[Davids-MBP-100:~/Dropbox/700.dir] 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> {topic: entering representations in atlas} atlas> {idea: Langlands classification says: repn <--> real Cartan subgroup H(R) \subset G(R), character of H(R).} atlas> {Many caveats: HC classification of discrete series: need COMPACT Cartan T(R), "character + rho": HC parameter { > is differential of a character (in X^*, character lattice, always Z^n in atlas) PLUS rho=half sum of pos roots); { > shift is in (1/2 Z)^n} atlas> set G=SO(2,1) Variable G: RealForm atlas> {compact Cartan subgroup is SO(2), characters <--> Z} atlas> set X=KGB(G) {K orbits of Borels; one corr. to compact Cartan, one to split Cartan SO(1,1) \simeq R^x \subset SO(2,1) { > } Variable X: [KGBElt] atlas> #X Value: 2 atlas> set d = parameter(X[0], [1], [0]) {try to use T(R), so FIRST (few) KGB element(s); [1] meant to be HC parameter lambda,} Error in 'set' command at :11:0-121: Not an integer vector {second argument lambda must be in Z^n (X^*(H)) PLUS rho!} Command 'set d' interrupted, nothing defined. atlas> rho(G) Value: [ 1 ]/2 atlas> set d = parameter(X[0], [1/2], [0]) {problem was lambda _wasn't_ shifted from X^* = Z by rho} Variable d: Param (overriding previous instance, which had type WeylElt) atlas> branch_irr(d,20) {meant to describe restriction to K=O(2) of this discrete series} Value: 1*parameter(x=0,lambda=[1]/2,nu=[0]/1) [1] 1*parameter(x=0,lambda=[3]/2,nu=[0]/1) [3] 1*parameter(x=0,lambda=[5]/2,nu=[0]/1) [5] 1*parameter(x=0,lambda=[7]/2,nu=[0]/1) [7] 1*parameter(x=0,lambda=[9]/2,nu=[0]/1) [9] 1*parameter(x=0,lambda=[11]/2,nu=[0]/1) [11] 1*parameter(x=0,lambda=[13]/2,nu=[0]/1) [13] 1*parameter(x=0,lambda=[15]/2,nu=[0]/1) [15] 1*parameter(x=0,lambda=[17]/2,nu=[0]/1) [17] 1*parameter(x=0,lambda=[19]/2,nu=[0]/1) [19] atlas> for mu in monomials($) do prints("dimension of K-type = ",dimension(mu)) od Runtime error: representation is infinite dimensional atlas> whattype dim {used "tab" to see all commands beginning with "dim"} dim_eigenspace dim_u_cap_k dim_u_cap_k_ge2 dim_u_cap_p_ge2 dimension_wedge_u_cap_s dim_nilpotent dim_u_cap_k_1 dim_u_cap_p dimension dimensions dim_u dim_u_cap_k_2 dim_u_cap_p_1 dimension_u_cap_s atlas> whattype dimension ? Overloaded instances of 'dimension' KGBElt->int RootDatum->int (RootDatum,ratvec)->int Param->int WeylClassTable->([int]->int) RealNilpotent->int KHighestWeight->int K_Type->int ParamPol->int (RootDatum,string,string,int,[int],string,string,string,[int])->int CharacterTable->(int->int) (int,[mat])->int {sought to use function dimension(K_Type), but output of monomials is just Param. One fix is explicitly to construct the K_Type (x,lambda) from the Param (x,lambda,nu):} for p in monomials(branch_irr(d,20)) do prints("p = ",p," dimension of K-type = ",dimension((x(p),lambda(p)))) od p = final parameter(x=0,lambda=[1]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[3]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[5]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[7]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[9]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[11]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[13]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[15]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[17]/2,nu=[0]/1) dimension of K-type = 2 p = final parameter(x=0,lambda=[19]/2,nu=[0]/1) dimension of K-type = 2 Value: [(),(),(),(),(),(),(),(),(),()] atlas> {general G: if there is a compact Cartan, corresponds to the first |W(G)/W(K)| entries of KGB. TO ENTER A DISCRETE SERIES:} atlas> {first argument of parameter: KGB element in that range (O \le x < |W(G)/W(K)|)} atlas> {second argument: ratvec lambda in X^*(H) + rho(G), HC parameter. NEED DOMINANT FOR IMAGINARY COROOTS (which is all pos coroots in ds case)} atlas> {third argument: ratvec nu in X^*(H)\otimes_Z Q, continuous parameter: irrelevant (normalized to zero)} atlas> set d3= parameter(X[0],[5/2],[11/3]) Variable d3: Param atlas> d3 Value: final parameter(x=0,lambda=[5]/2,nu=[0]/1) atlas> branch_irr(d3,20) {meant to describe restriction to K=O(2) of this discrete series} Value: 1*parameter(x=0,lambda=[5]/2,nu=[0]/1) [5] 1*parameter(x=0,lambda=[7]/2,nu=[0]/1) [7] 1*parameter(x=0,lambda=[9]/2,nu=[0]/1) [9] 1*parameter(x=0,lambda=[11]/2,nu=[0]/1) [11] 1*parameter(x=0,lambda=[13]/2,nu=[0]/1) [13] 1*parameter(x=0,lambda=[15]/2,nu=[0]/1) [15] 1*parameter(x=0,lambda=[17]/2,nu=[0]/1) [17] 1*parameter(x=0,lambda=[19]/2,nu=[0]/1) [19] atlas> height(d3) Value: 5 atlas> {bracket terms on right are HEIGHTS of irr reps of K listed} atlas> {height(parameter) =_{def} <(lambda+\theta(\lambda), (sum of pos coroots defined by LHS)>} atlas> atlas> set G=U(2,2) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> rho(G) Value: [ 3, 1, -1, -3 ]/2 atlas> set dU=parameter(KGB(G)[0],[5/2,1/2,-1/2,-5/2],[0,0,0,0]) Variable dU: Param atlas> height(dU) Value: 16 atlas> poscoroots Error during analysis of expression at :32:0-10 Undefined identifier 'poscoroots' Expression analysis failed atlas> poscoroots(G) Value: | 1, 0, 0, 1, 0, 1 | | -1, 1, 0, 0, 1, 0 | | 0, -1, 1, -1, 0, 0 | | 0, 0, -1, 0, -1, -1 | atlas> {dot them with lambda=[5/2,1/2,-1/2,-5/2]}{= (5/2-1/2)+(1/2-1/2)...=} 2+1+2 +3 +3 +5 Value: 16 atlas> {to get rep attached to general H(R): use general KGB elt. How do you know which you have?} atlas> whattype involution ? Overloaded instances of 'involution' (LieType,[int],string)->mat (LieType,mat,string)->mat CartanClass->mat KGBElt->mat (LieType,string)->mat Param->mat KGBElt_gen->mat atlas> {if x is a KGBElt, involution(x) is the (rk x rk integer) matrix defining the action of \theta on that CSG.} atlas> for x in KGB(G) do prints("x = ",x," involution(x) = ", involution(x)) od x = KGB element #0 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #1 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #2 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #3 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #4 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #5 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #6 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 0, 1 | | 0, 0, 1, 0 | x = KGB element #7 involution(x) = | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 0, 1 | | 0, 0, 1, 0 | x = KGB element #8 involution(x) = | 1, 0, 0, 0 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 0, 0, 0, 1 | x = KGB element #9 involution(x) = | 1, 0, 0, 0 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 0, 0, 0, 1 | x = KGB element #10 involution(x) = | 0, 1, 0, 0 | | 1, 0, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #11 involution(x) = | 0, 1, 0, 0 | | 1, 0, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | x = KGB element #12 involution(x) = | 0, 1, 0, 0 | | 1, 0, 0, 0 | | 0, 0, 0, 1 | | 0, 0, 1, 0 | x = KGB element #13 involution(x) = | 1, 0, 0, 0 | | 0, 0, 0, 1 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | x = KGB element #14 involution(x) = | 1, 0, 0, 0 | | 0, 0, 0, 1 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | x = KGB element #15 involution(x) = | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 1, 0, 0, 0 | | 0, 0, 0, 1 | x = KGB element #16 involution(x) = | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 1, 0, 0, 0 | | 0, 0, 0, 1 | x = KGB element #17 involution(x) = | 0, 0, 1, 0 | | 0, 0, 0, 1 | | 1, 0, 0, 0 | | 0, 1, 0, 0 | x = KGB element #18 involution(x) = | 0, 0, 0, 1 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 1, 0, 0, 0 | x = KGB element #19 involution(x) = | 0, 0, 0, 1 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 1, 0, 0, 0 | x = KGB element #20 involution(x) = | 0, 0, 0, 1 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 1, 0, 0, 0 | Value: [(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),()] atlas> {can write a Langlands parameter using ANY KGB element attached { > to the Cartan you want (here: number of transpositions, not caring { > transpositions of what); also need to get the imaginary pos roots { > you want.} atlas> print_KGB(G) kgbsize: 21 Base grading: [111]. 0: 0 [n,n,n] 1 2 3 10 8 6 (0,0,0,0)#0 e 1: 0 [n,c,n] 0 1 4 10 * 7 (1,1,0,0)#0 e 2: 0 [c,n,c] 2 0 2 * 8 * (0,1,1,0)#0 e 3: 0 [n,c,n] 4 3 0 11 * 6 (0,0,1,1)#0 e 4: 0 [n,n,n] 3 5 1 11 9 7 (1,1,1,1)#0 e 5: 0 [c,n,c] 5 4 5 * 9 * (1,0,0,1)#0 e 6: 1 [n,C,r] 7 13 6 12 * * (0,0,0,0) 1 3^e 7: 1 [n,C,r] 6 14 7 12 * * (1,1,0,0) 1 3^e 8: 1 [C,r,C] 15 8 13 * * * (0,0,0,0) 1 2^e 9: 1 [C,r,C] 16 9 14 * * * (1,0,0,1) 1 2^e 10: 1 [r,C,n] 10 15 11 * * 12 (0,0,0,0) 1 1^e 11: 1 [r,C,n] 11 16 10 * * 12 (0,0,1,1) 1 1^e 12: 2 [r,C,r] 12 17 12 * * * (0,0,0,0) 2 1^3^e 13: 2 [C,C,C] 18 6 8 * * * (0,0,0,0) 1 2x3^e 14: 2 [C,C,C] 19 7 9 * * * (1,0,1,0) 1 2x3^e 15: 2 [C,C,C] 8 10 18 * * * (0,0,0,0) 1 1x2^e 16: 2 [C,C,C] 9 11 19 * * * (0,1,0,1) 1 1x2^e 17: 3 [C,C,C] 20 12 20 * * * (0,0,0,0)#2 2x1^3^e 18: 3 [C,n,C] 13 19 15 * 20 * (0,0,0,0)#1 1x2x3^e 19: 3 [C,n,C] 14 18 16 * 20 * (0,1,1,0)#1 1x2x3^e 20: 4 [C,r,C] 17 20 17 * * * (0,0,0,0) 2 1x2x1^3^e atlas> {first 6 KGB elts 0-5 are for COMPACT Cartan; 0th has all { > three simple roots noncompact imaginary; 1st has simple { > roots noncompact, compact, noncompact; etc.} atlas> {KGBelts 2 and 5 corr to HOLOMORPHIC discrete series:} atlas> {ALL simple roots for K=U(2) x U(2) are simple for G} atlas> {FACT: in holomorphic discrete series, K mults are BOUNDED} atlas> set d0=parameter(KGB(G)[0],[3/2,1/2,-1/2,-3/2],[0,0,0,0]) Variable d0: Param atlas> set d2=parameter(KGB(G)[2],[3/2,1/2,-1/2,-3/2],[0,0,0,0]) Variable d2: Param atlas> set d1=parameter(KGB(G)[1],[3/2,1/2,-1/2,-3/2],[0,0,0,0]) Variable d1: Param atlas> print_branch_irr(d0,30) m x lambda 1 0 [ 3, 1, -1, -3 ]/2 1 0 [ 5, -1, -1, -3 ]/2 1 0 [ 3, 3, -3, -3 ]/2 1 0 [ 3, 1, 1, -5 ]/2 1 0 [ 5, 1, -3, -3 ]/2 1 0 [ 3, 3, -1, -5 ]/2 1 8 [ 7, -1, -3, -3 ]/2 1 8 [ 5, 1, -1, -5 ]/2 1 8 [ 3, 3, 1, -7 ]/2 1 12 [ 5, 3, -3, -5 ]/2 1 0 [ 7, -1, -3, -3 ]/2 2 0 [ 5, 1, -1, -5 ]/2 1 0 [ 3, 3, 1, -7 ]/2 1 10 [ 5, 3, -3, -5 ]/2 1 6 [ 5, 3, -3, -5 ]/2 1 2 [ 7, -1, -1, -5 ]/2 1 2 [ 5, 1, 1, -7 ]/2 1 0 [ 9, -3, -3, -3 ]/2 2 0 [ 7, -1, -1, -5 ]/2 2 0 [ 5, 3, -3, -5 ]/2 2 0 [ 5, 1, 1, -7 ]/2 1 0 [ 3, 3, 3, -9 ]/2 1 10 [ 5, 3, -1, -7 ]/2 1 6 [ 7, 1, -3, -5 ]/2 1 4 [ 5, 5, -5, -5 ]/2 1 3 [ 5, 5, -5, -5 ]/2 1 1 [ 5, 5, -5, -5 ]/2 2 0 [ 7, 1, -3, -5 ]/2 2 0 [ 5, 5, -5, -5 ]/2 2 0 [ 5, 3, -1, -7 ]/2 1 10 [ 5, 3, 1, -9 ]/2 2 8 [ 9, -1, -3, -5 ]/2 2 8 [ 7, 1, -1, -7 ]/2 2 8 [ 5, 3, 1, -9 ]/2 1 6 [ 9, -1, -3, -5 ]/2 1 3 [ 7, 3, -5, -5 ]/2 1 2 [ 9, -1, -3, -5 ]/2 1 2 [ 7, 1, -1, -7 ]/2 1 2 [ 5, 3, 1, -9 ]/2 1 1 [ 5, 5, -3, -7 ]/2 2 0 [ 9, -1, -3, -5 ]/2 2 0 [ 7, 3, -5, -5 ]/2 3 0 [ 7, 1, -1, -7 ]/2 2 0 [ 5, 5, -3, -7 ]/2 2 0 [ 5, 3, 1, -9 ]/2 1 10 [ 5, 3, 3, -11 ]/2 1 6 [ 11, -3, -3, -5 ]/2 2 12 [ 7, 5, -5, -7 ]/2 1 3 [ 9, 1, -5, -5 ]/2 1 2 [ 11, -3, -3, -5 ]/2 2 2 [ 9, -1, -1, -7 ]/2 2 2 [ 7, 1, 1, -9 ]/2 1 2 [ 5, 3, 3, -11 ]/2 1 1 [ 5, 5, -1, -9 ]/2 2 0 [ 11, -3, -3, -5 ]/2 2 0 [ 9, 1, -5, -5 ]/2 3 0 [ 9, -1, -1, -7 ]/2 3 0 [ 7, 3, -3, -7 ]/2 3 0 [ 7, 1, 1, -9 ]/2 2 0 [ 5, 5, -1, -9 ]/2 2 0 [ 5, 3, 3, -11 ]/2 1 11 [ 7, 5, -5, -7 ]/2 2 10 [ 7, 5, -5, -7 ]/2 1 7 [ 7, 5, -5, -7 ]/2 2 6 [ 7, 5, -5, -7 ]/2 1 4 [ 7, 5, -5, -7 ]/2 1 3 [ 11, -1, -5, -5 ]/2 1 3 [ 7, 5, -5, -7 ]/2 1 2 [ 9, 1, -3, -7 ]/2 1 2 [ 7, 3, -1, -9 ]/2 1 1 [ 7, 5, -5, -7 ]/2 1 1 [ 5, 5, 1, -11 ]/2 2 0 [ 11, -1, -5, -5 ]/2 3 0 [ 9, 1, -3, -7 ]/2 3 0 [ 7, 5, -5, -7 ]/2 3 0 [ 7, 3, -1, -9 ]/2 2 0 [ 5, 5, 1, -11 ]/2 2 10 [ 7, 5, -3, -9 ]/2 1 8 [ 13, -3, -5, -5 ]/2 3 8 [ 11, -1, -3, -7 ]/2 3 8 [ 9, 1, -1, -9 ]/2 3 8 [ 7, 3, 1, -11 ]/2 1 8 [ 5, 5, 3, -13 ]/2 2 6 [ 9, 3, -5, -7 ]/2 2 4 [ 7, 7, -7, -7 ]/2 1 3 [ 13, -3, -5, -5 ]/2 1 3 [ 9, 3, -5, -7 ]/2 2 3 [ 7, 7, -7, -7 ]/2 2 2 [ 11, -1, -3, -7 ]/2 2 2 [ 9, 1, -1, -9 ]/2 2 2 [ 7, 3, 1, -11 ]/2 2 1 [ 7, 7, -7, -7 ]/2 1 1 [ 7, 5, -3, -9 ]/2 1 1 [ 5, 5, 3, -13 ]/2 2 0 [ 13, -3, -5, -5 ]/2 3 0 [ 11, -1, -3, -7 ]/2 3 0 [ 9, 3, -5, -7 ]/2 4 0 [ 9, 1, -1, -9 ]/2 3 0 [ 7, 7, -7, -7 ]/2 3 0 [ 7, 5, -3, -9 ]/2 3 0 [ 7, 3, 1, -11 ]/2 2 0 [ 5, 5, 3, -13 ]/2 2 10 [ 7, 5, -1, -11 ]/2 2 6 [ 11, 1, -5, -7 ]/2 1 4 [ 9, 5, -7, -7 ]/2 1 4 [ 7, 7, -5, -9 ]/2 1 3 [ 11, 1, -5, -7 ]/2 2 3 [ 9, 5, -7, -7 ]/2 1 3 [ 7, 7, -5, -9 ]/2 2 2 [ 13, -3, -3, -7 ]/2 1 2 [ 11, 1, -5, -7 ]/2 3 2 [ 11, -1, -1, -9 ]/2 1 2 [ 9, 3, -3, -9 ]/2 3 2 [ 9, 1, 1, -11 ]/2 1 2 [ 7, 5, -1, -11 ]/2 2 2 [ 7, 3, 3, -13 ]/2 1 1 [ 9, 5, -7, -7 ]/2 2 1 [ 7, 7, -5, -9 ]/2 1 1 [ 7, 5, -1, -11 ]/2 1 0 [ 15, -5, -5, -5 ]/2 3 0 [ 13, -3, -3, -7 ]/2 3 0 [ 11, 1, -5, -7 ]/2 4 0 [ 11, -1, -1, -9 ]/2 3 0 [ 9, 5, -7, -7 ]/2 4 0 [ 9, 3, -3, -9 ]/2 4 0 [ 9, 1, 1, -11 ]/2 3 0 [ 7, 7, -5, -9 ]/2 3 0 [ 7, 5, -1, -11 ]/2 3 0 [ 7, 3, 3, -13 ]/2 1 0 [ 5, 5, 5, -15 ]/2 {note multiplicities (1st column of output) grow to FOUR in this range of heights} {use branch_irr rather than print_branch_irr to see HEIGHTS of K-types listed on the right} atlas> branch_irr(d0,40) Value: 1*parameter(x=0,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) [10] 1*parameter(x=0,lambda=[5,-1,-1,-3]/2,nu=[0,0,0,0]/1) [12] 1*parameter(x=0,lambda=[3,3,-3,-3]/2,nu=[0,0,0,0]/1) [12] 1*parameter(x=0,lambda=[3,1,1,-5]/2,nu=[0,0,0,0]/1) [12] 1*parameter(x=0,lambda=[5,1,-3,-3]/2,nu=[0,0,0,0]/1) [14] 1*parameter(x=0,lambda=[3,3,-1,-5]/2,nu=[0,0,0,0]/1) [14] 1*parameter(x=8,lambda=[7,-1,-3,-3]/2,nu=[0,0,0,0]/1) [15] 1*parameter(x=8,lambda=[5,1,-1,-5]/2,nu=[0,0,0,0]/1) [15] 1*parameter(x=8,lambda=[3,3,1,-7]/2,nu=[0,0,0,0]/1) [15] 1*parameter(x=12,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [16] 1*parameter(x=0,lambda=[7,-1,-3,-3]/2,nu=[0,0,0,0]/1) [16] 2*parameter(x=0,lambda=[5,1,-1,-5]/2,nu=[0,0,0,0]/1) [16] 1*parameter(x=0,lambda=[3,3,1,-7]/2,nu=[0,0,0,0]/1) [16] 1*parameter(x=10,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [17] 1*parameter(x=6,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [17] 1*parameter(x=2,lambda=[7,-1,-1,-5]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=2,lambda=[5,1,1,-7]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=0,lambda=[9,-3,-3,-3]/2,nu=[0,0,0,0]/1) [18] 2*parameter(x=0,lambda=[7,-1,-1,-5]/2,nu=[0,0,0,0]/1) [18] 2*parameter(x=0,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [18] 2*parameter(x=0,lambda=[5,1,1,-7]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=0,lambda=[3,3,3,-9]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=10,lambda=[5,3,-1,-7]/2,nu=[0,0,0,0]/1) [19] 1*parameter(x=6,lambda=[7,1,-3,-5]/2,nu=[0,0,0,0]/1) [19] 1*parameter(x=4,lambda=[5,5,-5,-5]/2,nu=[0,0,0,0]/1) [20] 1*parameter(x=3,lambda=[5,5,-5,-5]/2,nu=[0,0,0,0]/1) [20] 1*parameter(x=1,lambda=[5,5,-5,-5]/2,nu=[0,0,0,0]/1) [20] 2*parameter(x=0,lambda=[7,1,-3,-5]/2,nu=[0,0,0,0]/1) [20] 2*parameter(x=0,lambda=[5,5,-5,-5]/2,nu=[0,0,0,0]/1) [20] 2*parameter(x=0,lambda=[5,3,-1,-7]/2,nu=[0,0,0,0]/1) [20] 1*parameter(x=10,lambda=[5,3,1,-9]/2,nu=[0,0,0,0]/1) [21] 2*parameter(x=8,lambda=[9,-1,-3,-5]/2,nu=[0,0,0,0]/1) [21] 2*parameter(x=8,lambda=[7,1,-1,-7]/2,nu=[0,0,0,0]/1) [21] 2*parameter(x=8,lambda=[5,3,1,-9]/2,nu=[0,0,0,0]/1) [21] 1*parameter(x=6,lambda=[9,-1,-3,-5]/2,nu=[0,0,0,0]/1) [21] 1*parameter(x=3,lambda=[7,3,-5,-5]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=2,lambda=[9,-1,-3,-5]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=2,lambda=[7,1,-1,-7]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=2,lambda=[5,3,1,-9]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=1,lambda=[5,5,-3,-7]/2,nu=[0,0,0,0]/1) [22] 2*parameter(x=0,lambda=[9,-1,-3,-5]/2,nu=[0,0,0,0]/1) [22] 2*parameter(x=0,lambda=[7,3,-5,-5]/2,nu=[0,0,0,0]/1) [22] 3*parameter(x=0,lambda=[7,1,-1,-7]/2,nu=[0,0,0,0]/1) [22] 2*parameter(x=0,lambda=[5,5,-3,-7]/2,nu=[0,0,0,0]/1) [22] 2*parameter(x=0,lambda=[5,3,1,-9]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=10,lambda=[5,3,3,-11]/2,nu=[0,0,0,0]/1) [23] 1*parameter(x=6,lambda=[11,-3,-3,-5]/2,nu=[0,0,0,0]/1) [23] 2*parameter(x=12,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=3,lambda=[9,1,-5,-5]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=2,lambda=[11,-3,-3,-5]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=2,lambda=[9,-1,-1,-7]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=2,lambda=[7,1,1,-9]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=2,lambda=[5,3,3,-11]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=1,lambda=[5,5,-1,-9]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=0,lambda=[11,-3,-3,-5]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=0,lambda=[9,1,-5,-5]/2,nu=[0,0,0,0]/1) [24] 3*parameter(x=0,lambda=[9,-1,-1,-7]/2,nu=[0,0,0,0]/1) [24] 3*parameter(x=0,lambda=[7,3,-3,-7]/2,nu=[0,0,0,0]/1) [24] 3*parameter(x=0,lambda=[7,1,1,-9]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=0,lambda=[5,5,-1,-9]/2,nu=[0,0,0,0]/1) [24] 2*parameter(x=0,lambda=[5,3,3,-11]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=11,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [25] 2*parameter(x=10,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [25] 1*parameter(x=7,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [25] 2*parameter(x=6,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [25] 1*parameter(x=4,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=3,lambda=[11,-1,-5,-5]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=3,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=2,lambda=[9,1,-3,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=2,lambda=[7,3,-1,-9]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=1,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=1,lambda=[5,5,1,-11]/2,nu=[0,0,0,0]/1) [26] 2*parameter(x=0,lambda=[11,-1,-5,-5]/2,nu=[0,0,0,0]/1) [26] 3*parameter(x=0,lambda=[9,1,-3,-7]/2,nu=[0,0,0,0]/1) [26] 3*parameter(x=0,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 3*parameter(x=0,lambda=[7,3,-1,-9]/2,nu=[0,0,0,0]/1) [26] 2*parameter(x=0,lambda=[5,5,1,-11]/2,nu=[0,0,0,0]/1) [26] 2*parameter(x=10,lambda=[7,5,-3,-9]/2,nu=[0,0,0,0]/1) [27] 1*parameter(x=8,lambda=[13,-3,-5,-5]/2,nu=[0,0,0,0]/1) [27] 3*parameter(x=8,lambda=[11,-1,-3,-7]/2,nu=[0,0,0,0]/1) [27] 3*parameter(x=8,lambda=[9,1,-1,-9]/2,nu=[0,0,0,0]/1) [27] 3*parameter(x=8,lambda=[7,3,1,-11]/2,nu=[0,0,0,0]/1) [27] 1*parameter(x=8,lambda=[5,5,3,-13]/2,nu=[0,0,0,0]/1) [27] 2*parameter(x=6,lambda=[9,3,-5,-7]/2,nu=[0,0,0,0]/1) [27] 2*parameter(x=4,lambda=[7,7,-7,-7]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=3,lambda=[13,-3,-5,-5]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=3,lambda=[9,3,-5,-7]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=3,lambda=[7,7,-7,-7]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=2,lambda=[11,-1,-3,-7]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=2,lambda=[9,1,-1,-9]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=2,lambda=[7,3,1,-11]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=1,lambda=[7,7,-7,-7]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=1,lambda=[7,5,-3,-9]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=1,lambda=[5,5,3,-13]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=0,lambda=[13,-3,-5,-5]/2,nu=[0,0,0,0]/1) [28] 3*parameter(x=0,lambda=[11,-1,-3,-7]/2,nu=[0,0,0,0]/1) [28] 3*parameter(x=0,lambda=[9,3,-5,-7]/2,nu=[0,0,0,0]/1) [28] 4*parameter(x=0,lambda=[9,1,-1,-9]/2,nu=[0,0,0,0]/1) [28] 3*parameter(x=0,lambda=[7,7,-7,-7]/2,nu=[0,0,0,0]/1) [28] 3*parameter(x=0,lambda=[7,5,-3,-9]/2,nu=[0,0,0,0]/1) [28] 3*parameter(x=0,lambda=[7,3,1,-11]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=0,lambda=[5,5,3,-13]/2,nu=[0,0,0,0]/1) [28] 2*parameter(x=10,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [29] 2*parameter(x=6,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [29] 1*parameter(x=4,lambda=[9,5,-7,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=4,lambda=[7,7,-5,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=3,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [30] 2*parameter(x=3,lambda=[9,5,-7,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=3,lambda=[7,7,-5,-9]/2,nu=[0,0,0,0]/1) [30] 2*parameter(x=2,lambda=[13,-3,-3,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=2,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=2,lambda=[11,-1,-1,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=2,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=2,lambda=[9,1,1,-11]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=2,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [30] 2*parameter(x=2,lambda=[7,3,3,-13]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[9,5,-7,-7]/2,nu=[0,0,0,0]/1) [30] 2*parameter(x=1,lambda=[7,7,-5,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=0,lambda=[15,-5,-5,-5]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[13,-3,-3,-7]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [30] 4*parameter(x=0,lambda=[11,-1,-1,-9]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[9,5,-7,-7]/2,nu=[0,0,0,0]/1) [30] 4*parameter(x=0,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [30] 4*parameter(x=0,lambda=[9,1,1,-11]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[7,7,-5,-9]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [30] 3*parameter(x=0,lambda=[7,3,3,-13]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=0,lambda=[5,5,5,-15]/2,nu=[0,0,0,0]/1) [30] 2*parameter(x=10,lambda=[7,5,1,-13]/2,nu=[0,0,0,0]/1) [31] 2*parameter(x=6,lambda=[13,-1,-5,-7]/2,nu=[0,0,0,0]/1) [31] 3*parameter(x=12,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=4,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=3,lambda=[13,-1,-5,-7]/2,nu=[0,0,0,0]/1) [32] 2*parameter(x=3,lambda=[11,3,-7,-7]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=3,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=2,lambda=[13,-1,-5,-7]/2,nu=[0,0,0,0]/1) [32] 2*parameter(x=2,lambda=[11,1,-3,-9]/2,nu=[0,0,0,0]/1) [32] 2*parameter(x=2,lambda=[9,3,-1,-11]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=2,lambda=[7,5,1,-13]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 2*parameter(x=1,lambda=[7,7,-3,-11]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[7,5,1,-13]/2,nu=[0,0,0,0]/1) [32] 3*parameter(x=0,lambda=[13,-1,-5,-7]/2,nu=[0,0,0,0]/1) [32] 3*parameter(x=0,lambda=[11,3,-7,-7]/2,nu=[0,0,0,0]/1) [32] 4*parameter(x=0,lambda=[11,1,-3,-9]/2,nu=[0,0,0,0]/1) [32] 4*parameter(x=0,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 4*parameter(x=0,lambda=[9,3,-1,-11]/2,nu=[0,0,0,0]/1) [32] 3*parameter(x=0,lambda=[7,7,-3,-11]/2,nu=[0,0,0,0]/1) [32] 3*parameter(x=0,lambda=[7,5,1,-13]/2,nu=[0,0,0,0]/1) [32] 2*parameter(x=11,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [33] 3*parameter(x=10,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=10,lambda=[7,5,3,-15]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=8,lambda=[15,-3,-5,-7]/2,nu=[0,0,0,0]/1) [33] 4*parameter(x=8,lambda=[13,-1,-3,-9]/2,nu=[0,0,0,0]/1) [33] 4*parameter(x=8,lambda=[11,1,-1,-11]/2,nu=[0,0,0,0]/1) [33] 4*parameter(x=8,lambda=[9,3,1,-13]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=8,lambda=[7,5,3,-15]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=7,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=6,lambda=[15,-3,-5,-7]/2,nu=[0,0,0,0]/1) [33] 3*parameter(x=6,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [33] 2*parameter(x=4,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=3,lambda=[15,-3,-5,-7]/2,nu=[0,0,0,0]/1) [34] 2*parameter(x=3,lambda=[13,1,-7,-7]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=3,lambda=[11,3,-5,-9]/2,nu=[0,0,0,0]/1) [34] 2*parameter(x=3,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[15,-3,-5,-7]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=2,lambda=[13,-1,-3,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[11,3,-5,-9]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=2,lambda=[11,1,-1,-11]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[9,5,-3,-11]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=2,lambda=[9,3,1,-13]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[7,5,3,-15]/2,nu=[0,0,0,0]/1) [34] 2*parameter(x=1,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[9,5,-3,-11]/2,nu=[0,0,0,0]/1) [34] 2*parameter(x=1,lambda=[7,7,-1,-13]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[7,5,3,-15]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=0,lambda=[15,-3,-5,-7]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=0,lambda=[13,1,-7,-7]/2,nu=[0,0,0,0]/1) [34] 4*parameter(x=0,lambda=[13,-1,-3,-9]/2,nu=[0,0,0,0]/1) [34] 4*parameter(x=0,lambda=[11,3,-5,-9]/2,nu=[0,0,0,0]/1) [34] 5*parameter(x=0,lambda=[11,1,-1,-11]/2,nu=[0,0,0,0]/1) [34] 4*parameter(x=0,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 4*parameter(x=0,lambda=[9,5,-3,-11]/2,nu=[0,0,0,0]/1) [34] 4*parameter(x=0,lambda=[9,3,1,-13]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=0,lambda=[7,7,-1,-13]/2,nu=[0,0,0,0]/1) [34] 3*parameter(x=0,lambda=[7,5,3,-15]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=11,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [35] 3*parameter(x=10,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=10,lambda=[7,5,5,-17]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=7,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=6,lambda=[17,-5,-5,-7]/2,nu=[0,0,0,0]/1) [35] 3*parameter(x=6,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=4,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=4,lambda=[9,9,-9,-9]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=4,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=3,lambda=[15,-1,-7,-7]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=3,lambda=[13,1,-5,-9]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=3,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=3,lambda=[9,9,-9,-9]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=3,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=2,lambda=[17,-5,-5,-7]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=2,lambda=[15,-3,-3,-9]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=2,lambda=[13,1,-5,-9]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=2,lambda=[13,-1,-1,-11]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=2,lambda=[11,3,-3,-11]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=2,lambda=[11,1,1,-13]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=2,lambda=[9,5,-1,-13]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=2,lambda=[9,3,3,-15]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=2,lambda=[7,5,5,-17]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=1,lambda=[9,9,-9,-9]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=1,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[9,5,-1,-13]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=1,lambda=[7,7,1,-15]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=0,lambda=[17,-5,-5,-7]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=0,lambda=[15,-1,-7,-7]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[15,-3,-3,-9]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[13,1,-5,-9]/2,nu=[0,0,0,0]/1) [36] 5*parameter(x=0,lambda=[13,-1,-1,-11]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [36] 5*parameter(x=0,lambda=[11,3,-3,-11]/2,nu=[0,0,0,0]/1) [36] 5*parameter(x=0,lambda=[11,1,1,-13]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[9,9,-9,-9]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[9,5,-1,-13]/2,nu=[0,0,0,0]/1) [36] 4*parameter(x=0,lambda=[9,3,3,-15]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=0,lambda=[7,7,1,-15]/2,nu=[0,0,0,0]/1) [36] 2*parameter(x=0,lambda=[7,5,5,-17]/2,nu=[0,0,0,0]/1) [36] 3*parameter(x=10,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [37] 3*parameter(x=6,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [37] 2*parameter(x=4,lambda=[11,7,-9,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=4,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=4,lambda=[9,9,-7,-11]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=3,lambda=[17,-3,-7,-7]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=3,lambda=[15,-1,-5,-9]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=3,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=3,lambda=[11,7,-9,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=3,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=3,lambda=[9,9,-7,-11]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=2,lambda=[15,-1,-5,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=2,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=2,lambda=[13,1,-3,-11]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=2,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=2,lambda=[11,3,-1,-13]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=2,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=2,lambda=[9,5,1,-15]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=1,lambda=[11,7,-9,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=1,lambda=[9,9,-7,-11]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=1,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[9,5,1,-15]/2,nu=[0,0,0,0]/1) [38] 2*parameter(x=1,lambda=[7,7,3,-17]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=0,lambda=[17,-3,-7,-7]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[15,-1,-5,-9]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [38] 5*parameter(x=0,lambda=[13,1,-3,-11]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[11,7,-9,-9]/2,nu=[0,0,0,0]/1) [38] 5*parameter(x=0,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 5*parameter(x=0,lambda=[11,3,-1,-13]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[9,9,-7,-11]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [38] 4*parameter(x=0,lambda=[9,5,1,-15]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=0,lambda=[7,7,3,-17]/2,nu=[0,0,0,0]/1) [38] 3*parameter(x=10,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [39] 1*parameter(x=8,lambda=[19,-5,-7,-7]/2,nu=[0,0,0,0]/1) [39] 3*parameter(x=8,lambda=[17,-3,-5,-9]/2,nu=[0,0,0,0]/1) [39] 5*parameter(x=8,lambda=[15,-1,-3,-11]/2,nu=[0,0,0,0]/1) [39] 5*parameter(x=8,lambda=[13,1,-1,-13]/2,nu=[0,0,0,0]/1) [39] 5*parameter(x=8,lambda=[11,3,1,-15]/2,nu=[0,0,0,0]/1) [39] 3*parameter(x=8,lambda=[9,5,3,-17]/2,nu=[0,0,0,0]/1) [39] 1*parameter(x=8,lambda=[7,7,5,-19]/2,nu=[0,0,0,0]/1) [39] 3*parameter(x=6,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [39] 4*parameter(x=12,lambda=[11,9,-9,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=4,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=4,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=4,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=3,lambda=[19,-5,-7,-7]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=3,lambda=[17,-3,-5,-9]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=3,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [40] 3*parameter(x=3,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=3,lambda=[13,3,-5,-11]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=3,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=3,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=2,lambda=[17,-3,-5,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=2,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=2,lambda=[15,-1,-3,-11]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=2,lambda=[13,3,-5,-11]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=2,lambda=[13,1,-1,-13]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=2,lambda=[11,5,-3,-13]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=2,lambda=[11,3,1,-15]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=2,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=2,lambda=[9,5,3,-17]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=1,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[11,5,-3,-13]/2,nu=[0,0,0,0]/1) [40] 3*parameter(x=1,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=1,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[9,5,3,-17]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[7,7,5,-19]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=0,lambda=[19,-5,-7,-7]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[17,-3,-5,-9]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [40] 5*parameter(x=0,lambda=[15,-1,-3,-11]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 5*parameter(x=0,lambda=[13,3,-5,-11]/2,nu=[0,0,0,0]/1) [40] 6*parameter(x=0,lambda=[13,1,-1,-13]/2,nu=[0,0,0,0]/1) [40] 5*parameter(x=0,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] 5*parameter(x=0,lambda=[11,5,-3,-13]/2,nu=[0,0,0,0]/1) [40] 5*parameter(x=0,lambda=[11,3,1,-15]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [40] 4*parameter(x=0,lambda=[9,5,3,-17]/2,nu=[0,0,0,0]/1) [40] 2*parameter(x=0,lambda=[7,7,5,-19]/2,nu=[0,0,0,0]/1) [40] {same thing for HOLOMORPHIC discrete series d2. Not only are multiplicities all 1, but SET of K-types is MUCH smaller.} atlas> branch_irr(d2,40) Value: 1*parameter(x=2,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) [10] 1*parameter(x=2,lambda=[5,1,-1,-5]/2,nu=[0,0,0,0]/1) [16] 1*parameter(x=2,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=2,lambda=[7,1,-1,-7]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=2,lambda=[7,3,-3,-7]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=2,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=2,lambda=[9,1,-1,-9]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=2,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=2,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=2,lambda=[11,1,-1,-11]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=2,lambda=[11,3,-3,-11]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=2,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=2,lambda=[13,1,-1,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=2,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] {now same thing for DIFFERENT, but still non-holomorphic, discrete series d1} atlas> branch_irr(d1,50) Value: 1*parameter(x=1,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) [10] 1*parameter(x=1,lambda=[5,1,-3,-3]/2,nu=[0,0,0,0]/1) [14] 1*parameter(x=1,lambda=[3,3,-1,-5]/2,nu=[0,0,0,0]/1) [14] 1*parameter(x=1,lambda=[5,1,-1,-5]/2,nu=[0,0,0,0]/1) [16] 1*parameter(x=1,lambda=[5,3,-3,-5]/2,nu=[0,0,0,0]/1) [18] 1*parameter(x=10,lambda=[5,3,-1,-7]/2,nu=[0,0,0,0]/1) [19] 1*parameter(x=7,lambda=[7,1,-3,-5]/2,nu=[0,0,0,0]/1) [19] 1*parameter(x=1,lambda=[7,1,-3,-5]/2,nu=[0,0,0,0]/1) [20] 1*parameter(x=1,lambda=[5,3,-1,-7]/2,nu=[0,0,0,0]/1) [20] 1*parameter(x=1,lambda=[7,3,-5,-5]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=1,lambda=[7,1,-1,-7]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=1,lambda=[5,5,-3,-7]/2,nu=[0,0,0,0]/1) [22] 1*parameter(x=4,lambda=[9,1,-5,-5]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=1,lambda=[9,1,-5,-5]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=1,lambda=[7,3,-3,-7]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=1,lambda=[5,5,-1,-9]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=0,lambda=[5,5,-1,-9]/2,nu=[0,0,0,0]/1) [24] 1*parameter(x=1,lambda=[9,1,-3,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=1,lambda=[7,5,-5,-7]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=1,lambda=[7,3,-1,-9]/2,nu=[0,0,0,0]/1) [26] 1*parameter(x=10,lambda=[7,5,-3,-9]/2,nu=[0,0,0,0]/1) [27] 1*parameter(x=7,lambda=[9,3,-5,-7]/2,nu=[0,0,0,0]/1) [27] 1*parameter(x=1,lambda=[9,3,-5,-7]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=1,lambda=[9,1,-1,-9]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=1,lambda=[7,5,-3,-9]/2,nu=[0,0,0,0]/1) [28] 1*parameter(x=10,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [29] 1*parameter(x=7,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [29] 1*parameter(x=4,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[11,1,-5,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[9,5,-7,-7]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[7,7,-5,-9]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=1,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=0,lambda=[7,5,-1,-11]/2,nu=[0,0,0,0]/1) [30] 1*parameter(x=4,lambda=[11,3,-7,-7]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[11,3,-7,-7]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[11,1,-3,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[9,5,-5,-9]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[9,3,-1,-11]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=1,lambda=[7,7,-3,-11]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=0,lambda=[7,7,-3,-11]/2,nu=[0,0,0,0]/1) [32] 1*parameter(x=4,lambda=[13,1,-7,-7]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[13,1,-7,-7]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[11,3,-5,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[11,1,-1,-11]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[9,7,-7,-9]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[9,5,-3,-11]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=1,lambda=[7,7,-1,-13]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=0,lambda=[7,7,-1,-13]/2,nu=[0,0,0,0]/1) [34] 1*parameter(x=10,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=7,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [35] 1*parameter(x=4,lambda=[13,1,-5,-9]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[13,1,-5,-9]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[11,5,-7,-9]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[11,3,-3,-11]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[9,7,-5,-11]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=1,lambda=[9,5,-1,-13]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=0,lambda=[9,5,-1,-13]/2,nu=[0,0,0,0]/1) [36] 1*parameter(x=10,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [37] 1*parameter(x=7,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [37] 1*parameter(x=4,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[13,3,-7,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[13,1,-3,-11]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[11,7,-9,-9]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[11,5,-5,-11]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[11,3,-1,-13]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[9,9,-7,-11]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=1,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=0,lambda=[9,7,-3,-13]/2,nu=[0,0,0,0]/1) [38] 1*parameter(x=10,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [39] 1*parameter(x=7,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [39] 1*parameter(x=4,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=4,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[15,1,-7,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[13,5,-9,-9]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[13,3,-5,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[13,1,-1,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[11,7,-7,-11]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[11,5,-3,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=1,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=0,lambda=[9,9,-5,-13]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=0,lambda=[9,7,-1,-15]/2,nu=[0,0,0,0]/1) [40] 1*parameter(x=4,lambda=[15,3,-9,-9]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=4,lambda=[15,1,-5,-11]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[15,3,-9,-9]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[15,1,-5,-11]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[13,5,-7,-11]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[13,3,-3,-13]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[11,9,-9,-11]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[11,7,-5,-13]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[11,5,-1,-15]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=1,lambda=[9,9,-3,-15]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=0,lambda=[11,5,-1,-15]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=0,lambda=[9,9,-3,-15]/2,nu=[0,0,0,0]/1) [42] 1*parameter(x=10,lambda=[11,9,-7,-13]/2,nu=[0,0,0,0]/1) [43] 1*parameter(x=7,lambda=[13,7,-9,-11]/2,nu=[0,0,0,0]/1) [43] 1*parameter(x=4,lambda=[17,1,-9,-9]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=4,lambda=[15,3,-7,-11]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[17,1,-9,-9]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[15,3,-7,-11]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[15,1,-3,-13]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[13,7,-9,-11]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[13,5,-5,-13]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[13,3,-1,-15]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[11,9,-7,-13]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[11,7,-3,-15]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=1,lambda=[9,9,-1,-17]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=0,lambda=[11,7,-3,-15]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=0,lambda=[9,9,-1,-17]/2,nu=[0,0,0,0]/1) [44] 1*parameter(x=10,lambda=[11,9,-5,-15]/2,nu=[0,0,0,0]/1) [45] 1*parameter(x=7,lambda=[15,5,-9,-11]/2,nu=[0,0,0,0]/1) [45] 1*parameter(x=4,lambda=[17,1,-7,-11]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=4,lambda=[15,5,-9,-11]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[17,1,-7,-11]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[15,5,-9,-11]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[15,3,-5,-13]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[15,1,-1,-15]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[13,9,-11,-11]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[13,7,-7,-13]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[13,5,-3,-15]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[11,11,-9,-13]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[11,9,-5,-15]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=1,lambda=[11,7,-1,-17]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=0,lambda=[11,9,-5,-15]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=0,lambda=[11,7,-1,-17]/2,nu=[0,0,0,0]/1) [46] 1*parameter(x=10,lambda=[11,9,-3,-17]/2,nu=[0,0,0,0]/1) [47] 1*parameter(x=7,lambda=[17,3,-9,-11]/2,nu=[0,0,0,0]/1) [47] 1*parameter(x=4,lambda=[17,3,-9,-11]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=4,lambda=[17,1,-5,-13]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=4,lambda=[15,7,-11,-11]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[17,3,-9,-11]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[17,1,-5,-13]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[15,7,-11,-11]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[15,5,-7,-13]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[15,3,-3,-15]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[13,9,-9,-13]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[13,7,-5,-15]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[13,5,-1,-17]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[11,11,-7,-15]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=1,lambda=[11,9,-3,-17]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=0,lambda=[13,5,-1,-17]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=0,lambda=[11,11,-7,-15]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=0,lambda=[11,9,-3,-17]/2,nu=[0,0,0,0]/1) [48] 1*parameter(x=10,lambda=[11,9,-1,-19]/2,nu=[0,0,0,0]/1) [49] 1*parameter(x=7,lambda=[19,1,-9,-11]/2,nu=[0,0,0,0]/1) [49] 1*parameter(x=4,lambda=[19,1,-9,-11]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=4,lambda=[17,5,-11,-11]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=4,lambda=[17,3,-7,-13]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[19,1,-9,-11]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[17,5,-11,-11]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[17,3,-7,-13]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[17,1,-3,-15]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[15,7,-9,-13]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[15,5,-5,-15]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[15,3,-1,-17]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[13,11,-11,-13]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[13,9,-7,-15]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[13,7,-3,-17]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[11,11,-5,-17]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=1,lambda=[11,9,-1,-19]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=0,lambda=[13,7,-3,-17]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=0,lambda=[11,11,-5,-17]/2,nu=[0,0,0,0]/1) [50] 1*parameter(x=0,lambda=[11,9,-1,-19]/2,nu=[0,0,0,0]/1) [50] atlas> print_KGB(G) kgbsize: 21 Base grading: [111]. 0: 0 [n,n,n] 1 2 3 10 8 6 (0,0,0,0)#0 e 1: 0 [n,c,n] 0 1 4 10 * 7 (1,1,0,0)#0 e 2: 0 [c,n,c] 2 0 2 * 8 * (0,1,1,0)#0 e 3: 0 [n,c,n] 4 3 0 11 * 6 (0,0,1,1)#0 e 4: 0 [n,n,n] 3 5 1 11 9 7 (1,1,1,1)#0 e 5: 0 [c,n,c] 5 4 5 * 9 * (1,0,0,1)#0 e 6: 1 [n,C,r] 7 13 6 12 * * (0,0,0,0) 1 3^e 7: 1 [n,C,r] 6 14 7 12 * * (1,1,0,0) 1 3^e 8: 1 [C,r,C] 15 8 13 * * * (0,0,0,0) 1 2^e 9: 1 [C,r,C] 16 9 14 * * * (1,0,0,1) 1 2^e 10: 1 [r,C,n] 10 15 11 * * 12 (0,0,0,0) 1 1^e 11: 1 [r,C,n] 11 16 10 * * 12 (0,0,1,1) 1 1^e 12: 2 [r,C,r] 12 17 12 * * * (0,0,0,0) 2 1^3^e 13: 2 [C,C,C] 18 6 8 * * * (0,0,0,0) 1 2x3^e 14: 2 [C,C,C] 19 7 9 * * * (1,0,1,0) 1 2x3^e 15: 2 [C,C,C] 8 10 18 * * * (0,0,0,0) 1 1x2^e 16: 2 [C,C,C] 9 11 19 * * * (0,1,0,1) 1 1x2^e 17: 3 [C,C,C] 20 12 20 * * * (0,0,0,0)#2 2x1^3^e 18: 3 [C,n,C] 13 19 15 * 20 * (0,0,0,0)#1 1x2x3^e 19: 3 [C,n,C] 14 18 16 * 20 * (0,1,1,0)#1 1x2x3^e 20: 4 [C,r,C] 17 20 17 * * * (0,0,0,0) 2 1x2x1^3^e atlas> involution(KGB(G)[20]) Value: | 0, 0, 0, 1 | | 0, 0, 1, 0 | | 0, 1, 0, 0 | | 1, 0, 0, 0 | {LAST KGB element corresponds to MOST SPLIT Cartan. Corr reps are PRINCIPAL SERIES} atlas> set ps=parameter(KGB(G)[20],[3/2,1/2,-1/2,-3/2],[3/2,1/2,-1/2,-3/2]) Variable ps: Param atlas> ps Value: final parameter(x=20,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2) atlas> infinitesimal_character (ps) Value: [ 3, 1, -1, -3 ]/2 {This is Langlands parameter for trivial rep of G: last KGBElt, lambda = rho(G), nu = rho(G). atlas> dimension(ps) Value: 1 atlas> set ps2=parameter(KGB(G)[20],[3/2,1/2,-1/2,-3/2],2*[3/2,1/2,-1/2,-3/2]) Variable ps2: Param atlas> dimension(ps2) Runtime error: representation is infinite dimensional Evaluation aborted. atlas> set ps2=parameter(KGB(G)[20],3*[3/2,1/2,-1/2,-3/2],2*[3/2,1/2,-1/2,-3/2]) Variable ps2: Param (overriding previous instance, which had type Param) atlas> dimension(ps2) Runtime error: representation is infinite dimensional Evaluation aborted. atlas> set ps2=parameter(KGB(G)[20],2*[3/2,1/2,-1/2,-3/2],2*[3/2,1/2,-1/2,-3/2]) Error in 'set' command at :77:0-74: Not an integer vector {since rho not integral, multiplying lambda by two moves it from X^*+rho to X^*.} atlas> set ps3=parameter(KGB(G)[20],3*[3/2,1/2,-1/2,-3/2],3*[3/2,1/2,-1/2,-3/2]) Variable ps3: Param atlas> dimension(ps3) Value: 729 atlas> {finite-dimensional of highest weight m\rho has dimension (m+1)^{#pos roots}} atlas> infinitesimal_character (p3) Error during analysis of expression at :82:0-28 Undefined identifier 'p3' Expression analysis failed atlas> infinitesimal_character (ps3) Value: [ 9, 3, -3, -9 ]/2 atlas> {=3\rho so highest weight is 2\rho, dim 3^6=729} atlas> set ps5=parameter(KGB(G)[20],5*[3/2,1/2,-1/2,-3/2],5*[3/2,1/2,-1/2,-3/2]) Variable ps5: Param atlas> dimension(ps5) Value: 15625 atlas> {= 5^6} atlas> {Next week: is_standard, is_nonzero, is_final...} atlas> {that is, more about entering parameters}