atlas> set G=Sp(4,R) Identifier G: RealForm atlas> print_block(trivial(G)) Parameter defines element 10 of the following block: 0: 0 [i1,i1] 1 2 ( 6, *) ( 4, *) *(x= 0,lam=rho+ [0,0], nu= [0,0]/1) e 1: 0 [i1,i1] 0 3 ( 6, *) ( 5, *) *(x= 1,lam=rho+ [0,0], nu= [0,0]/1) e 2: 0 [ic,i1] 2 0 ( *, *) ( 4, *) *(x= 2,lam= {The terms in brackets give the types of roots:} atlas> {i*: imaginary; ic: compact; i1/i2: noncompact} atlas> {r*: real; never mind r1/r2/rn} atlas> {C*: complex} atlas> {the next two columns give the cross action} atlas> {The next two columns give Cayley transforms, defined if a simple root is i1/i2/r1/r2} atlas> Identifier G: RealForm atlas> set p=trivial(G) Identifier p: Param atlas> set x=KGB(G,2) Identifier x: KGBElt atlas> print_K_std(p,x,30) 1*(KGB element #2,[ 0, 0 ]) 1*(KGB element #2,[ 1, -1 ]) 2*(KGB element #2,[ 2, 0 ]) 2*(KGB element #2,[ 0, -2 ]) 1*(KGB element #2,[ 2, 2 ]) 1*(KGB element #2,[ -2, -2 ]) 1*(KGB element #2,[ 3, 1 ]) 1*(KGB element #2,[ 1, -3 ]) 3*(KGB element #2,[ 2, -2 ]) 2*(KGB element #2,[ 3, -1 ]) 2*(KGB element #2,[ 1, -3 ]) 3*(KGB element #2,[ 4, 0 ]) 3*(KGB element #2,[ 0, -4 ]) 2*(KGB element #2,[ 4, 2 ]) 2*(KGB element #2,[ -2, -4 ]) 3*(KGB element #2,[ 3, -3 ]) 4*(KGB element #2,[ 4, -2 ]) 4*(KGB element #2,[ 2, -4 ]) 1*(KGB element #2,[ 4, 4 ]) 1*(KGB element #2,[ -4, -4 ]) 2*(KGB element #2,[ 5, 1 ]) 2*(KGB element #2,[ 1, -5 ]) 3*(KGB element #2,[ 5, -1 ]) 3*(KGB element #2,[ 1, -5 ]) 1*(KGB element #2,[ 5, 3 ]) 1*(KGB element #2,[ -3, -5 ]) 5*(KGB element #2,[ 4, -4 ]) atlas> set p=trivial(G) Identifier p: Param atlas> set x=KGB(G,2) Identifier x: KGBElt atlas> print_K_std(p,x,30) 1*(KGB element #2,[ 0, 0 ]) 1*(KGB element #2,[ 1, -1 ]) 2*(KGB element #2,[ 2, 0 ]) 2*(KGB element #2,[ 0, -2 ]) 1*(KGB element #2,[ 2, 2 ]) 1*(KGB element #2,[ -2, -2 ]) 1*(KGB element #2,[ 3, 1 ]) 1*(KGB element #2,[ 1, -3 ]) 3*(KGB element #2,[ 2, -2 ]) 2*(KGB element #2,[ 3, -1 ]) 2*(KGB element #2,[ 1, -3 ]) 3*(KGB element #2,[ 4, 0 ]) 3*(KGB element #2,[ 0, -4 ]) 2*(KGB element #2,[ 4, 2 ]) 2*(KGB element #2,[ -2, -4 ]) 3*(KGB element #2,[ 3, -3 ]) 4*(KGB element #2,[ 4, -2 ]) 4*(KGB element #2,[ 2, -4 ]) 1*(KGB element #2,[ 4, 4 ]) 1*(KGB element #2,[ -4, -4 ]) 2*(KGB element #2,[ 5, 1 ]) 2*(KGB element #2,[ 1, -5 ]) 3*(KGB element #2,[ 5, -1 ]) 3*(KGB element #2,[ 1, -5 ]) 1*(KGB element #2,[ 5, 3 ]) 1*(KGB element #2,[ -3, -5 ]) 5*(KGB element #2,[ 4, -4 ]) 4*(KGB element #2,[ 5, -3 ]) 4*(KGB element #2,[ 6, 0 ]) 4*(KGB element #2,[ 3, -5 ]) 4*(KGB element #2,[ 0, -6 ]) 3*(KGB element #2,[ 6, 2 ]) 3*(KGB element #2,[ -2, -6 ]) atlas> set b=block_of (trivial(G)) Identifier b: [Param] atlas> set p=b[0] Identifier p: Param (hiding previous one of type Param) atlas> p Value: final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) atlas> print_K_std(p,x,40) 1*(KGB element #2,[ 3, -1 ]) 1*(KGB element #2,[ 4, 0 ]) 1*(KGB element #2,[ 3, -3 ]) 1*(KGB element #2,[ 4, -2 ]) 1*(KGB element #2,[ 5, 1 ]) 2*(KGB element #2,[ 5, -1 ]) 1*(KGB element #2,[ 4, -4 ]) 2*(KGB element #2,[ 5, -3 ]) 2*(KGB element #2,[ 6, 0 ]) 1*(KGB element #2,[ 3, -5 ]) 1*(KGB element #2,[ 6, 2 ]) 2*(KGB element #2,[ 6, -2 ]) 2*(KGB element #2,[ 7, 1 ]) 2*(KGB element #2,[ 5, -5 ]) 2*(KGB element #2,[ 6, -4 ]) 3*(KGB element #2,[ 7, -1 ]) 1*(KGB element #2,[ 4, -6 ]) 1*(KGB element #2,[ 7, 3 ]) atlas> set p=b[2] Identifier p: Param (hiding previous one of type Param) atlas> p Value: final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) atlas> print_K_std(p,x,70) 1*(KGB element #2,[ 3, 3 ]) 1*(KGB element #2,[ 5, 3 ]) 1*(KGB element #2,[ 5, 5 ]) 1*(KGB element #2,[ 7, 3 ]) 1*(KGB element #2,[ 7, 5 ]) 1*(KGB element #2,[ 7, 7 ]) 1*(KGB element #2,[ 9, 3 ]) 1*(KGB element #2,[ 9, 5 ]) 1*(KGB element #2,[ 9, 7 ]) 1*(KGB element #2,[ 9, 9 ]) 1*(KGB element #2,[ 11, 3 ]) 1*(KGB element #2,[ 11, 5 ]) 1*(KGB element #2,[ 11, 7 ]) atlas> {Here is some information on how to make sense of atlas parameters} atlas> {It is helpful to check theta_x, the Cartan involution for this Cartan} atlas> set p=trivial(G) Identifier p: Param (hiding previous one of type Param) atlas> p Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) atlas> involution(x(p)) Value: | -1, 0 | | 0, -1 | atlas> {theta_x=-1: the split Cartan} atlas> set p=large_discrete_series (G) Identifier p: Param (hiding previous one of type Param) atlas> involution(x(p)) Value: | 1, 0 | | 0, 1 | atlas> {theta_x=1: the compact Cartan} ] atlas> for i:4 do prints(b[i], involution(x(b[i]))) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) | 1, 0 | | 0, 1 | final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) | 1, 0 | | 0, 1 | final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) | 1, 0 | | 0, 1 | final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) | 1, 0 | | 0, 1 | atlas> {Each theta=1: these are the four discrete series representations of Sp(4,R) with infinitesimal character rho=[2,1]} atlas> {They all have the same lambda=rho (and of course nu=0)} atlas> {x, in addition to encoding the Cartan, specifies a positive chamber} atlas> {In the usual notation, the Harish-Chandra parameters of the discrete series are:} atlas> {x=0: [2,-1] (large)} atlas> {x=1: [1,-2] (large)} atlas> {x=2: [2,1] (holomorphic)} atlas> {x=3: [2,-1] (anti-holomorphic)} atlas> print_block(trivial(G)) Parameter defines element 10 of the following block: 0: 0 [i1,i1] 1 2 ( 6, *) ( 4, *) *(x= 0,lam=rho+ [0,0], nu= [0,0]/1) e 1: 0 [i1,i1] 0 3 ( 6, *) ( 5, *) *(x= 1,lam=rho+ [0,0], nu= [0,0]/1) e 2: 0 [ic,i1] 2 0 ( *, *) ( 4, *) *(x= 2,lam=rho+ [0,0], nu= [0,0]/1) e 3: 0 [ic,i1] 3 1 ( *, *) ( 5, *) *(x= 3,lam=rho+ [0,0], nu= [0,0]/1) e ... atlas> {The type of discrete series is determined by the simple roots: atlas> {both non-compact (large), or one compact/one noncompact (holomorphic/anti-holomorphic)} atlas> {The command cuspidal_data gives the data (P,pi_M) so that the } atlas {standard module is Ind_P^G(pi_M)} atlas> atlas> atlas> set G=Sp(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> set p=trivial(G) Identifier p: Param (hiding previous one of type Param) atlas> set (P,q)=cuspidal_data (p) Identifiers P: ([int],KGBElt) (hiding previous one of type ([int],KGBElt)), q: Param (hiding previous one of type Param) atlas> Levi(P) Value: disconnected split real group with Lie algebra 'gl(1,R).gl(1,R)' atlas> q Value: final parameter (x=0,lambda=[0,0]/1,nu=[2,1]/1) atlas> atlas> set L=Levi(P) Identifier L: RealForm atlas> induce_irreducible (q,P,G) Value: 1*final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) 1*final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) 2*final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2) 1*final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1) 1*final parameter (x=6,lambda=[2,1]/1,nu=[0,1]/1) 1*final parameter (x=7,lambda=[2,1]/1,nu=[2,0]/1) 1*final parameter (x=8,lambda=[2,1]/1,nu=[2,0]/1) 1*final parameter (x=9,lambda=[2,1]/1,nu=[3,3]/2) 1*final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) atlas> induce_irreducible (q,P,G)=composition_series (p) Value: true atlas> atlas> set b=block_of (trivial(G)) Identifier b: [Param] (hiding previous one of type [Param]) atlas> set p=b[0] Identifier p: Param (hiding previous one of type Param) atlas> set (P,q)=cuspidal_data (p) Identifiers P: ([int],KGBElt) (hiding previous one of type ([int],KGBElt)), q: Param (hiding previous one of type Param) atlas> Levi(p) Error in expression Levi(p) at :85:0-7 Failed to match 'Levi' with argument type Param Type check failed atlas> Levi(P) Value: connected split real group with Lie algebra 'sp(4,R)' atlas> q Value: final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) atlas> for p in block_of (trivial(G)) do let (P,q)=cuspidal_data (p) in prints("");prints(p);prints(Levi(P), " ", q) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) connected split real group with Lie algebra 'sp(4,R)' final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) connected split real group with Lie algebra 'sp(4,R)' final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) connected split real group with Lie algebra 'sp(4,R)' final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) connected split real group with Lie algebra 'sp(4,R)' final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=0,lambda=[1,2]/1,nu=[1,0]/1) final parameter (x=6,lambda=[2,1]/1,nu=[0,1]/1) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=1,lambda=[1,2]/1,nu=[1,0]/1) final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=0,lambda=[3,-3]/2,nu=[1,1]/2) final parameter (x=7,lambda=[2,1]/1,nu=[2,0]/1) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=0,lambda=[0,1]/1,nu=[2,0]/1) final parameter (x=8,lambda=[2,1]/1,nu=[2,0]/1) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=1,lambda=[0,1]/1,nu=[2,0]/1) final parameter (x=9,lambda=[2,1]/1,nu=[3,3]/2) disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' final parameter (x=0,lambda=[1,-1]/2,nu=[3,3]/2) final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) disconnected split real group with Lie algebra 'gl(1,R).gl(1,R)' final parameter (x=0,lambda=[0,0]/1,nu=[2,1]/1) final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1) disconnected split real group with Lie algebra 'gl(1,R).gl(1,R)' final parameter (x=0,lambda=[1,1]/1,nu=[2,1]/1) atlas> {we can also induce any representation of any Levi factor} atlas> {here is the split parabolic with simple root #0, e_1-e_2, i.e. GL(2,R)} atlas> set P=parabolic([0],KGB(G,10)) Identifier P: ([int],KGBElt) (hiding previous one of type ([int],KGBElt)) atlas> Levi(P) Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' atlas> set L=Levi(P) Identifier L: RealForm (hiding previous one of type RealForm) atlas> set pl=trivial(L) Identifier pl: Param atlas> induce_irreducible (pl,P,G) Value: 1*final parameter (x=10,lambda=[2,1]/1,nu=[1,1]/2) atlsa> {This is the middle of the edge of the square unitary region in the spherical unitary dual} atlas> is_unitary(induce_irreducible (pl,P,G)) Value: true atlas> atlas> {Here is the split parabolic with root 1, i.e. SL(2,R)xR^*} atlas> set P=parabolic([1],KGB(G,10)) Identifier P: ([int],KGBElt) (hiding previous one of type ([int],KGBElt)) atlas> Levi(P) Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' atlas> set L=Levi(P) Identifier L: RealForm (hiding previous one of type RealForm) atlas> set pl=trivial(L) Identifier pl: Param (hiding previous one of type Param) atlas> induce_irreducible (pl,P,G) Value: 1*final parameter (x=4,lambda=[1,0]/1,nu=[1,-1]/2) 1*final parameter (x=10,lambda=[2,1]/1,nu=[1,0]/1) atlas> {Unlike the previous case, this induction is reducible} atlas> {It is the direct sum of two irreducible unitary representations} atlas> {The second term is the corner of the square of the spherical unitary dual} atlas> atlas> atlas> {Some examples from Kobayashi's 1991 Memoirs Article, for G=Sp(p,q)} atlas> {A(p,q,r,s,lambda_p,lambda_q)} atlas> {G=Sp(p,q), L=U(1)^rxU(p-r,q-s)xU(1)^s} atlas> {lambda=(lambda_p,lambda_q) on the U(1) factors} atlas> {A(p,q,r,lambda)=A(p,q,r,s=0,lambda_p=lambda,lambda_q=0)} atlas> A(2,2,1,[4]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' Good infinitesimal character:[ 4, 3, 2, 1 ]/1 LKT:[(KGB element #0,[ 4, 0, 0, 0 ])] dimensions of LKT:[35] final parameter (x=27,lambda=[4,3,2,1]/1,nu=[0,5,5,0]/2) Value: final parameter (x=27,lambda=[4,3,2,1]/1,nu=[0,5,5,0]/2) atlas> A(2,2,1,[3]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' Weakly good infinitesimal character:[ 3, 3, 2, 1 ]/1 LKT:[(KGB element #0,[ 3, 0, 0, 0 ])] dimensions of LKT:[20] final parameter (x=27,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) Value: final parameter (x=27,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) atlas> A(2,2,1,[2]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' Weakly fair infinitesimal character:[ 3, 2, 2, 1 ]/1 LKT:[(KGB element #0,[ 2, 0, 0, 0 ])] dimensions of LKT:[10] final parameter (x=33,lambda=[4,2,3,1]/1,nu=[5,0,5,0]/2) Value: final parameter (x=33,lambda=[4,2,3,1]/1,nu=[5,0,5,0]/2) atlas> A(2,2,1,[1]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' Weakly fair infinitesimal character:[ 3, 2, 1, 1 ]/1 LKT:[(KGB element #0,[ 1, 0, 0, 0 ])] dimensions of LKT:[4] final parameter (x=37,lambda=[4,3,1,1]/1,nu=[5,5,0,0]/2) Value: final parameter (x=37,lambda=[4,3,1,1]/1,nu=[5,5,0,0]/2) atlas> A(2,2,1,[0]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' Weakly fair infinitesimal character:[ 3, 2, 1, 0 ]/1 LKT:[(KGB element #0,[ 0, 0, 0, 0 ])] dimensions of LKT:[1] final parameter (x=39,lambda=[4,3,1,0]/1,nu=[5,5,1,-1]/2) Value: final parameter (x=39,lambda=[4,3,1,0]/1,nu=[5,5,1,-1]/2) atlas> A(2,2,1,[-1]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' None infinitesimal character:[ 3, 2, 1, 1 ]/1 LKT:[(KGB element #1,[ 1, 0, 0, 0 ])] dimensions of LKT:[4] final parameter (x=38,lambda=[4,3,1,1]/1,nu=[5,5,0,0]/2) Value: final parameter (x=38,lambda=[4,3,1,1]/1,nu=[5,5,0,0]/2) atlas> A(2,2,1,[-2]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' None infinitesimal character:[ 3, 2, 2, 1 ]/1 LKT:[(KGB element #1,[ 2, 0, 0, 0 ])] dimensions of LKT:[10] final parameter (x=34,lambda=[4,2,3,1]/1,nu=[5,0,5,0]/2) Value: final parameter (x=34,lambda=[4,2,3,1]/1,nu=[5,0,5,0]/2) atlas> {Here is the last point where the module is irreducible, according to Kobayashi} atlas> A(2,2,1,[-3]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' None infinitesimal character:[ 3, 3, 2, 1 ]/1 LKT:[(KGB element #1,[ 3, 0, 0, 0 ])] dimensions of LKT:[20] final parameter (x=28,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) Value: final parameter (x=28,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) atlas> A(2,2,1,[-4]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' None Runtime error: Aq is not irreducible. Use Aq_param_pol(x,lambda) instead (in call of error, built-in) (in call of Aq@(KGBElt,ratvec,ratvec), defined at thetastable.at:171:4--177:12) (in call of A@(int,int,int,[int]), defined at sppq.at:10:4--24:19) Evaluation aborted. atlas> {run an alternative command to see the reducible module} atlas> A_red(2,2,1,[-4]) lambda_parabolic=[ 1, 0, 0, 0 ]/1 Q=3 L=connected real group with Lie algebra 'sp(2,1).u(1)' None infinitesimal character:[ 4, 3, 2, 1 ]/1 1*final parameter (x=28,lambda=[4,3,2,1]/1,nu=[0,5,5,0]/2) -1*final parameter (x=41,lambda=[4,3,2,1]/1,nu=[7,7,3,3]/2) Value: 1*final parameter (x=28,lambda=[4,3,2,1]/1,nu=[0,5,5,0]/2) -1*final parameter (x=41,lambda=[4,3,2,1]/1,nu=[7,7,3,3]/2) atlas> {The sign indicates that vanishing outside degree S fails; and the second term is in degree S\pm 1} atlas> atlas> atlas> {Here are some examples where r=s=1} atlas> {I fixed a bug with the good/weakly good etc. reporting in these cases} atlas> A(2,2,1,1,[5],[4]) lambda=[ 5, 0, 4, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 5, 0, 6, 1 ]/1 Good infinitesimal character:[ 5, 4, 2, 1 ]/1 LKT:[(KGB element #0,[ 5, 3, 0, 0 ])] dimensions of LKT:[1120] final parameter (x=17,lambda=[5,4,2,1]/1,nu=[0,0,3,3]/2) Value: final parameter (x=17,lambda=[5,4,2,1]/1,nu=[0,0,3,3]/2) atlas> A(2,2,1,1,[4],[3]) lambda=[ 4, 0, 3, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' Good infinitesimal character:[ 4, 3, 2, 1 ]/1 LKT:[(KGB element #0,[ 4, 2, 0, 0 ])] dimensions of LKT:[350] final parameter (x=17,lambda=[4,3,2,1]/1,nu=[0,0,3,3]/2) Value: final parameter (x=17,lambda=[4,3,2,1]/1,nu=[0,0,3,3]/2) atlas> A(2,2,1,1,[3],[2]) lambda=[ 3, 0, 2, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' Weakly good infinitesimal character:[ 3, 2, 2, 1 ]/1 LKT:[(KGB element #0,[ 3, 1, 0, 0 ])] dimensions of LKT:[80] final parameter (x=17,lambda=[3,2,2,1]/1,nu=[0,0,3,3]/2) Value: final parameter (x=17,lambda=[3,2,2,1]/1,nu=[0,0,3,3]/2) atlas> A(2,2,1,1,[3],[1]) lambda=[ 3, 0, 1, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' Weakly fair infinitesimal character:[ 3, 2, 1, 1 ]/1 LKT:[(KGB element #0,[ 3, 0, 0, 0 ])] dimensions of LKT:[20] final parameter (x=23,lambda=[3,3,1,2]/1,nu=[0,3,0,3]/2) Value: final parameter (x=23,lambda=[3,3,1,2]/1,nu=[0,3,0,3]/2) atlas> A(2,2,1,1,[3],[0]) lambda=[ 3, 0, 0, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 3, 0, 2, 1 ]/1 Weakly fair infinitesimal character:[ 3, 2, 1, 0 ]/1 LKT:[(KGB element #0,[ 3, 0, 1, 0 ])] dimensions of LKT:[35] final parameter (x=15,lambda=[3,2,1,0]/1,nu=[0,1,0,-1]/1) Value: final parameter (x=15,lambda=[3,2,1,0]/1,nu=[0,1,0,-1]/1) atlas> A(2,2,1,1,[3],[-1]) lambda=[ 3, 0, -1, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' None infinitesimal character:[ 3, 2, 1, 1 ]/1 LKT:[(KGB element #0,[ 3, 0, 2, 0 ])] dimensions of LKT:[40] final parameter (x=8,lambda=[3,2,1,1]/1,nu=[0,1,-1,0]/2) Value: final parameter (x=8,lambda=[3,2,1,1]/1,nu=[0,1,-1,0]/2) atlas> A(2,2,1,1,[3],[-2]) lambda=[ 3, 0, -2, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' None infinitesimal character:[ 3, 2, 2, 1 ]/1 LKT:[(KGB element #2,[ 3, 3, 0, 0 ])] dimensions of LKT:[30] final parameter (x=2,lambda=[3,2,2,1]/1,nu=[0,0,0,0]/1) Value: final parameter (x=2,lambda=[3,2,2,1]/1,nu=[0,0,0,0]/1) atlas> A(2,2,1,1,[3],[-3]) lambda=[ 3, 0, -3, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' None Runtime error: Aq has multiplicity. Use Aq_param_pol(x,lambda) instead (in call of error, built-in) (in call of Aq@(KGBElt,ratvec,ratvec), defined at thetastable.at:171:4--177:12) (in call of A@(int,int,int,int,[int],[int]), defined at sppq.at:41:4--60:19) Evaluation aborted. atlas> A_red(2,2,1,1,[3],[-3]) lambda=[ 3, 0, -3, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' None infinitesimal character:[ 3, 3, 2, 1 ]/1 -1*final parameter (x=27,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) Value: -1*final parameter (x=27,lambda=[3,3,2,1]/1,nu=[0,5,5,0]/2) atlas> {In this example the result is an irreducible module in degree S\pm 1} atlas> A(2,2,1,1,[0],[3]) lambda=[ 0, 0, 3, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' None Runtime error: Aq is not irreducible. Use Aq_param_pol(x,lambda) instead (in call of error, built-in) (in call of Aq@(KGBElt,ratvec,ratvec), defined at thetastable.at:171:4--177:12) (in call of A@(int,int,int,int,[int],[int]), defined at sppq.at:41:4--60:19) Evaluation aborted. atlas> A(2,2,1,1,[3],[3]) lambda=[ 3, 0, 3, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' Weakly good infinitesimal character:[ 3, 3, 2, 1 ]/1 LKT:[(KGB element #0,[ 3, 2, 0, 0 ])] dimensions of LKT:[200] final parameter (x=17,lambda=[3,3,2,1]/1,nu=[0,0,3,3]/2) Value: final parameter (x=17,lambda=[3,3,2,1]/1,nu=[0,0,3,3]/2) atlas> A(2,2,1,1,[2],[2]) lambda=[ 2, 0, 2, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 2, 0, 4, 1 ]/1 Weakly good infinitesimal character:[ 2, 2, 2, 1 ]/1 LKT:[(KGB element #0,[ 2, 1, 0, 0 ])] dimensions of LKT:[40] final parameter (x=17,lambda=[2,2,2,1]/1,nu=[0,0,3,3]/2) Value: final parameter (x=17,lambda=[2,2,2,1]/1,nu=[0,0,3,3]/2) atlas> A(2,2,1,1,[2],[1]) lambda=[ 2, 0, 1, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 2, 0, 3, 1 ]/1 Weakly fair infinitesimal character:[ 2, 2, 1, 1 ]/1 LKT:[(KGB element #0,[ 2, 0, 0, 0 ])] dimensions of LKT:[10] final parameter (x=23,lambda=[2,3,1,2]/1,nu=[0,3,0,3]/2) Value: final parameter (x=23,lambda=[2,3,1,2]/1,nu=[0,3,0,3]/2) atlas> A(2,2,1,1,[2],[0]) lambda=[ 2, 0, 0, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 2, 0, 2, 1 ]/1 Weakly fair infinitesimal character:[ 2, 2, 1, 0 ]/1 LKT:[(KGB element #0,[ 2, 0, 1, 0 ])] dimensions of LKT:[16] final parameter (x=15,lambda=[2,2,1,0]/1,nu=[0,1,0,-1]/1) Value: final parameter (x=15,lambda=[2,2,1,0]/1,nu=[0,1,0,-1]/1) atlas> A(2,2,1,1,[2],[-1]) lambda=[ 2, 0, -1, 0 ] lambda_parabolic=[ 2, 0, 1, 0 ]/1 Q=2 L=connected real group with Lie algebra 'sp(1,1).u(1).u(1)' rho_L:[ 0, 0, 2, 1 ]/1 rho_L+lambda:[ 2, 0, 1, 1 ]/1 None infinitesimal character:[ 2, 2, 1, 1 ]/1 LKT:[(KGB element #0,[ 2, 0, 2, 0 ])] dimensions of LKT:[14] final parameter (x=8,lambda=[2,2,1,1]/1,nu=[0,1,-1,0]/2) Value: final parameter (x=8,lambda=[2,2,1,1]/1,nu=[0,1,-1,0]/2)