The "block" command prints out a list of representations in a single block.
A block is an equivalence class of irreducible representations, where the
equivalence relation is generated by the relation of two representations
having a non-split extension. At present we can consider only blocks of
representations having regular integral infinitesimal character.
The user must specify a (real form defining a) real group and a dual real form
(a real form of the dual complex group); unless these were already specified
before or only one choice is available, the program will prompt for a
selection. These data determine a block (up to the equivalences of categories
provided by the Jantzen-Zuckerman translation principle). For a given real
group $G$, the largest and most interesting block is the one containing a
finite-dimensional representation of $G$; it is selected by choosing in the
dual group the quasisplit real form (always the last choice provided).
Each row of the output corresponds to a single irreducible representation in
the block. The output may be saved to a file, or viewed on the terminal.
The first column is a numbering of the elements, from 0 to $n-1$ (with $n$ the
number of elements in the block), each followed by a pair $(x,y)$. Here $x$ is
the number of an orbit of $K$ on $G/B$, as enumerated by the 'kgb' command.
Similarly $y$ is the number of an orbit of $K^\vee$ on $G^\vee/B^\vee$, as
enumerated by the 'kgb' command for the dual group and the chosen dual real
form (or more simply by the 'dualkgb' command). The pair $(x,y)$ is a
parameter for a representation of $G$ (so the negative transpose of the
involution of $H$ associated to $x$ is the one of $H^\vee$ associated to $y$.)
The next column gives the length of the parameter $(x,y)$.
Then comes, in brackets, a sequence giving the types of all simple roots:
i1: imaginary, non-compact, type 1 (single valued Cayley transform)
i2: imaginary, non-compact, type 2 (double valued Cayley transform)
ic: imaginary compact (no Cayley or inverse Cayley transforms)
r1: real, type 1 (double valued inverse Cayley transform)
r2: real, type 2 (single valued inverse Cayley transform)
rn: real, non-parity (no inverse Cayley or Cayley transforms)
C+: complex, cross action increases length
C-: complex, cross action decreases length
Type 1 real and imaginary roots correspond to a subgroup $SL(2,\R)$.
The imaginary Cayley transforms are single valued, and the inverse real
Cayley transforms are double valued. Type 2 real and imaginary roots
correspond to a subgroup $PGL(2,\R)$, in which the situation is reversed.
A slightly different formulation of these explanations may be found in
the help file for the blocku command.
The next sequence of $r$ columns (where $r$ is the number of simple roots)
gives the cross actions of the simple roots: number $s$ of the sequence gives
the number of the parameter to which the current parameter is sent under the
cross action by simple root $s$.
The next $r$ columns give Cayley transform(s) or inverse Cayley transform(s)
of the current parameter by the respective simple root, if any are defined. In
case of a non-compact imaginary root there are one or two Cayley transforms,
and in the case of real type 1 or 2 root there are two or one inverse Cayley
transforms, and in other cases none are defined. The format used is $(k,k')$
where $k$ and $k'$ are either the number of a parameter reached from the
current parameter by the Cayley transform or inverse Cayley transform, or '*'
if at that position no such transform is defined.
The next column gives the number of the Cartan class determined by the
above-mentioned involution $x$ of $H$ (the numbering matches the one
produced by the 'cartan' command).
The final column gives the twisted involution of the Weyl group that specifies
the involution of $H$ (see the help for the 'kgb' command).
Example: Here is the output of block for the (big) block of Sp(4,R)
containing the trivial representation:
empty: block
Lie type: C2 sc s
(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(3,2)
enter your choice: 2
Name an output file (return for stdout, ? to abandon):
0( 0,6): 0 [i1,i1] 1 2 ( 4, *) ( 5, *) 0 e
1( 1,6): 0 [i1,i1] 0 3 ( 4, *) ( 6, *) 0 e
2( 2,6): 0 [ic,i1] 2 0 ( *, *) ( 5, *) 0 e
3( 3,6): 0 [ic,i1] 3 1 ( *, *) ( 6, *) 0 e
4( 4,5): 1 [r1,C+] 4 9 ( 0, 1) ( *, *) 1 1
5( 5,4): 1 [C+,r1] 7 5 ( *, *) ( 0, 2) 2 2
6( 6,4): 1 [C+,r1] 8 6 ( *, *) ( 1, 3) 2 2
7( 7,3): 2 [C-,i1] 5 8 ( *, *) (10, *) 2 1,2,1
8( 8,3): 2 [C-,i1] 6 7 ( *, *) (10, *) 2 1,2,1
9( 9,2): 2 [i2,C-] 9 4 (10,11) ( *, *) 1 2,1,2
10(10,0): 3 [r2,r1] 11 10 ( 9, *) ( 7, 8) 3 2,1,2,1
11(10,1): 3 [r2,rn] 10 11 ( 9, *) ( *, *) 3 2,1,2,1