The "branch" command is part of the K-type suite of commands:
KTypeform, Ktypemat, srtest and branch. It describes the irreducible
representations of K "K-types". See [David Vogan, Branching to a
Maximal Compact Subgroup, in proceedings of the conference for Roger
Howe, Singapore 2007].
branch computes the K-types of a given final standard module, up to
a given height. Each such K-type is the lowest K-type of a final
standard module.
This information is precisely the first column of the second matrix in
the output of the Ktypemat command. See the help file for Ktypemat for
more information.
See the help file for Ktypeform for an overview of K-types and normal
final standard modules.
Example 1: K-types of the holomorphic discrete series of SL(2,R):
real: branch
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: 0
Height of representation [ 2 ]@(0)#0 is 2.
Give height bound: 20
Name an output file (return for stdout, ? to abandon):
+ [ 2 ]@(0)#0 + [ 6 ]@(0)#0 + [10 ]@(0)#0 + [14 ]@(0)#0 + [18 ]@(0)#0
The given standard module is the holomorphic discrete series
representation of SL(2,R) with infinitesimal character rho. The
formula says the K-types up to height 20 are (the lowest K-types of
the given standard modules with multipicity one, in this case
holomorphic discrete series with infinitesimal character 1,3,5,7,9.
See Example 1 in the help file for Ktypemat.