We can also look at the lowest \(K\) types of a
representation. For this we need the command highest_weights:
atlas> whattype highest_weights ?
Overloaded instances of 'highest_weights'
(KGBElt,ratvec)->[(KGBElt,vec)]
((KGBElt,ratvec),KGBElt)->[(KGBElt,vec)]
Param->[(KGBElt,vec)]
(Param,KGBElt)->[(KGBElt,vec)]
atlas>
We will use the first instance of the usage of this function in this case.
A good reference on how to obtain the highest weights of the lowest \(K\)-types of a representation is Anthony Knapp’s paper, “Minimal \(K\)-type formula”. Noncommutative harmonic analysis and Lie groups (Marseille, 1982), 107-118.
To learn about the reverse process of attaching a series of representations to a given \(K\)-type see David Vogan’s book, “Representations of real reductive Lie groups”. Birkhäusser, 1981
Let’s find the lowest \(K\)-types of each minimal principal series of \(Sp(4,\mathbb R )\). We proceed as follows
atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set ps=all_minimal_principal_series(G,rho(G))
Variable ps: [Param] (overriding previous instance, which had type [Param])
atlas>
atlas> void: for p in ps do prints(p, " ", highest_weights (p, KGB(G,2))) od
final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) [(KGB element #2,[ 0, 0 ])]
final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1) [(KGB element #2,[ 1, 0 ]),
(KGB element #2,[ 0, -1 ])]
final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1) [(KGB element #2,[ 1, 0 ]),
(KGB element #2\ ,[ 0, -1 ])
final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1) [(KGB element #2,[ 1, 1 ]),
(KGB element #2\ ,[ -1, -1 ])]
atlas>
The first representation, the trivial one, has lowest \(K\)-type
[0,0]. The next two have lowest \(K\)-types [1,0] and
[0,-1] and the last one has \(K\)-types [1,1] and [-1,-1].
COMMENT: The choice of 2 in the input KGB(G,2) is so that the
output of the \(K\)-types is given in the more familiar
coordinates. We will see more about this when we discuss KGB
elements in more detail.