Packets of unipotent representations in big block of Sp(4,R) Map between cells and Special Orbits for big block of Sp(4,R) G=Sp(4,R) G^v=SO(3,2) Sp(4,R) SO(3,2) Special Special Orbit Cells Dual Orbit Cells #O_R lambda 4 0,1 (LDS) 11111 4,5 1 (0,0) 22 2,3,4 311 1,2,3 2 (1,0) 1111 5 (trivial) 5 0 1 (2,1) (211 not special 221 not special) #O_R= number of real forms of dual orbit Sp(4,R) Translation Table between atlas and Dan's parameters Atlas Dan Description 0 (1,-2) large DS 1 (2,-1) large DS 2 (-1,-2) anti-hol DS 3 (2,1) hol DS 4 (-2,1^+) A(0,-2) 5 (2,1^+) A(2,0) 6 (2_1) A(3/2,-3/2) 7 (2_-1) 8 (2^-,-1) 9 (2^-,1) 10 (2^+,1^+) trivial 11 (2^-,1^-) non-spherical PS 2^+ means "\underline 2^+" in Dan's notation 2_1 means "\underline{2,1} in Dan's notation Duality of Cells Sp(4,R) SO(3,2) 0 LDS 4 C 1 LDS 5 sgn 2 2 3 3 4 1 5 C 0 LDS All associated varieties for SO(3,2) are irreducible cell Aq 0: 0 1: 1,2 2: 3 3: 4 4: 8 5: 9 ================================================================= Unipotent Packets for Sp(4,R) ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 4 0,1 (LDS) 11111 4,5 1 (0,0) cells stable sums 0,1 0+1 (sum of large discrete series) %stable -c 0,1 -S 1,2 lambda is singular at simple roots: 1,2 cells:0,1 Parameters (living at lambda): 0,1 0( 0,6): 0 0 [i1,i1] 1 2 ( 6, *) ( 4, *) 1( 1,6): 0 0 [i1,i1] 0 3 ( 6, *) ( 5, *) Dual parameters (to those living at lambda): 8,9 8(6, 0): 3 3 [r2,r2] 9 10 ( 7, *) ( 5, *) 2,1,2,1 9(6, 1): 3 3 [r2,r2] 8 11 ( 7, *) ( 6, *) 2,1,2,1 Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 0,1: 1 1 ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 22 2,3,4 311 1,2,3 2 (1,0) lambda=(1,0) Atlas Dan at rho Dan at lambda LKT at lambda 2 (-1,-2) (-1,0) (-2,-2) 3 (2,1) (1,0) (2,2) 6 (2_1) (1_0) (1,-1) 8 (2^-,-1) (1^-,0-) (-1,-1) 9 (2^-,1) (1^-,0+) (1,1) 11 (2^-,1^-) (1^+,0^+) (0,0) at lambda: 0+ means S^1 factor, limit HDS (0- = anti HDS) Different from 0^+ which means R^x factor See ABV Example 17.14, especially 27.17 cells stable sums 2,3 2+3+8+9 4 6+11 2,3,4 2+3+6 or -2-3+11 ...(extra?) duality of cells: 2,3,4 -> 2,3,1 respctively Associated varieties for SO(3,2): AV(cell 1) = real form #1 of 311 AV(cell 2) = real form #2 of 311 AV(cell 3) = real form #2 of 311 %stable -c 2,3,4 -S 2 lambda is singular at simple roots: 2 cells:2,3,4 Parameters (living at lambda): 2,3,6,8,9,11 2( 2,6): 0 0 [ic,i1] 2 0 ( *, *) ( 4, *) 3( 3,6): 0 0 [ic,i1] 3 1 ( *, *) ( 5, *) 6( 6,5): 1 1 [r1,C+] 6 7 ( 0, 1) ( *, *) 1 8( 8,3): 2 2 [C-,i1] 4 9 ( *, *) (10, *) 1,2,1 9( 9,3): 2 2 [C-,i1] 5 8 ( *, *) (10, *) 1,2,1 11(10,1): 3 3 [r2,rn] 10 11 ( 7, *) ( *, *) 1,2,1,2 Dual parameters (to those living at lambda): 10,11,7,3,4,1 10(6, 2): 3 3 [rn,r2] 10 8 ( *, *) ( 5, *) 2,1,2,1 11(6, 3): 3 3 [rn,r2] 11 9 ( *, *) ( 6, *) 2,1,2,1 7(5, 6): 2 1 [i2,C-] 7 2 ( 8, 9) ( *, *) 2,1,2 3(3, 8): 1 2 [C+,r2] 5 4 ( *, *) ( 0, *) 2 4(3, 9): 1 2 [C+,r2] 6 3 ( *, *) ( 0, *) 2 1(1,10): 0 0 [i1,ic] 0 1 ( 2, *) ( *, *) Dimension of space of stable characters: 3 Basis of stable characters expressed as sums of irreducibles 2,3,6,8,9,11: 1 1 0 0 0 1 1 1 0 1 1 0 -1 - 1 1 0 0 0 ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 1111 5 (trivial) 5 0 1 (2,1) cells stable sums 5 10 (trivial representation) %stable -c 5 cells:5 Parameters (living at lambda): 10 10(10,0): 3 3 [r2,r1] 11 10 ( 7, *) ( 8, 9) 1,2,1,2 Dual parameters (to those living at lambda): 0 0(0,10): 0 0 [i1,i2] 1 0 ( 2, *) ( 3, 4) Dimension of space of stable characters: 1 Basis of stable characters expressed as sums of irreducibles 10: 1