Nilpotent Orbit and Arthur packet information for E8_s i: number of orbit H: semisimple element of orbit diagram: diagram of orbit dim: dimension BC Levi: Bala Carter Levi Cent_0: centralizer(orbit)_0 (reductive, connected) Z: Z(Cent(O)_0)=Z(Cent_0) A(O): Cent(O)/Cent(O)_0 #RF: number of real forms of O in split dual group C2: number of conjugacy classes of involutions in Cent #A: number of Arthur parameters supported on this nilpotent Cent(O): explicit description of (disconnected) centralizer(O) [a]: see notes following the table ?: question about precise structure no question means the structure is precisely known .2 means semidirect Z/2Z (non-trivial action) #reps: number of unipotent representations attached to this orbit *: the Arthur packets for this orbit are a priori disjoint (Aq-cells or codimension) a=b+c+...: packets are disjoint, sizes adding up to #packets i diagram dim BC Levi Cent_0 Z A(O) #RF C2 #A Cent(O) #reps E8_q E8_c 0 [0,0,0,0,0,0,0,0] 0 8T1 E8 1 [1] 1 3 3 E8 3#=1+1+1 0 0 1 [0,0,0,0,0,0,0,1] 58 A1+7T1 E7 2 [1] 1 4 4 E7 4*=1+1+1+1 0 0 2 [1,0,0,0,0,0,0,0] 92 2A1+6T1 B6 2 [1] 1 5 5 Spin(13) 5*=1+1+1+1+1 0 0 3 [0,0,0,0,0,0,1,0] 112 3A1+5T1 A1+F4 2 [1] 1 6 6 SL(2)xF4 6*=1+1+1+1+1+1 0 0 4 [0,0,0,0,0,0,0,2] 114 A2+6T1 E6 3 [1,2] 2 3 5 E6|2 [a] 10 0 0 5 [0,1,0,0,0,0,0,0] 128 4A1+4T1 C4 2 [1] 1 5 5 Sp(8) 5*=1+1+1+1+1 0 0 6 [1,0,0,0,0,0,0,1] 136 A1+A2+5T1 A5 6 [1,2] 2 4 6 SL(6) [b] 12* 0 0 7 [0,0,0,0,0,1,0,0] 146 2A1+A2+4T1 A1+B3 2 [1] 2 5 5 SL(2)xSpin(7)/(-1,-1)[c] 6* 0 0 8 [1,0,0,0,0,0,0,2] 148 A3+5T1 B5 2 [1] 1 4 4 Spin(11) 4=1+1+1+1 0 0 9 [0,0,1,0,0,0,0,0] 154 3A1+A2+3T1 A1+G2 2 [1] 2 4 4 SL(2)xG2 4*=1+1+1+1 0 0 10 [2,0,0,0,0,0,0,0] 156 2A2+4T1 2G2 1 [1,2] 3 4 4 [2G2]|2 [d] 7 0 0 11 [1,0,0,0,0,0,1,0] 162 A1+2A2+3T1 A1+G2 2 [1] 1 4 4 SL(2)xG2 4*=1+1+1+1 0 0 12 [0,0,0,0,0,1,0,1] 164 A1+A3+4T1 A1+B3 4 [1] 1 6 6 SL(2)xSpin(7) 6=1+1+1+1+1+1 0 0 13 [0,0,0,0,0,0,2,0] 166 D4+4T1 D4 4 [1,2,3] 2 5 5 Spin(8)|S3 [e] 12 0 0 14 [0,0,0,0,0,0,2,2] 168 D4+4T1 F4 1 [1] 1 3 3 F4 3 2 0 15 [0,0,0,0,1,0,0,0] 168 2A1+2A2+2T1 B2 2 [1] 1 3 3 Spin(5) 3 0 0 16 [0,0,1,0,0,0,0,1] 172 2A1+A3+3T1 A1+B2 4 [1] 1 6 6 SL(2)xSpin(5) 6 0 0 17 [0,1,0,0,0,0,1,0] 176 A1+D4+3T1 3A1 8 [1,2,3] 2 8 6 [SL(2)^3]|S3 [f] 14* 0 0 18 [1,0,0,0,0,1,0,0] 178 A2+A3+3T1 B2+T1 2 [1,2] 3 4 6 GSpin(5)|2 [g] 8 0 0 19 [2,0,0,0,0,0,0,2] 180 A4+4T1 A4 5 [1,2] 2 3 4 SL(5).2 [h] 8 0 0 20 [0,0,0,1,0,0,0,0] 182 A1+A2+A3+2T1 2A1 2 [1] 2 4 4 SL(2)xSL(2) 4=1+1+1+1 0 0 21 [0,1,0,0,0,0,1,2] 184 A1+D4+3T1 C3 2 [1] 1 4 4 Sp(6) 4=1+1+1+1 2 0 22 [0,2,0,0,0,0,0,0] 184 A2+D4+2T1 A2 1 [1,2] 3 2 3 PSL(3)|2 [i] 6 0 0 23 [1,0,0,0,0,1,0,1] 188 A1+A4+3T1 A2+T1 3 [1,2] 2 4 4 GL(3).2 [j] 8 0 0 24 [1,0,0,0,1,0,0,0] 188 2A3+2T1 C2 2 [1] 1 3 3 Sp(4) 3=1+1+1 0 0 25 [1,0,0,0,0,1,0,2] 190 D5+3T1 A3 4 [1,2] 2 3 5 SL(4)|2 [k] 10 2 0 26 [0,0,0,1,0,0,0,1] 192 2A1+A4+2T1 A1+T1 2 [1,2] 3 3 4 GL(2).2 [l] 8 0 0 27 [0,0,0,0,0,2,0,0] 194 A2+A4+2T1 2A1 2 [1] 2 3 3 SL(2)xSL(2)/<-1,-1> [m] 5 0 0 28 [2,0,0,0,0,1,0,1] 196 A5+3T1 A1+G2 2 [1] 1 4 4 SL(2)xG2 4=1+1+1+1 0 0 29 [0,0,0,1,0,0,0,2] 196 A1+D5+2T1 2A1 2 [1] 2 4 4 SL(2)xPSL(2) [n] 5 0 0 30 [0,0,1,0,0,1,0,0] 196 A1+A2+A4+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 0 0 31 [2,0,0,0,0,0,2,0] 198 E6+2T1 G2 1 [1,2] 2 2 4 G2xZ2 8 0 0 32 [0,2,0,0,0,0,0,2] 198 A2+D4+2T1 A2 3 [1,2] 3 2 3 SL(3)|2 [o] 4 1 0 33 [2,0,0,0,0,0,2,2] 200 D5+3T1 B3 2 [1] 1 3 3 Spin(7) 3=1+1+1 3 0 34 [0,0,0,1,0,0,1,0] 200 A3+A4+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 0 0 35 [1,0,0,1,0,0,0,1] 202 A1+A5+2T1 2A1 4 [1] 1 4 4 SL(2)xSL(2) 4=1+1+1+1 0 0 36 [0,0,1,0,0,1,0,1] 202 A2+D5+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 0 0 37 [0,1,1,0,0,0,1,0] 204 D6+2T1 2A1 4 [1,2] 2 4 4 [SL(2)xSL(2)].2 [p] 7 0 0 38 [1,0,0,0,1,0,1,0] 204 A1+E6+T1 A1 2 [1,2] 2 2 4 SL(2)xZ2 8 0 0 39 [0,0,0,1,0,1,0,0] 206 E7+T1 A1 2 [1,2,3] 2 2 4 SL(2)xS3 10 0 0 40 [1,0,0,0,1,0,1,2] 208 A1+D5+2T1 2A1 4 [1] 1 4 4 SL(2)xSL(2) 4=1+1+1+1 2 0 41 [0,0,0,0,2,0,0,0] 208 E8 e 1 [S5] 3 1 3 S5 18*=6+7+5 0 0 42 [2,0,0,0,0,2,0,0] 210 A6+2T1 2A1 2 [1] 2 3 3 SL(2)xSL(2)/<-1,-1> [q] 5 0 0 43 [0,1,1,0,0,0,1,2] 210 D6+2T1 2A1 4 [1,2] 2 4 4 [SL(2)xSL(2)].2 [r] 7 4 0 44 [0,0,0,1,0,1,0,2] 212 E7+T1 A1 2 [1,2] 2 2 4 SL(2)xZ2 5 0 0 45 [1,0,0,1,0,1,0,0] 212 A1+A6+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 0 0 46 [2,0,0,0,0,2,0,2] 214 E6+2T1 A2 3 [1,2] 2 2 3 SL(3)|2 [s] 6 1 0 47 [0,0,0,0,2,0,0,2] 214 A2+D5+T1 T1 1 [1,2] 3 2 3 GL(1)|2 [t] 4 1 0 48 [2,0,0,0,0,2,2,2] 216 E6+2T1 G2 1 [1] 1 2 2 G2 2=1+1 3 0 49 [2,1,1,0,0,0,1,2] 216 D6+2T1 B2 2 [1] 1 3 3 Spin(5) 3=1+1+1 3 0 50 [1,0,0,1,0,1,0,1] 216 D7+T1 T1 1 [1,2] 2 2 2 GL(1).2 [u] 4 0 0 51 [1,0,0,1,0,1,0,2] 218 A1+E6+T1 T1 1 [1,2] 2 2 2 GL(1).2 [u] 4 0 0 52 [1,0,0,1,0,1,1,0] 218 A7+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 0 0 53 [2,0,0,1,0,1,0,2] 220 E7+T1 A1 2 [1,2] 2 2 4 SL(2)xZ2 8 2 0 54 [0,0,0,2,0,0,0,2] 220 E8 e 1 [1,2,3] 2 1 2 S3 6 0 0 55 [1,0,0,1,0,1,2,2] 222 A1+E6+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 2 0 56 [2,0,0,0,2,0,0,2] 222 D7+T1 T1 1 [1,2] 3 2 3 GL(1)|2 [v] 5 1 0 57 [0,1,1,0,1,0,2,2] 224 E7+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 3 0 58 [0,0,0,2,0,0,2,0] 224 E8 e 1 [1,2,3] 2 1 2 S3 5 0 0 59 [2,1,1,0,1,1,0,1] 226 D7+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 1 0 60 [0,0,0,2,0,0,2,2] 226 E8 e 1 [1,2,3] 2 1 2 S3 5 7 0 61 [2,1,1,0,1,0,2,2] 228 E7+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 1 0 62 [2,0,0,2,0,0,2,0] 228 E8 e 1 [1,2] 2 1 2 Z2 6 2 0 63 [2,0,0,2,0,0,2,2] 230 E8 e 1 [1,2] 2 1 2 Z2 3 2 0 64 [2,1,1,0,1,2,2,2] 232 E7+T1 A1 2 [1] 1 2 2 SL(2) 2=1+1 2 0 65 [2,0,0,2,0,2,0,2] 232 E8 e 1 [1,2] 2 1 2 Z2 4 1 0 66 [2,0,0,2,0,2,2,2] 234 E8 e 1 [1,2] 2 1 2 Z2 4 6 0 67 [2,2,2,0,2,0,2,2] 236 E8 e 1 [1] 1 1 1 1 1=1 2 0 68 [2,2,2,0,2,2,2,2] 238 E8 e 1 [1] 1 1 1 1 1=1 1 0 69 [2,2,2,2,2,2,2,2] 240 E8 e 1 [1] 1 1 1 1 1=1 1 1 Notes: Lemma; 1 -> G -> H -> Z2 -> 1 where G is connected reductive If the action on G is trivial (whether or not the extension split) then the number of conjugacy classes of involutions of H is strictly greater than those of G. In other words #AP > C_2. [a] extension split (E8_centralizer_components.txt), #AP\ne 2*C2 => nontrivial action [b] extension is split (E8_centralizer_components.txt), must be nontrivial action since #AP\ne 2C_2 [c] E8_centralizer_isogenies.txt [d] extension splits (E8_centralizer_components.txt) nontrivial action (#AP \ne 2C_2) [G2xG2]\rtimes Z2 (swap action) [e] only way to get #AP=5 [f] extension splits (E8_centralizer_components.txt) only way to get #AP=6 [SL(2)^3]\rtimes S3 S3 acting by triality [g] extension splits (E8_centralizer_components.txt) center of identity component is GL(1) (E8_centralizer_isogenies.txt) action nontrivial (#AP \ne 2C_2) => GSpin(5)\rtimes Z2 acting by inverse on the center [h] E8_centralizer_isogenies.txt: possibly non-split extension 1 -> SL(5) -> Cent -> Z2 -> 1 Cent= delta^2=g\in SL(5), \delta(g)=transpose(g)^{-1}, \delta^2(g)=g => g\in Z also delta(delta^2)delta^{-1}=delta^2 => transpose(g)^{-1}=g => g=1. So the extension does in fact split => action can't be nontrivial (#AP \ne 2*|C_2|) => SL(5)\rtimes Z2, acting by transpose-inverse Note: this is a rare example where the order predicted by E8_centralizer_isogenies.txt is too big (4 not 2) [i] extension splits (E8_centralizer_components.txt) => PSL(2)\rtimes Z2, acting by transpose-inverse (can't be trivial action #AP\ne 2|C_2| 1 -> PSL(2)-> Cent -> Z2 -> 1 [h] extension necessarily split, #AP\ne 2C2 => PSL(2)\rtimes Z2 acting by outer automorphism [j] center is GL(1) (E8_centralizer_isogenies.txt) 1 -> GL(3) -> Cent -> Z2 -> 1 \delta^2\in Z^\delta if \delta acts trivially on Z then this automatically splits # (#AP \ne 2|C_2|) => \delta acts by inverse on Z, and therefore by outer automorphism of GL(3), i.e. delta acts by transpose-inverse, therefore trivially on elements of order 2 in Z if sequence splits then there are at least 5 conjugacy classes of involutions # Cent= delta^2=-I, delta acts by transpose-inverse [k] extension splits (E8_centralizer_components.txt) nontrivial action # => SL(5)\rtimes Z2 acting by transpose-inverse [l] If action is trivial the sequence splits, a contradiction => delta acts by transpose-inverse if delta^2=1 => #AP>4 a contradiction, so GL(2).Z2 nonsplit extension, delta acty by transpose-inverse [m,n] E8_centralizer_isogenies.txt [o] extensions splits (automatic if action is trivial; also see E8_centralizer_components.txt) trivial/split extension => # => SL(3)\rtimes Z2, non-trivial action [p] Action is nontrivial by the Lemma E8_centralizer_isogenies.txt: possibly non-split extension If the extension splits then #AP=5 a contradiction => [SL(2)xSL(2)].2 (nontrivial action, delta^2=[-1,-1] [q] E8_centralizer_isogenies.txt (derived group is simply connected) [r] Extensions split (E8_centralizer_components.txt), action nontrivial by the Lemma => [SL(2)xSL(2)/<-1,-1>].2 (nontrivial action, delta^2=[-1,-1] [s] E8_centralizer_isogenies.txt: possibly non-split extension but obviously it splits (no elements of order 2 in Z) action not nontrivial => SL(3)\rtimes Z2 (outer action) [t] E8_centralizer_components.txt=> extension splits, obviously non-trivial action GL(1)\rtimes Z2 (acting by inverse) [u] E8_centralizer_components.txt => extension might not split 1 -> GL(1) -> Cent -> Z2 -> 1 action trivial => splits + Lemma => contradiction so action is by inverse Lemma => #AP>C2 contradiction => nonsplit extension: jzj^{-1}=1/z, j^2=-1 aka normalizer of a torus in SL(2) [v] action trivial => split => #AP=4 contradiction action by inverse, E8_centralizer_isogenies.txt: extension splits GL(1)\rtimes Z2 (acting by inverse)