packets for big block of split F4 Map between cells and special orbits for big block of split F4 Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit F4 0(LDS) 0 24 0000 1 F4(a1) 1,3 A1~ 22,23 0001 F4(a2) 4,5,11 A1+A1~ 17,19,20 0100 F4(a3) 9,13,14 F4(a3) 9,13,14 0200 3 C3 12 A2~ 18 0002 1 B3 2,6,7,8 A2 10,15,16,21 2000 3 A2 10,15,16,21 B3 2,6,7,8 2200 2 A2~ 18 C3 12 1012 A1+A1~ 17,19,20 F4(a2) 4,5,11 0202 2 A1~ 22,23 F4(a1) 1,3 2202 2 0 24(trivial) F4 0 2222 1 Note: B3 is dual to A2; C3 is dual to A2~ Appears to contradict duality (Carter pg 440) but must be explained by G -> G^\vee This is the *only* table so that every even orbit with k real forms has >= k cells Duality of Cells and Orbits Cell Orbit Dual_Orbit Dual Cell 0 F4 0 24 1 F4(a1) A1~ 23 2 B3 A2 21 3 F4(a1) A1~ 22 4 F4(a2) A1+A1~ 20 5 F4(a2) A1+A1~ 17 6 B3 A2 16 7 B3 A2 10 8 B3 A2 15 9 F4(a3) F4(a3) 9 10 A2 B3 7 11 F4(a2) A1+A1~ 19 12 C3 A2~ 18 13 F4(a3) F4(a3) 13 14 F4(a3) F4(a3) 14 15 A2 B3 8 16 A2 B3 6 17 A1+A1~ F4(a2) 5 18 A2~ C3 12 19 A1+A1~ F4(a2) 11 20 A1+A1~ F4(a2) 4 21 A2 B3 2 22 A1~ F4(a1) 3 23 A1~ F4(a1) 1 24 0 F4 0 Cells and Associated Varieties cell A(lambda) Associated Variety of cell 0: 7 F4 (1 real form) 1: 4,10,12,19 real form #1 of F4(a1) 3: 11,16 real form #2 of F4(a1) 2: 1,2,15,20 real form #1 of B3 6: 29 real form #2 of B3 7: real form #2 of B3 8: real form #2 of B3 4: 9,14,21,26 real form #1 of F4(a2) 11: 6,8,17,23,24,28 real form #2 of F4(a2) 5: both real forms of F4(a2) 10: 0,22,39 real form #1 of A2 15: 80 real form #2 of A2 16: 123 real form #3 of A2 21: all three real forms of A2 9: 31,44 real form #1 of F4(a3) 13: 3,18,40,46 real form #2 of F4(a3) 14: 5,...,50 real form #3 of F4(a3) 18: 128 A2~ (1 real form) 24: 331 0 (1 real form) 12: 41 C3 not even 19: A1+A1~ not even 20: A1+A1~ not even 17: 113 A1+A1~ not even 22: A1~ not even 23: A1~ not even ===================================================================================== Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit F4 0(LDS) 0 24 0000 1 %stable -d -S 1,2,3,4 -c 24 lambda is singular at simple roots: 1,2,3,4 cells:24 Parameters (living at lambda): 7 7( 7,228): 0 0 [i1,i1,i1,i1] 8 11 10 9 ( 12, *) ( 15, *) ( 19, *) ( 23, *) Dual parameters (to those living at lambda): 331 331(228, 7): 14 7 [r2,r2,r2,r2] 332 335 334 333 (310, *) (313, *) (317, *) (321, *) 1,2,1,3,2,1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 Dimension of space of stable characters: 1 Everything is stable ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit F4(a3) 9,13,14 F4(a3) 9,13,14 0200 3 interesting case Cells Dimension of space of stable sums 14 1 9 2 13 2 13,14 4 (13:2 + 14:1 + 1 extra) 9,14 4 (14:1 + 9:2 + 1 extra) 9,13 5 (9:2 + 13:2 + 1 extra) 9,13,14 9 (all others + 1 extra) Arthur packets real form #1 of F4(a3) -> cell 9 (dual side) -> cell 9 (G side) 191+92+295 81+192+194 all nonzero: for example 81 + 191 + 2x192 + 194 + 295 real form #2 of F4(a3) -> cell 13 (dual side) -> cell 13 (G side) Basis of stable characters expressed as sums of irreducibles 34,147,193,246,327: 34 - 193 + 327 147+193+246 all nonzero: for example 34 - 147 -2 x 193 - 246 + 327 real form #2 of F4(a3) -> cell 14 (dual side) -> cell 14 (G side) unique stable sum in this cell: 98 + 161 + 225 + 285 %stable -d -S 1,3,4 -c 9,13,14 lambda is singular at simple roots: 1,3,4 cells:9,13,14 Parameters (living at lambda): 34,81,98,147,161,191,192,193,194,225,246,285,295,327 34( 34,218): 2 2 [C+,C-,i1,C+] 52 21 33 59 ( *, *) ( *, *) ( 44, *) ( *, *) 2,3,2 81( 75,195): 4 1 [i2,C-,i1,C+] 81 55 79 120 ( 96, 97) ( *, *) (100, *) ( *, *) 2,3,2,1,2,3,2 98( 90,186): 5 4 [C+,C-,C+,C+] 129 72 130 137 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,2,1,2,3,2 147(126,153): 6 2 [i1,C-,C+,i1] 149 117 181 149 (160, *) ( *, *) ( *, *) (169, *) 2,3,4,1,2,3,4,2,3,1,2 161(135,140): 7 4 [C+,C-,C+,C+] 190 132 196 203 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,2,1,2,3,4,2,3,1,2 191(156,116): 8 3 [C+,C-,C+,i1] 224 163 231 193 ( *, *) ( *, *) ( *, *) (220, *) 2,3,4,2,3,2,1,2,3,4,2,3,1,2 192(156,117): 8 3 [C+,C-,C+,i1] 223 164 230 194 ( *, *) ( *, *) ( *, *) (219, *) 2,3,4,2,3,2,1,2,3,4,2,3,1,2 193(157,116): 8 3 [C+,C-,C+,i1] 222 165 229 191 ( *, *) ( *, *) ( *, *) (220, *) 2,3,4,2,3,2,1,2,3,4,2,3,1,2 194(157,117): 8 3 [C+,C-,C+,i1] 221 166 228 192 ( *, *) ( *, *) ( *, *) (219, *) 2,3,4,2,3,2,1,2,3,4,2,3,1,2 225(179, 95): 9 4 [C+,C-,i2,C+] 257 198 225 263 ( *, *) ( *, *) (244,245) ( *, *) 2,3,4,2,3,1,2,3,4,2,3,2,1,2,3,2 246(193, 83): 10 6 [C+,C-,rn,C+] 273 214 246 278 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,3,4,2,3,1,2,3,4,2,3,2,1,2,3,2 285(211, 47): 11 4 [i2,C-,C+,C+] 285 262 308 309 (293,294) ( *, *) ( *, *) ( *, *) 2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2 295(217, 31): 12 5 [rn,C-,C+,C+] 295 275 312 313 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 327(228, 3): 14 7 [rn,r2,rn,rn] 327 335 327 327 ( *, *) (316, *) ( *, *) ( *, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Dual parameters (to those living at lambda): 298,252,234,184,168,133,135,134,136,103,89,49,31,3 298(218, 34): 12 6 [C-,C+,r2,C-] 272 319 297 279 ( *, *) ( *, *) (288, *) ( *, *) 1,2,3,2,1,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 252(195, 75): 10 5 [r1,C+,r2,C-] 252 275 250 219 (222,224) ( *, *) (233, *) ( *, *) 1,3,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 234(186, 90): 9 4 [C-,C+,C-,C-] 190 262 205 203 ( *, *) ( *, *) ( *, *) ( *, *) 1,3,2,3,4,3,2,1,3,2,3,4,3,2,1,3 184(153,126): 8 6 [r2,C+,C-,r2] 186 216 158 186 (160, *) ( *, *) ( *, *) (169, *) 1,3,2,1,4,3,2,1,3,2,3,4,3 168(140,135): 7 4 [C-,C+,C-,C-] 129 198 138 137 ( *, *) ( *, *) ( *, *) ( *, *) 1,3,2,4,3,2,1,3,2,3,4,3 133(116,156): 6 3 [C-,C+,C-,r2] 97 163 107 134 ( *, *) ( *, *) ( *, *) (121, *) 1,3,2,3,4,3,2,1,3,4 135(117,156): 6 3 [C-,C+,C-,r2] 95 165 105 136 ( *, *) ( *, *) ( *, *) (120, *) 1,3,2,3,4,3,2,1,3,4 134(116,157): 6 3 [C-,C+,C-,r2] 96 164 106 133 ( *, *) ( *, *) ( *, *) (121, *) 1,3,2,3,4,3,2,1,3,4 136(117,157): 6 3 [C-,C+,C-,r2] 94 166 104 135 ( *, *) ( *, *) ( *, *) (120, *) 1,3,2,3,4,3,2,1,3,4 103( 95,179): 5 4 [C-,C+,r1,C-] 65 132 103 73 ( *, *) ( *, *) ( 87, 88) ( *, *) 1,3,4,3,2,1,3,4 89( 83,193): 4 1 [C-,C+,ic,C-] 53 115 89 58 ( *, *) ( *, *) ( *, *) ( *, *) 1,4,3,2,1,3,4 49( 47,211): 3 4 [r1,C+,C-,C-] 49 72 27 28 ( 29, 30) ( *, *) ( *, *) ( *, *) 1,3,4,3 31( 31,217): 2 2 [ic,C+,C-,C-] 31 55 14 15 ( *, *) ( *, *) ( *, *) ( *, *) 3,4,3 3( 3,228): 0 0 [ic,i1,ic,ic] 3 11 3 3 ( *, *) ( 18, *) ( *, *) ( *, *) Dimension of space of stable characters: 9 Basis of stable characters expressed as sums of irreducibles 34,81,98,147,161,191,192,193,194,225,246,285,295,327: 0 -1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 -1 0 0 0 0 1 0 0 0 1 0 0 0 0 -1 -1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 -1 -1 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 -1 1 0 0 0 0 0 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit C3 12 A2~ 18 0002 1 %stable -d -S 1,2,3 -c 18 lambda is singular at simple roots: 1,2,3 cells:18 Parameters (living at lambda): 213 213(174,111): 8 2 [i1,i1,i1,C-] 211 212 211 180 (235, *) (236, *) (238, *) ( *, *) 4,3,2,3,4,1,2,3,4,2,3,1,2,3,4 Dual parameters (to those living at lambda): 128 128(111,174): 6 6 [r2,r2,r2,C+] 126 127 126 157 (102, *) (109, *) (111, *) ( *, *) 1,2,1,3,2,1,3,2,3 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 213: 1 ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit B3 2,6,7,8 A2 10,15,16,21 2000 3 %stable -d -S 2,3,4 -c 10,15,16,21 Stable sums: cells dual cells stable sums 2,6 16,21 68+208b 6,7 10,16 68+324 2,8 15,21 68+251 6,8 15,16 -208+251 2,7 10,21 68+324 7,8 10,15 -251+324 2,6,7,8 10,15,16,21 68+324 68+251 68+208 cell 10,15,16 contains A(lambda) cell 21 does not AV(cell 10)= real form #1 of A2 AV(cell 15)= real form #1 of A2 AV(cell 16)= real form #1 of A2 AV(cell 21)= all 3 real forms of A2 lambda is singular at simple roots: 2,3,4 cells:10,15,16,21 Parameters (living at lambda): 68,208,251,324 68( 65,199): 4 3 [C-,C+,rn,C+] 47 97 68 105 ( *, *) ( *, *) ( *, *) ( *, *) 3,1,2,3,2,1 208(169,110): 8 1 [C-,i2,i1,i1] 176 208 207 209 ( *, *) (221,222) (226, *) (233, *) 1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 251(195, 74): 10 5 [C-,rn,C+,rn] 218 251 276 251 ( *, *) ( *, *) ( *, *) ( *, *) 4,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 324(228, 0): 14 7 [r2,rn,rn,rn] 325 324 324 324 (320, *) ( *, *) ( *, *) ( *, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Dual parameters (to those living at lambda): 260,123,80,0 260(199, 65): 10 3 [C+,C-,ic,C-] 291 224 260 230 ( *, *) ( *, *) ( *, *) ( *, *) 2,3,2,4,3,2,1,3,2,3,4,3,2,1,3,2,3,4 123(110,169): 6 5 [C+,r1,r2,r2] 153 123 122 124 ( *, *) ( 94, 96) ( 99, *) (100, *) 2,3,2,3,4,3,2,3,4 80( 74,195): 4 2 [C+,ic,C-,ic] 119 80 56 80 ( *, *) ( *, *) ( *, *) ( *, *) 3,2,3,4,3,2,3 0( 0,228): 0 0 [i1,ic,ic,ic] 1 0 0 0 ( 22, *) ( *, *) ( *, *) ( *, *) Dimension of space of stable characters: 3 Basis of stable characters expressed as sums of irreducibles 68,208,251,324: 1 0 0 1 1 0 1 0 1 1 0 0 ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit A2 10,15,16,21 B3 2,6,7,8 2200 2 stable: cells dual cells stable sums 21 325 10,15,16 146+257+293 Associated varieties for F4(split): AV(cell 2) = real form #1 of B3 AV(cell 6)= real form #2 of B3 AV(cell 7)= real form #2 of B3 AV(cell 8) = real form #2 of B3 %stable -d -S 3,4 -c 2,6,7,8 lambda is singular at simple roots: 3,4 cells:2,6,7,8 Parameters (living at lambda): 146,257,293,325 146(125,153): 6 2 [ic,C-,C+,i1] 146 118 182 148 ( *, *) ( *, *) ( *, *) (171, *) 2,3,4,1,2,3,4,2,3,1,2 257(197, 63): 10 4 [C-,C-,i2,C+] 225 226 257 287 ( *, *) ( *, *) (272,274) ( *, *) 2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 293(217, 29): 12 5 [r2,C-,C+,C+] 294 277 310 314 (285, *) ( *, *) ( *, *) ( *, *) 2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 325(228, 1): 14 7 [r2,r2,rn,rn] 324 326 325 325 (320, *) (318, *) ( *, *) ( *, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Dual parameters (to those living at lambda): 183,65,29,1 183(153,125): 8 6 [rn,C+,C-,r2] 183 217 159 185 ( *, *) ( *, *) ( *, *) (170, *) 1,3,2,1,4,3,2,1,3,2,3,4,3 65( 63,197): 4 4 [C+,C+,r1,C-] 103 99 65 43 ( *, *) ( *, *) ( 52, 54) ( *, *) 3,4,3,2,3,4 29( 29,217): 2 2 [i1,C+,C-,C-] 30 57 12 16 ( 49, *) ( *, *) ( *, *) ( *, *) 3,4,3 1( 1,228): 0 0 [i1,i1,ic,ic] 0 2 1 1 ( 22, *) ( 20, *) ( *, *) ( *, *) Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 146,257,293,325: 0 0 0 1 1 1 1 0 ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit A1+A1~ 17,19,20 F4(a2) 4,5,11 0202 2 stable: cells stable sums 4,5 267+333 5,11 267+309 Note: cells 17,19,20 dual to 5,11,4 cell 5 does not have A(lambda) Associated varieties for F4(split): AV(cell 4 )= real form #1 of F4(a2) AV(cell 11)= real form #2 of F4(a2) AV(cell 5) = both real forms #1,#2 of F4(a2) %stable -d -S 1,3 -c 4,5,11 lambda is singular at simple roots: 1,3 cells:4,5,11 Parameters (living at lambda): 267,309,333 267(204, 70): 10 3 [C+,C-,i1,C-] 291 240 265 231 ( *, *) ( *, *) (284, *) ( *, *) 4,3,2,3,4,2,3,2,1,2,3,4,2,3,1,2,3,4 309(223, 28): 12 4 [i2,C-,i2,C-] 309 292 309 285 (314,315) ( *, *) (321,322) ( *, *) 4,3,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2 333(228, 9): 14 7 [rn,r2,rn,r2] 333 331 333 328 ( *, *) (319, *) ( *, *) (312, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Dual parameters (to those living at lambda): 75,28,9 75( 70,204): 4 3 [C-,C+,r2,C+] 47 112 74 107 ( *, *) ( *, *) ( 64, *) ( *, *) 1,2,3,2,1,3 28( 28,223): 2 4 [r1,C+,r1,C+] 28 48 28 49 ( 16, 17) ( *, *) ( 23, 24) ( *, *) 1,3 9( 9,228): 0 0 [ic,i1,ic,i1] 9 7 9 4 ( *, *) ( 21, *) ( *, *) ( 14, *) Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 267,309,333: 1 0 1 1 1 0 ------------------------------------------------------------------------------------------------------------------ Special Special #real forms of Orbit Cells Dual Orbit Cells diagram #(even) dual orbit A1~ 22,23 F4(a1) 1,3 2202 2 stable: cells stable sums 22 291+314 23 297+334 %stable -d -S 3 -c 1,3 lambda is singular at simple roots: 3 cells:1,3 Parameters (living at lambda): 291,297,314,334 291(215, 45): 11 3 [C-,ic,i1,C-] 267 291 289 259 ( *, *) ( *, *) (300, *) ( *, *) 4,3,2,3,4,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 297(218, 33): 12 6 [C-,r2,C+,C-] 279 296 317 272 ( *, *) (288, *) ( *, *) ( *, *) 4,3,2,3,4,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 314(225, 16): 13 5 [r2,C-,i2,C-] 315 300 314 293 (309, *) ( *, *) (331,335) ( *, *) 4,3,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 334(228, 10): 14 7 [r2,r2,rn,r2] 329 328 334 331 (323, *) (317, *) ( *, *) (310, *) 4,3,4,2,3,4,2,3,2,1,2,3,4,2,3,2,1,2,3,4,2,3,2,1 Dual parameters (to those living at lambda): 47,33,16,10 47( 45,215): 3 3 [C+,rn,r2,C+] 75 47 46 69 ( *, *) ( *, *) ( 36, *) ( *, *) 2,3,2,3 33( 33,218): 2 1 [C+,i1,C-,C+] 59 32 19 52 ( *, *) ( 44, *) ( *, *) ( *, *) 3,2,3 16( 16,225): 1 2 [i1,C+,r1,C+] 17 36 16 29 ( 28, *) ( *, *) ( 7, 11) ( *, *) 3 10( 10,228): 0 0 [i1,i1,ic,i1] 5 4 10 7 ( 25, *) ( 19, *) ( *, *) ( 12, *) Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 291,297,314,334: 0 1 0 1 1 0 1 0