Lowest :math:`K`-types of a Representation =========================================== We can also look at the lowest :math:`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 :math:`K`-types of a representation is Anthony Knapp's paper, "Minimal :math:`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 :math:`K`-type see David Vogan's book, "Representations of real reductive Lie groups". BirkhĂ¤usser, 1981 Let's find the lowest :math:`K`-types of each minimal principal series of :math:`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 :math:`K`-type ``[0,0]``. The next two have lowest :math:`K`-types ``[1,0]`` and ``[0,-1]`` and the last one has :math:`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 :math:`K`-types is given in the more familiar coordinates. We will see more about this when we discuss ``KGB`` elements in more detail.