atlas> {Some examples of representations of SL(2,R) and friends} atlas> set G=SL(2,R) Identifier G: RealForm atlas> G Value: connected split real group with Lie algebra 'sl(2,R)' atlas> print_KGB (G) kgbsize: 3 Base grading: [1]. 0: 0 [n] 1 2 (0)#0 e 1: 0 [n] 0 2 (1)#0 e 2: 1 [r] 2 * (0)#1 1^e atlas> atlas> set p=trivial(G) Identifier p: Param atlas> p Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1) atlas> print_block(p) Parameter defines element 2 of the following block: 0: 0 [i1] 1 (2,*) *(x=0,lam=rho+ [0], nu= [0]/1) e 1: 0 [i1] 0 (2,*) *(x=1,lam=rho+ [0], nu= [0]/1) e 2: 1 [r1] 2 (0,1) *(x=2,lam=rho+ [0], nu= [1]/1) 1^e atlas> set Gd=dual_quasisplit_form (G) Identifier Gd: RealForm atlas> G Value: connected split real group with Lie algebra 'sl(2,R)' atlas> Gd Value: disconnected split real group with Lie algebra 'sl(2,R)' atlas> print_block(trivial(Gd)) Parameter defines element 1 of the following block: 0: 0 [i2] 0 (1,2) *(x=0,lam=rho+ [0], nu= [0]/1) e 1: 1 [r2] 2 (0,*) *(x=1,lam=rho+ [0], nu= [1]/2) 1^e 2: 1 [r2] 1 (0,*) *(x=1,lam=rho+ [-1], nu= [1]/2) 1^e atlas> p Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1) atlas> set std=I(p) Identifier std: (Param,string) (hiding previous one of type string (constant)) atlas> show(std) I(x=2,lambda=[1/1],nu=[1/1]) atlas> set pi=J(p) Identifier pi: (Param,string) atlas> show(pi) J(x=2,lambda=[1/1],nu=[1/1]) atlas> show(composition_series (std)) 1*J(x=0,lambda=[1/1],nu=[0/1]) 1*J(x=1,lambda=[1/1],nu=[0/1]) 1*J(x=2,lambda=[1/1],nu=[1/1]) atlas> show(character_formula (pi)) -1*I(x=0,lambda=[1/1],nu=[0/1]) -1*I(x=1,lambda=[1/1],nu=[0/1]) 1*I(x=2,lambda=[1/1],nu=[1/1]) atlas> p Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1) atlas> set q=parameter(KGB(G,2),[0],[1]) Identifier q: Param (hiding previous one of type vec (constant)) atlas> show(composition_series (I(p))) 1*J(x=0,lambda=[1/1],nu=[0/1]) 1*J(x=1,lambda=[1/1],nu=[0/1]) 1*J(x=2,lambda=[1/1],nu=[1/1]) atlas> show(composition_series (I(q))) 1*J(x=2,lambda=[2/1],nu=[1/1]) atlas> print_block(q) Parameter defines element 0 of the following block: 0: 0 [rn] 0 (*,*) *(x=2,lam=rho+ [-1], nu= [1]/1) 1^e atlas> branch(p,20) Value: 1*final parameter (x=0,lambda=[1]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[3]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[5]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[7]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[9]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[1]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[3]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[5]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[7]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[9]/1,nu=[0]/1) 1*final parameter (x=2,lambda=[1]/1,nu=[0]/1) atlas> branch(I(p),20) Value: ( 1*final parameter (x=0,lambda=[1]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[3]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[5]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[7]/1,nu=[0]/1) 1*final parameter (x=0,lambda=[9]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[1]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[3]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[5]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[7]/1,nu=[0]/1) 1*final parameter (x=1,lambda=[9]/1,nu=[0]/1) 1*final parameter (x=2,lambda=[1]/1,nu=[0]/1),"K_types") atlas> show(branch(I(p),20)) 1*J_K(x=0,lambda=[1/1]) 1*J_K(x=0,lambda=[3/1]) 1*J_K(x=0,lambda=[5/1]) 1*J_K(x=0,lambda=[7/1]) 1*J_K(x=0,lambda=[9/1]) 1*J_K(x=1,lambda=[1/1]) 1*J_K(x=1,lambda=[3/1]) 1*J_K(x=1,lambda=[5/1]) 1*J_K(x=1,lambda=[7/1]) 1*J_K(x=1,lambda=[9/1]) 1*J_K(x=2,lambda=[1/1]) atlas> show(branch(J(p),20)) 1*J_K(x=2,lambda=[1/1]) atlas> show(branch(I(q),20)) 1*J_K(x=0,lambda=[0/1]) 1*J_K(x=0,lambda=[2/1]) 1*J_K(x=0,lambda=[4/1]) 1*J_K(x=0,lambda=[6/1]) 1*J_K(x=0,lambda=[8/1]) 1*J_K(x=0,lambda=[10/1]) 1*J_K(x=1,lambda=[0/1]) 1*J_K(x=1,lambda=[2/1]) 1*J_K(x=1,lambda=[4/1]) 1*J_K(x=1,lambda=[6/1]) 1*J_K(x=1,lambda=[8/1]) 1*J_K(x=1,lambda=[10/1]) atlas> show(branch(J(q),20)) 1*J_K(x=0,lambda=[0/1]) 1*J_K(x=0,lambda=[2/1]) 1*J_K(x=0,lambda=[4/1]) 1*J_K(x=0,lambda=[6/1]) 1*J_K(x=0,lambda=[8/1]) 1*J_K(x=0,lambda=[10/1]) 1*J_K(x=1,lambda=[0/1]) 1*J_K(x=1,lambda=[2/1]) 1*J_K(x=1,lambda=[4/1]) 1*J_K(x=1,lambda=[6/1]) 1*J_K(x=1,lambda=[8/1]) 1*J_K(x=1,lambda=[10/1]) atlas> set G=Sp(4,R) Identifier G: RealForm (hiding previous one of type 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=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 4: 1 [C+,r1] 7 4 ( *, *) ( 0, 2) *(x= 5,lam=rho+ [0,0], nu= [0,1]/1) 2^e 5: 1 [C+,r1] 8 5 ( *, *) ( 1, 3) *(x= 6,lam=rho+ [0,0], nu= [0,1]/1) 2^e 6: 1 [r1,C+] 6 9 ( 0, 1) ( *, *) *(x= 4,lam=rho+ [0,0], nu= [1,-1]/2) 1^e 7: 2 [C-,i1] 4 8 ( *, *) (10, *) *(x= 7,lam=rho+ [0,0], nu= [2,0]/1) 1x2^e 8: 2 [C-,i1] 5 7 ( *, *) (10, *) *(x= 8,lam=rho+ [0,0], nu= [2,0]/1) 1x2^e 9: 2 [i2,C-] 9 6 (10,11) ( *, *) *(x= 9,lam=rho+ [0,0], nu= [3,3]/2) 2x1^e 10: 3 [r2,r1] 11 10 ( 9, *) ( 7, 8) *(x=10,lam=rho+ [0,0], nu= [2,1]/1) 1^2x1^e 11: 3 [r2,rn] 10 11 ( 9, *) ( *, *) *(x=10,lam=rho+[-1,-1], nu= [2,1]/1) 1^2x1^e 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 4: 1 [C+,r1] 7 4 ( *, *) ( 0, 2) *(x= 5,lam=rho+ [0,0], nu= [0,1]/1) 2^e 5: 1 [C+,r1] 8 5 ( *, *) ( 1, 3) *(x= 6,lam=rho+ [0,0], nu= [0,1]/1) 2^e 6: 1 [r1,C+] 6 9 ( 0, 1) ( *, *) *(x= 4,lam=rho+ [0,0], nu= [1,-1]/2) 1^e 7: 2 [C-,i1] 4 8 ( *, *) (10, *) *(x= 7,lam=rho+ [0,0], nu= [2,0]/1) 1x2^e 8: 2 [C-,i1] 5 7 ( *, *) (10, *) *(x= 8,lam=rho+ [0,0], nu= [2,0]/1) 1x2^e 9: 2 [i2,C-] 9 6 (10,11) ( *, *) *(x= 9,lam=rho+ [0,0], nu= [3,3]/2) 2x1^e 10: 3 [r2,r1] 11 10 ( 9, *) ( 7, 8) *(x=10,lam=rho+ [0,0], nu= [2,1]/1) 1^2x1^e 11: 3 [r2,rn] 10 11 ( 9, *) ( *, *) *(x=10,lam=rho+[-1,-1], nu= [2,1]/1) 1^2x1^e atlas> block_sizes (G) Value: | 0, 0, 1 | | 0, 0, 4 | | 1, 5, 12 | atlas> for H in real_forms (G) do prints(H) od compact connected real group with Lie algebra 'sp(2)' connected real group with Lie algebra 'sp(1,1)' connected split real group with Lie algebra 'sp(4,R)' Value: [(),(),()] 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> set std=I(p) Identifier std: (Param,string) (hiding previous one of type (Param,string)) atlas> set pi=J(p) Identifier pi: (Param,string) (hiding previous one of type (Param,string)) atlas> show(composition_series (std)) 1*J(x=0,lambda=[2/1,1/1],nu=[0/1,0/1]) 1*J(x=1,lambda=[2/1,1/1],nu=[0/1,0/1]) 2*J(x=4,lambda=[2/1,1/1],nu=[1/2,-1/2]) 1*J(x=5,lambda=[2/1,1/1],nu=[0/1,1/1]) 1*J(x=6,lambda=[2/1,1/1],nu=[0/1,1/1]) 1*J(x=7,lambda=[2/1,1/1],nu=[2/1,0/1]) 1*J(x=8,lambda=[2/1,1/1],nu=[2/1,0/1]) 1*J(x=9,lambda=[2/1,1/1],nu=[3/2,3/2]) 1*J(x=10,lambda=[2/1,1/1],nu=[2/1,1/1]) atlas> show(character_formula (pi)) -1*I(x=0,lambda=[2/1,1/1],nu=[0/1,0/1]) -1*I(x=1,lambda=[2/1,1/1],nu=[0/1,0/1]) -1*I(x=2,lambda=[2/1,1/1],nu=[0/1,0/1]) -1*I(x=3,lambda=[2/1,1/1],nu=[0/1,0/1]) 1*I(x=4,lambda=[2/1,1/1],nu=[1/2,-1/2]) 1*I(x=5,lambda=[2/1,1/1],nu=[0/1,1/1]) 1*I(x=6,lambda=[2/1,1/1],nu=[0/1,1/1]) -1*I(x=7,lambda=[2/1,1/1],nu=[2/1,0/1]) -1*I(x=8,lambda=[2/1,1/1],nu=[2/1,0/1]) -1*I(x=9,lambda=[2/1,1/1],nu=[3/2,3/2]) 1*I(x=10,lambda=[2/1,1/1],nu=[2/1,1/1]) atlas> set p=large_discrete_series (G) Identifier p: Param (hiding previous one of type Param) atlas> show(branch(J(p),20)) 1*J_K(x=0,lambda=[2/1,1/1]) 1*J_K(x=0,lambda=[3/1,0/1]) 1*J_K(x=9,lambda=[4/1,-1/1]) atlas> show(branch(J(p),40)) 1*J_K(x=0,lambda=[2/1,1/1]) 1*J_K(x=0,lambda=[3/1,0/1]) 1*J_K(x=0,lambda=[3/1,2/1]) 2*J_K(x=0,lambda=[4/1,1/1]) 2*J_K(x=0,lambda=[4/1,3/1]) 2*J_K(x=0,lambda=[5/1,0/1]) 2*J_K(x=0,lambda=[5/1,2/1]) 2*J_K(x=0,lambda=[5/1,4/1]) 3*J_K(x=0,lambda=[6/1,1/1]) 1*J_K(x=1,lambda=[4/1,3/1]) 1*J_K(x=1,lambda=[5/1,4/1]) 1*J_K(x=2,lambda=[5/1,0/1]) 1*J_K(x=2,lambda=[6/1,1/1]) 1*J_K(x=7,lambda=[3/1,4/1]) 2*J_K(x=7,lambda=[3/1,6/1]) 1*J_K(x=9,lambda=[4/1,-1/1]) 1*J_K(x=9,lambda=[5/1,-2/1]) 2*J_K(x=9,lambda=[6/1,-3/1]) atlas> set a=branch(p,40) Identifier a: ParamPol atlas> a Value: 1*final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) 1*final parameter (x=0,lambda=[3,0]/1,nu=[0,0]/1) 1*final parameter (x=0,lambda=[3,2]/1,nu=[0,0]/1) 2*final parameter (x=0,lambda=[4,1]/1,nu=[0,0]/1) 2*final parameter (x=0,lambda=[4,3]/1,nu=[0,0]/1) 2*final parameter (x=0,lambda=[5,0]/1,nu=[0,0]/1) 2*final parameter (x=0,lambda=[5,2]/1,nu=[0,0]/1) 2*final parameter (x=0,lambda=[5,4]/1,nu=[0,0]/1) 3*final parameter (x=0,lambda=[6,1]/1,nu=[0,0]/1) 1*final parameter (x=1,lambda=[4,3]/1,nu=[0,0]/1) 1*final parameter (x=1,lambda=[5,4]/1,nu=[0,0]/1) 1*final parameter (x=2,lambda=[5,0]/1,nu=[0,0]/1) 1*final parameter (x=2,lambda=[6,1]/1,nu=[0,0]/1) 1*final parameter (x=7,lambda=[3,4]/1,nu=[0,0]/1) 2*final parameter (x=7,lambda=[3,6]/1,nu=[0,0]/1) 1*final parameter (x=9,lambda=[4,-1]/1,nu=[0,0]/1) 1*final parameter (x=9,lambda=[5,-2]/1,nu=[0,0]/1) 2*final parameter (x=9,lambda=[6,-3]/1,nu=[0,0]/1) atlas> set b=monomials(a) Identifier b: [Param] atlas> for p in b do prints(LKT_highest_weights(p,KGB(G,2))) od mu_G=[ 3, 1 ] mu_G=[ 4, 0 ] mu_G=[ 4, 2 ] mu_G=[ 5, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 0 ] mu_G=[ 6, 2 ] mu_G=[ 6, 4 ] mu_G=[ 7, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 4 ] mu_G=[ 6, 2 ] mu_G=[ 7, 3 ] mu_G=[ 5, -1 ] mu_G=[ 7, -1 ] mu_G=[ 3, 3 ] mu_G=[ 4, 4 ] mu_G=[ 5, 5 ] mu_G=[ 3, 1 ] mu_G=[ 4, 0 ] mu_G=[ 4, 2 ] mu_G=[ 5, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 0 ] mu_G=[ 6, 2 ] mu_G=[ 6, 4 ] mu_G=[ 7, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 4 ] mu_G=[ 6, 2 ] mu_G=[ 7, 3 ] mu_G=[ 5, -1 ] mu_G=[ 7, -1 ] mu_G=[ 3, 3 ] mu_G=[ 4, 4 ] mu_G=[ 5, 5 ] mu_G=[ 3, 1 ] mu_G=[ 4, 0 ] mu_G=[ 4, 2 ] mu_G=[ 5, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 0 ] mu_G=[ 6, 2 ] mu_G=[ 6, 4 ] mu_G=[ 7, 1 ] mu_G=[ 5, 3 ] mu_G=[ 6, 4 ] mu_G=[ 7, 3 ] mu_G=[ 5, -1 ] mu_G=[ 7, -1 ] mu_G=[ 3, 3 ] mu_G=[ 4, 4 ] mu_G=[ 5, 5 ] atlas> set G=split_form(E8) Identifier G: RealForm (hiding previous one of type RealForm) atlas> block_sizes (G) Value: | 0, 0, 1 | | 0, 3150, 73410 | | 1, 73410, 453060 | atlas> set G=Spin(4,4) Identifier G: RealForm (hiding previous one of type RealForm) atlas> set G=Spin(8,8) Identifier G: RealForm (hiding previous one of type RealForm) atlas> block_sizes (G) Value: | 0, 0, 0, 0, 0, 0, 1 | | 0, 0, 0, 0, 0, 28, 156 | | 0, 0, 70, 0, 0, 1288, 3626 | | 0, 0, 0, 840, 0, 0, 16200 | | 0, 0, 0, 0, 840, 0, 16200 | | 0, 28, 1288, 0, 0, 13720, 27692 | | 1, 120, 3640, 9360, 9360, 32536, 58275 | atlas> for a in real_forms (G) do prints(a) od compact connected real group with Lie algebra 'so(16)' connected real group with Lie algebra 'so(14,2)' connected real group with Lie algebra 'so(12,4)' connected real group with Lie algebra 'so*(16)[0,1]' connected real group with Lie algebra 'so*(16)[1,0]' connected real group with Lie algebra 'so(10,6)' connected split real group with Lie algebra 'so(8,8)' Value: [(),(),(),(),(),(),()] 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> p Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) atlas> set q=parameter(KGB(G,1),[2,1],[2,1]/10) Identifier q: Param (hiding previous one of type Param) atlas> infinitesimal_character (q) Value: [ 2, 1 ]/1 atlas> q Value: final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) atlas> set q=parameter(KGB(G,10),[2,1],[2,1]/10) Identifier q: Param (hiding previous one of type Param) atlas> infinitesimal_character (q) Value: [ 2, 1 ]/10 atlas> q Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/10) atlas> show(composition_series (I(q)) ( > ) 1*J(x=10,lambda=[2/1,1/1],nu=[1/5,1/10]) atlas> is_unitary (q) Value: true Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) atlas> {This tests unitarity of the representations parameter(x=10,lambda=[2,1],nu=t[2,1]) for 0\le t\le 1} Identifier G: RealForm (hiding previous one of type RealForm) atlas> {This recovers the pictures of the spherical unitary dual for split G2, from Vogan, Inventiones 1994} atlas> test_line (trivial(G)) testing line through final parameter (x=9,lambda=[1,1]/1,nu=[1,1]/1) reducibility points: [1/5,1/4,1/3,1/2,3/5,3/4,1/1] integrality points (for 2*nu): [1/10,1/8,1/6,1/5,1/4,3/10,1/3,3/8,2/5,1/2,3/5,5/8,2/3,7/10,3/4,4/5,5/6,7/8,9/10,1/1] [ 1, 1 ]/10: true [ 1, 1 ]/5: true [ 9, 9 ]/40: false [ 1, 1 ]/4: true [ 7, 7 ]/24: true [ 1, 1 ]/3: true [ 5, 5 ]/12: false [ 1, 1 ]/2: false [ 11, 11 ]/20: false [ 3, 3 ]/5: false [ 27, 27 ]/40: false [ 3, 3 ]/4: false [ 7, 7 ]/8: false [ 1, 1 ]/1: true atlas> atlas> set G=Sp(8,8) Identifier G: RealForm (hiding previous one of type RealForm) atlas> {The next example, due to Alex Leontiev, illustrates an overflow error in the software: these numbers must all be positive!} atlas> block_sizes (G) Value: | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25320 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1245440 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32761820 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 463427328 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -899166888 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -902920448 | | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1375183046 | | 1, 33, 768, 12416, 160200, 1660296, 14320768, 103185408, 627588156, -1075809796, 1039683072, -1038613632, -2130824136, 1228453112, 2104637312, 129180416, -1827859930 |