Given a block for a real group, the "dualblock" command constructs a
block for a real form of the dual group which is dual in the sense of
Vogan to the given block.
Fix a block B of representations of a real form G(R) of G, with
regular integral infinitesimal character. (This is a finite set of
irreducible representations; the the help file for the block
command). According to Vogan {IC4] there is a real form G^v(R) of the
dual group G^v, and a block B^v of representations of G^v(R) with
regular integral infinitesimal character, which is "dual" to B. This
means that there is a bijection B -> B^v, written pi -> pi^v, with a
number of properties.
In the context of Atlas a block B is determined by a real form of G
and a real form of G^v. Associated to these choices are K and K^v, the
(complexified) maximal compact subgroups of the real form of G and
G^v, respectively. The K-orbits on G/B are parametrized by integers
0<=x<=m, and those of K^v on G^v/B^v by integers 0<=y<=n. The B is
parametrized by a subset S of pairs (x,y). See the help file for the
block command. The numbering of the kgb elements for G^v is that of
the dualkgb command.
The dual block B^v is parametrized by {(y,x)| (x,y) in S}.
The dualmap command gives the bijection between B and B^v. Also see
dualkgb.
Example: principal block for Sp(4,R) and its dual
The principal block of Sp(4,R) (containing a finite dimensional
representation) has 12 elements. The corresponding real form of the
dual group is the split one, SO(4,3). There are 11 orbits (0..10) of K
on G/B, and 7 (0..6) of K^v on G^v/B^v.
The block B contains 4 discrete series representations, numbers 0-4,
with parameters (0,6),(1,6),(2,6) and (3,6). Dual to these are 4
principal series representations of SO(4,3), numbers 8-11, with
parameters (6,0),(6,1), (6,2) and (6,3).
Lie type: C2 sc s
main: block
(weak) real forms are:
0: sp(2)
1: sp(1,1)
2: sp(4,R)
enter your choice: 2
possible (weak) dual real forms are:
0: so(5)
1: so(4,1)
2: so(2,3)
enter your choice: 2
Name an output file (return for stdout, ? to abandon):
0( 0,6): 0 0 [i1,i1] 1 2 ( 6, *) ( 4, *)
1( 1,6): 0 0 [i1,i1] 0 3 ( 6, *) ( 5, *)
2( 2,6): 0 0 [ic,i1] 2 0 ( *, *) ( 4, *)
3( 3,6): 0 0 [ic,i1] 3 1 ( *, *) ( 5, *)
4( 4,4): 1 2 [C+,r1] 8 4 ( *, *) ( 0, 2) 2
5( 5,4): 1 2 [C+,r1] 9 5 ( *, *) ( 1, 3) 2
6( 6,5): 1 1 [r1,C+] 6 7 ( 0, 1) ( *, *) 1
7( 7,2): 2 1 [i2,C-] 7 6 (10,11) ( *, *) 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
10(10,0): 3 3 [r2,r1] 11 10 ( 7, *) ( 8, 9) 1,2,1,2
11(10,1): 3 3 [r2,rn] 10 11 ( 7, *) ( *, *) 1,2,1,2
block: dualblock
Name an output file (return for stdout, ? to abandon):
0(0,10): 0 0 [i1,i2] 1 0 ( 2, *) ( 3, 4)
1(1,10): 0 0 [i1,ic] 0 1 ( 2, *) ( *, *)
2(2, 7): 1 1 [r1,C+] 2 7 ( 0, 1) ( *, *) 1
3(3, 8): 1 2 [C+,r2] 5 4 ( *, *) ( 0, *) 2
4(3, 9): 1 2 [C+,r2] 6 3 ( *, *) ( 0, *) 2
5(4, 4): 2 2 [C-,i2] 3 5 ( *, *) ( 8,10) 1,2,1
6(4, 5): 2 2 [C-,i2] 4 6 ( *, *) ( 9,11) 1,2,1
7(5, 6): 2 1 [i2,C-] 7 2 ( 8, 9) ( *, *) 2,1,2
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
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
Example: Block of the irreducible principal series of Sp(4,R)
Take G(R)=Sp(4,R) and G^v(R)=SO(5) (compact). Then K\G/B has 11
elements as in the previous example, and K^v\G^v/B^v is a
singleton. In this case the block B consists of a single irreducible
principal series representation of Sp(4,R), parametrized by
(10,0). The dual block consists of the trivial representation of
SO(5), parametrized by (0,10).
Lie type: C2 sc s
main: block
(weak) real forms are:
0: sp(2)
1: sp(1,1)
2: sp(4,R)
enter your choice: 2
possible (weak) dual real forms are:
0: so(5)
1: so(4,1)
2: so(2,3)
enter your choice: 0
Name an output file (return for stdout, ? to abandon):
0(10,0): 3 3 [rn,rn] 0 0 (*,*) (*,*) 1,2,1,2
block: dualblock
Name an output file (return for stdout, ? to abandon):
0(0,10): 0 0 [ic,ic] 0 0 (*,*) (*,*)
[IC4] David A. Vogan, Jr, Irreducible characters of semisimple Lie groups IV:
Character-multiplicity duality. Duke Math. J. 49 (1982), no. 4, pp. 943--1073.