Unipotent Packtets for SO(3,2)

Map between cells and Special Orbits for big block of SO(3,2)
G=SO(3,2) G^v=Sp(6,R)

Special		     Special
Orbit	Cells	     Dual Orbit  Cells  #O_R   lambda
5	0 (LDS)	     1111	 5	 1     (0,0)
311	1,2,3	     22		 2,3,4	 3     (1/2,1/2)		
11111	4,5 (C,sgn)  4		 0,1	 2     (3/2,1/2)=rho	     

(221 not special	211 not special)
#O_R= number of real forms of dual orbit


cell    A_q for Sp(4,R)
0:      0
1:      1
2:      2,4
3:      3,5
4:      6
5:      10
-------------------------------------------------------------------
Special		     Special
Orbit	Cells	     Dual Orbit  Cells  #O_R   lambda
5	0 (LDS)	     1111	 5	 1     (0,0)

Unique large discrete series of SO(3,2) is stable at (0,0)

%stable -d -c 5 -S 1,2

lambda is singular at simple roots: 1,2
cells:5
Parameters (living at lambda): 0
 0(0,10):  0  0  [i1,i2]   1   0   ( 2, *)  ( 3, 4)

Dual parameters (to those living at lambda): 10
10(10,0):  3  3  [r2,r1]  11  10   ( 7, *)  ( 8, 9)   1,2,1,2

Dimension of space of stable characters: 1
Everything is stable
-------------------------------------------------------------------
Special		     Special
Orbit	Cells	     Dual Orbit  Cells  #O_R   lambda
311	1,2,3	     22		 2,3,4	 3     (1/2,1/2)		

cells   dual cells    stable sums
3	2	      3+10
2	3	      4+11
1	4	      1+7
one extra stable sum, for example:
2,3     2,3	      3+11 or -3+4 
or
1,2	3,4	      1+11 or -1+4 
etc.

AV(cell 2) = real form #1 of 22
AV(cell 3) = real form #2 of 22
AV(cell 4) = real form #3 of 22

sophus-t43:so32-d-new% stable -d -c 2,3,4 -S 1
%stable -d -c 2,3,4 -S 1

lambda is singular at simple roots: 1
cells:2,3,4
Parameters (living at lambda): 1,3,4,7,10,11
 1(1,10):  0  0  [i1,ic]   0   1   ( 2, *)  ( *, *)
 3(3, 8):  1  2  [C+,r2]   5   4   ( *, *)  ( 0, *)   2
 4(3, 9):  1  2  [C+,r2]   6   3   ( *, *)  ( 0, *)   2
 7(5, 6):  2  1  [i2,C-]   7   2   ( 8, 9)  ( *, *)   2,1,2
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

Dual parameters (to those living at lambda): 11,8,9,6,2,3
11(10,1):  3  3  [r2,rn]  10  11   ( 7, *)  ( *, *)   1,2,1,2
 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
 6( 6,5):  1  1  [r1,C+]   6   7   ( 0, 1)  ( *, *)   1
 2( 2,6):  0  0  [ic,i1]   2   0   ( *, *)  ( 4, *)
 3( 3,6):  0  0  [ic,i1]   3   1   ( *, *)  ( 5, *)

Dimension of space of stable characters: 5
Basis of stable characters expressed as sums of  irreducibles 1,3,4,7,10,11:
1    0    0    0    0    1

1    0    0    0    1    0

1    0    0    1    0    0

-1   0    1    0    0    0

-1   1    0    0    0    0

-------------------------------------------------------------------
Special		     Special
Orbit	Cells	     Dual Orbit  Cells  #O_R   lambda
11111	4,5 (C,sgn)  4		 0,1	 2     (3/2,1/2)=rho	     

trivial, sgn are stable at rho

sophus-t43:so32-d-new% stable -d -c 0,1
%stable -d -c 0,1

cells:0,1
Parameters (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

Dual parameters (to those 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, *)

Dimension of space of stable characters: 2
Basis of stable characters expressed as sums of  irreducibles 8,9:
0   1

1   0