[Davids-MBP-100:~] dav% atlas all
This is 'atlas' (version 1.0.7, axis language version 0.9.9),
the Atlas of Lie Groups and Representations interpreter,
compiled on Oct 15 2019 at 08:22:04.   http://www.liegroups.org/
atlas> set S=SL(2,R)
Variable S: RealForm
atlas> set XS=KGB(S)
Variable XS: [KGBElt] (overriding previous instance, which had type [KGBElt])
atlas> #XS
Value: 3
atlas> set xs=XS[2] {real Borel in SL(2,R)}

{The LAST element of KGB(G) is always the unique open K orbit of Borel
subgroups, corresponding to a (complex) Borel subgroup of the
(complexified) minimal real parabolic subgroup.}

Variable xs: KGBElt
atlas> set qf=parameter(xs,[1]{character of M},[1]{nu, character of A})
Variable qf: Param

{This is the spherical principal series of SL(2,R) (spherical because
the lambda parameter is shifted by rho=1 from the trivial character)
containing the trivial representation of SL(2,R).

atlas> qf
Value: final parameter(x=2,lambda=[1]/1,nu=[1]/1)
atlas> rho(S)
Value: [ 1 ]/1
atlas> composition_series (qf)
Value: 
1*parameter(x=2,lambda=[1]/1,nu=[1]/1) [0]
1*parameter(x=1,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=0,lambda=[1]/1,nu=[0]/1) [1]

{First of these (which atlas labels by the same parameter as the
spherical rep we're studying) is the Langlands quotient of this
spherical rep, namely the trivial rep of SL(2,R). Last two are
discrete series.}

atlas> set qu {unitary} =parameter(xs,[0]{NONTRIV character of M},[0]{nu, UNITARYcharacter of A})

{NONSPHERICAL principal series at nu=0, therefore unitary; this is the
"reducible unitary principal series" for SL(2,R)}

Variable qu: Param

atlas> composition_series (qu)
Value: 
1*parameter(x=1,lambda=[0]/1,nu=[0]/1) [0]
1*parameter(x=0,lambda=[0]/1,nu=[0]/1) [0]
atlas> {NO factor with x=2: NO Langlands quotient}
atlas> {two "limits of discrete series": UNITARY}
atlas> set qu2 {unitary} =parameter(xs,[1]{TRIV character of
M},[0]{nu, UNITARYcharacter of A})

{SPHERICAL principal series at nu=0, therefore unitary; this is an
"irreducible unitary principal series" for SL(2,R)}

Variable qu2: Param
atlas> composition_series (qu2)
Value: 
1*parameter(x=2,lambda=[1]/1,nu=[0]/1) [0]
atlas> {IRREDUCIBLE}
atlas> qu2
Value: final parameter(x=2,lambda=[1]/1,nu=[0]/1)
atlas> {same as unique composition factor: this is Langlands quotient}

atlas> set G=Sp(4,R)
{linear auts of R^4 preserving SYMPLECTIC form}

Variable G: RealForm
atlas> set X=KGB(G)
Variable X: [KGBElt]
atlas> X
Value: [KGB element #0,KGB element #1,KGB element #2,KGB element #3,KGB element #4,KGB element #5,KGB element #6,KGB element #7,KGB element #8,KGB element #9,KGB element #10]

atlas> set p=parameter(X[10]{real Borel in Sp(4,R)},[2,1]{triv on M},[2,1]{nu})
Variable p: Param
atlas> composition_series (p)
Value: 
1*parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1) [0]
1*parameter(x=9,lambda=[2,1]/1,nu=[3,3]/2) [2]
1*parameter(x=8,lambda=[2,1]/1,nu=[2,0]/1) [3]
1*parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1) [3]
1*parameter(x=6,lambda=[2,1]/1,nu=[0,1]/1) [6]
1*parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1) [6]
2*parameter(x=4,lambda=[2,1]/1,nu=[1,-1]/2) [6]
1*parameter(x=1,lambda=[2,1]/1,nu=[0,0]/1) [7]
1*parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1) [7]
atlas> {line by line description of some composition factors}
atlas> {Langlands quotient}
atlas> {last two lines: discrete series reps}
atlas> {other 6 lines: complicated: notice 2*}
atlas> {That comp factor occurs TWICE}

atlas> set p2=parameter(X[10]{real Borel in Sp(4,R)},[1,0]{nontriv on M},[2,1]{nu})
Variable p2: Param

{This is a NONSPHERICAL principal series representation with the same
central and infinitesimal character as p}

atlas> composition_series (p2)
Value: 
1*parameter(x=10,lambda=[3,2]/1,nu=[2,1]/1) [0]
1*parameter(x=9,lambda=[2,1]/1,nu=[3,3]/2) [2]
1*parameter(x=6,lambda=[2,1]/1,nu=[0,1]/1) [6]
1*parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1) [6]
1*parameter(x=4,lambda=[2,1]/1,nu=[1,-1]/2) [6]
1*parameter(x=3,lambda=[2,1]/1,nu=[0,0]/1) [7]
1*parameter(x=2,lambda=[2,1]/1,nu=[0,0]/1) [7]
1*parameter(x=1,lambda=[2,1]/1,nu=[0,0]/1) [7]
1*parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1) [7]
atlas> {1st line is Langlands quotient}
atlas> {last FOUR lines are discrete series}
atlas> {other four comp factors also appear in 1st princ series}

atlas> {Next week: parameters for general G in atlas}
atlas> quit