The action (Sp(4)xSp(4)xU(2)xT):M8,2(C) Chal Benson and Gail Ratcliff This Maple worksheet provides details concerning an example presented in section 5.6 of our paper Spaces of Bounded Spherical Functions on Heisenberg Groups: Part II. At issue is the (multiplicity free) action of the compact group Sp(4)xSp(4)xU(2)xT on the space V=M8,2(C) of 8x2 complex matrices via (k1, k2, k3, c). Z = diag(ck1,k2) Z k3^t. We will show that this action is well-behaved as defined in Section 2 of the paper. restart:with(linalg): Functions dot and sym below implement the inner product of 8x2 matrices and the symplectic product of vectors in C^4. In the inner product complex conjugation should be applied to the matrix entries of the second input. But for our purposes it will suffice to restrict attention to matrices all of whose entries are real. dot:=(z,w)->trace(z&*transpose(w)); Zio2JEkiekc2Ikkid0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLV9JJ2xpbmFsZ0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUkmdHJhY2VHRiU2Iy1JIyYqR0YlNiQ5JC1fRixJKnRyYW5zcG9zZUdGJTYjOSVGJUYlRiU= sym:=(a,b)->sum(a['i']*b['i'+2]-a['i'+2]*b['i'],'i'=1..2); Zio2JEkiYUc2IkkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc3VtR0YlNiQsJiomJjkkNiMuSSJpR0YlIiIiJjklNiMsJkYyRjQiIiNGNEY0RjQqJiZGMEY3RjQmRjZGMUY0ISIiL0YyO0Y0RjlGJUYlRiU= So for example... z:=matrix(8,2): dot(z,z); sym(col(submatrix(z,5..8,1..2),1),col(submatrix(z,5..8,1..2),2)); LEIqJCZJInpHNiI2JCIiIkYoIiIjRigqJCZGJTYkRihGKUYpRigqJCZGJTYkRilGKEYpRigqJCZGJTYkRilGKUYpRigqJCZGJTYkIiIkRihGKUYoKiQmRiU2JEY2RilGKUYoKiQmRiU2JCIiJUYoRilGKCokJkYlNiRGPUYpRilGKCokJkYlNiQiIiZGKEYpRigqJCZGJTYkRkRGKUYpRigqJCZGJTYkIiInRihGKUYoKiQmRiU2JEZLRilGKUYoKiQmRiU2JCIiKEYoRilGKCokJkYlNiRGUkYpRilGKCokJkYlNiQiIilGKEYpRigqJCZGJTYkRllGKUYpRig= LCoqJiZJInpHNiI2JCIiJiIiIkYpJkYlNiQiIigiIiNGKUYpKiYmRiU2JEYsRilGKSZGJTYkRihGLUYpISIiKiYmRiU2JCIiJ0YpRikmRiU2JCIiKUYtRilGKSomJkYlNiRGOkYpRikmRiU2JEY3Ri1GKUYz This multiplicity free action has rank 7. Fundamental highest weights and highest weight vectors are implement below as A1,...,A7 (highest weights) and h1,...h7 (highest weight vectors). We list the weights as 7-tuples "[ sp1, sp1, sp2, sp2, t, u, u]". The first 2 entries give minus the weight on the first Sp(4) factor, entries 3,4 minus the weight on the second Sp(4) factor. Entry 5 is minus the weight from the scalar action and the last two entries give minus the weight on the U(2) factor. h1:=z->z[1,1]: h1(z); A1:=[1,0,0,0,1,1,0]; Jkkiekc2IjYkIiIiRiY= NykiIiIiIiFGJEYkRiNGI0Yk Qyg+SSNoMkc2ImYqNiNJInpHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSY5JDYkIiImIiIiRiVGJUYlISIiLUYkRidGMD5JI0EyR0YlNykiIiFGNkYwRjZGNkYwRjZGMA== Jkkiekc2IjYkIiImIiIi NykiIiFGIyIiIkYjRiNGJEYj h3:=z->z[1,1]*z[2,2]-z[1,2]*z[2,1]: h3(z); A3:=[1,1,0,0,2,1,1]; LCYqJiZJInpHNiI2JCIiIkYoRigmRiU2JCIiI0YrRihGKComJkYlNiRGKEYrRigmRiU2JEYrRihGKCEiIg== NykiIiJGIyIiIUYkIiIjRiNGIw== h4:=z->z[5,1]*z[6,2]-z[5,2]*z[6,1]: h4(z); A4:=[0,0,1,1,0,1,1]; LCYqJiZJInpHNiI2JCIiJiIiIkYpJkYlNiQiIiciIiNGKUYpKiYmRiU2JEYoRi1GKSZGJTYkRixGKUYpISIi NykiIiFGIyIiIkYkRiNGJEYk h5:=z->z[1,1]*z[5,2]-z[1,2]*z[5,1]: h5(z); A5:=[1,0,1,0,1,1,1]; LCYqJiZJInpHNiI2JCIiIkYoRigmRiU2JCIiJiIiI0YoRigqJiZGJTYkRihGLEYoJkYlNiRGK0YoRighIiI= NykiIiIiIiFGI0YkRiNGI0Yj h6:=z->sym(col(submatrix(z,1..4,1..2),1),col(submatrix(z,1..4,1..2),2)): h6(z); A6:=[0,0,0,0,2,1,1]; LCoqJiZJInpHNiI2JCIiIkYoRigmRiU2JCIiJCIiI0YoRigqJiZGJTYkRitGKEYoJkYlNiRGKEYsRighIiIqJiZGJTYkRixGKEYoJkYlNiQiIiVGLEYoRigqJiZGJTYkRjhGKEYoJkYlNiRGLEYsRihGMg== NykiIiFGI0YjRiMiIiMiIiJGJQ== h7:=z->sym(col(submatrix(z,5..8,1..2),1),col(submatrix(z,5..8,1..2),2)): h7(z); A7:=[0,0,0,0,0,1,1]; LCoqJiZJInpHNiI2JCIiJiIiIkYpJkYlNiQiIigiIiNGKUYpKiYmRiU2JEYsRilGKSZGJTYkRihGLUYpISIiKiYmRiU2JCIiJ0YpRikmRiU2JCIiKUYtRilGKSomJkYlNiRGOkYpRikmRiU2JEY3Ri1GKUYz NykiIiFGI0YjRiNGIyIiIkYk Matrix X will be an arbitrary element of the Lie algebra sp(4,C)xsp(4,C).... X := matrix(8, 8):for m to 4 do for n from 5 to 8 do X[m, n] := 0; X[n, m] := 0 end do end do:X[1, 4] := X[2, 3]: X[3, 3] := -X[1, 1]: X[4, 3] := -X[1, 2]: X[3, 4] := -X[2, 1]: X[4, 4] := -X[2, 2]: X[4, 1] := X[3, 2]: X[5, 8] := X[6, 7]: X[7, 7] := -X[5, 5]: X[8, 7] := -X[5, 6]: X[7, 8] := -X[6, 5]: X[8, 8] := -X[6, 6]: X[8, 5] := X[7, 6]:evalm(X); 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 Matrix Y will be an arbitrary element of gl(2,C).... Y:=matrix(2,2): evalm(Y); PTYiNiQ7IiIiIiIjRiVFXFtsJTYkRiZGJiZJIllHRiNGKTYkRidGJyZGK0YsNiRGJ0YmJkYrRi42JEYmRicmRitGMA== Matrix T will be be an arbitrary scalar acting on the first 4 rows.... Qyg+SSJURzYiLUknbWF0cml4R0YlNiUiIilGKSIiISEiIj8oSSJuR0YlIiIiRi4iIiVJJXRydWVHJSpwcm90ZWN0ZWRHPiZGJDYkRi1GLUkjVDFHRiVGKy1JJmV2YWxtRzYkRjFJKF9zeXNsaWJHRiU2I0YkRi4= 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 The moment map takes V to the dual of sp(4)xsp(4)xu(2)x(iR). We implement this below as function mom. mom := z->simplify(expand( dot(evalm(X&*z),z)+dot(evalm(T&*z),z)+dot(evalm(z&*transpose(Y)),z))) ; Zio2I0kiekc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkpc2ltcGxpZnlHRiU2Iy1JJ2V4cGFuZEclKnByb3RlY3RlZEc2IywoLUkkZG90R0YlNiQtSSZldmFsbUc2JEYuSShfc3lzbGliR0YlNiMtSSMmKkdGJTYkSSJYR0YlOSRGPSIiIi1GMjYkLUY1NiMtRjo2JEkiVEdGJUY9Rj1GPi1GMjYkLUY1NiMtRjo2JEY9LV9JJ2xpbmFsZ0dGNkkqdHJhbnNwb3NlR0YlNiNJIllHRiVGPUY+RiVGJUYl We claim that the following is a generic spherical point for the weight QyQtSSZldmFsbUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLDAqJkkiYUdGKCIiIkkjQTFHRihGLUYtKiZJImJHRihGLUkjQTJHRihGLUYtKiZJImNHRihGLUkjQTNHRihGLUYtKiZJImRHRihGLUkjQTRHRihGLUYtKiZJImVHRihGLUkjQTVHRihGLUYtKiZJImZHRihGLUkjQTZHRihGLUYtKiZJImdHRihGLUkjQTdHRihGLUYtRi0= PTYiNiM7IiIiIiIoRVxbbChGJiwoSSJhR0YjRiZJImNHRiNGJkkiZUdGI0YmIiIjRisiIiQsKEkiYkdGI0YmSSJkR0YjRiZGLEYmIiIlRjEiIiYsKkYqRiZGK0YtRixGJkkiZkdGI0YtIiInLDBGKkYmRjBGJkYrRiZGMUYmRixGJkY1RiZJImdHRiNGJkYnLCxGK0YmRjFGJkYsRiZGNUYmRjhGJg== namely ...... QyQ+SSRzcHRHNiItSSdNYXRyaXhHRiU2Iy9JJCVpZEdGJSIrQyEpNCpSJSEiIg== For example...... QygtSSZldmFsbUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLDBJI0ExR0YoIiIiSSNBMkdGKCIiI0kjQTNHRigiIiRJI0E0R0YoIiIlSSNBNUdGKCIiJkkjQTZHRigiIidJI0E3R0YoIiIoRiwtSSVzdWJzR0YmNiQ8KS9JImFHRihGLC9JImJHRihGLi9JImNHRihGMC9JImRHRihGMi9JImVHRihGNC9JImZHRihGNi9JImdHRihGOEkkc3B0R0YoRiwtSSRtb21HRig2I0kiJUdGKEYs PTYiNiM7IiIiIiIoRVxbbChGJiIiKiIiIyIiJEYrIiM2IiIlRi0iIiYiI0MiIiciI0dGJyIjRA== LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciK284YzVW LDAmSSJYRzYiNiQiIiJGJyIiKiZGJDYkIiIjRisiIiQmRiQ2JCIiJkYvIiM2JkYkNiQiIidGMyIiJUkjVDFHRiUiI0MmSSJZR0YlRiYiI0cmRjhGKiIjRA== %%%%%%%%%%%%%%%%%%%%%%% We discovered the formula given for spt above by generalizing from numerical examples. It is not too hard to check by hand that this is indeed a generic generalized spherical point. But we prove this below by using Maple to apply the moment map (mom) to spt to verify that it maps to the weight aA1+bA2+cA3+dA4+eA5+fA6+gA7. First we will rewrite spt in a form that enables Maple to simplify the result produced by mom. getallterms:=proc(sp) local ntermsarray, dtermsarray, Trms, i, j; ntermsarray:=map(x->{op(convert(numer(x^2), list))}, sp); dtermsarray:=map(x->{op(convert(denom(x^2), list))}, sp); Trms:={}; for i from 1 to 8 do for j from 1 to 2 do Trms:=Trms union ntermsarray[i,j] union dtermsarray[i,j] od od; convert(Trms, list) end: Terms:=getallterms(spt); nops(Terms); NzNJImFHNiJJImJHRiRJImNHRiRJImRHRiRJImVHRiRJImZHRiRJImdHRiQsJkYjIiIiRiVGLCwmRiNGLEYoRiwsJkYoRixGJUYsLChGI0YsRiZGLEYoRiwsKEYjRixGJiIiI0YoRiwsKEYjRixGKEYsRiVGLCwoRiVGLEYnRixGKEYsLChGJUYsRidGMUYoRiwsKkYjRixGJkYxRihGLEYpRiwsKkYlRixGJ0YxRihGLEYqRiw= IiM8 rewrite := proc(q) local num, den, numr, denr, j: global Terms; num:={op(convert(numer(q), list))}; den:={op(convert(denom(q), list))}; numr:=1; denr:=1; for j from 1 to nops(Terms) do if Terms[j] in num then numr:=numr*t[j] fi; if Terms[j] in den then denr:=denr*t[j] fi od; numr/denr end: The matrix zt is the spherical point expressed in terms of the t's. zt:=zip((x,y)->sign(x)*(rewrite(y^2)), spt, spt); LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciK1doWDhX To convert back from "t-variables" to parameters (a,b,...) use this substitution...... Sub:={seq(t[j]=sqrt(Terms[j]), j=1..nops(Terms))}; 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 We apply the moment map to zt, convert back to parameters (a,...,g) , simplify and collect terms... mom(zt): subs(Sub,%): simplify(%): collect(%,[X[1,1],X[2,2],X[5,5],X[6,6],T1,Y[1,1],Y[2,2]]); LDAqJiwoSSJhRzYiIiIiSSJjR0YmRidJImVHRiZGJ0YnJkkiWEdGJjYkRidGJ0YnRicqJkYoRicmRis2JCIiI0YwRidGJyomLChJImJHRiZGJ0kiZEdGJkYnRilGJ0YnJkYrNiQiIiZGN0YnRicqJkY0RicmRis2JCIiJ0Y7RidGJyomLCpGJUYnRihGMEYpRidJImZHRiZGMEYnSSNUMUdGJkYnRicqJiwwRiVGJ0YzRidGKEYnRjRGJ0YpRidGPkYnSSJnR0YmRidGJyZJIllHRiZGLEYnRicqJiwsRihGJ0Y0RidGKUYnRj5GJ0ZCRidGJyZGREYvRidGJw== So this is a "diagonal functional" and the weight for aA1+bA2+cA3+dA4+eA5+fA6+gA7 is: evalm(a*A1+b*A2+c*A3+d*A4+e*A5+f*A6+g*A7); PTYiNiM7IiIiIiIoRVxbbChGJiwoSSJhR0YjRiZJImNHRiNGJkkiZUdGI0YmIiIjRisiIiQsKEkiYkdGI0YmSSJkR0YjRiZGLEYmIiIlRjEiIiYsKkYqRiZGK0YtRixGJkkiZkdGI0YtIiInLDBGKkYmRjBGJkYrRiZGMUYmRixGJkY1RiZJImdHRiNGJkYnLCxGK0YmRjFGJkYsRiZGNUYmRjhGJg== which agrees with the moment map output above. This completes the justification that z:=spt(a,b,c,d,e,f,g) is indeed a (generalized) spherical point for the weight aA1+bA2+cA3+dA4+eA5+fA6+gA7. %%%%%%%%%%%%%%%%%%%%%%%%%%%% To show that this multilicity free action is well-behaved we will apply Lemma 2.5 from the paper. Next we evaluate the highest weight vectors h1,...,h7 at our general spherical point spt. For this we work in terms of the t's and simplify. h1(zt):h1n:=subs(Sub,%); LCQqMEkiYUc2IiMiIiIiIiMsKEYkRidJImNHRiVGJ0kiZUdGJUYnRiYsKEYkRidGK0YnSSJiR0YlRidGJiwqRiRGJ0YqRihGK0YnSSJmR0YlRidGJiwmRiRGJ0YtRicjISIiRigsJkYkRidGK0YnRjEsKEYkRidGKkYoRitGJ0YxRjI= h2(zt):subs(Sub,%):h2n:=factor(%); LCQqMEkiYkc2IiMiIiIiIiMsKEkiYUdGJUYnSSJlR0YlRidGJEYnRiYsKEYkRidJImRHRiVGJ0YrRidGJiwqRiRGJ0YtRihGK0YnSSJnR0YlRidGJiwmRipGJ0YkRicjISIiRigsJkYrRidGJEYnRjEsKEYkRidGLUYoRitGJ0YxRjI= h3(zt):subs(Sub,%):h3n:=factor(%); KiosKEkiYUc2IiIiIkkiY0dGJUYmSSJlR0YlRiYjRiYiIiMsKkYkRiZGJ0YqRihGJkkiZkdGJUYmRiZGJ0YpLChGJEYmRidGKkYoRiYhIiI= h4(zt):subs(Sub,%):h4n:=factor(%); LCQqKiwoSSJiRzYiIiIiSSJkR0YmRidJImVHRiZGJyNGJyIiIywqRiVGJ0YoRitGKUYnSSJnR0YmRidGJ0YoRiosKEYlRidGKEYrRilGJyEiIkYv h5(zt):subs(Sub,%):h5n:=factor(%); KjYsKEkiYUc2IiIiIkkiY0dGJUYmSSJlR0YlRiYjRiYiIiMsKEYkRiZGKEYmSSJiR0YlRiZGKSwqRiRGJkYnRipGKEYmSSJmR0YlRiZGKUYoRiksKEYsRiZJImRHRiVGJkYoRiZGKSwqRixGJkYwRipGKEYmSSJnR0YlRiZGKSwoRixGJkYwRipGKEYmIyEiIkYqLCZGKEYmRixGJkY0LChGJEYmRidGKkYoRiZGNCwmRiRGJkYoRiZGNA== h6(zt):subs(Sub,%):h6n:=factor(%); LCQqJiwqSSJhRzYiIiIiSSJjR0YmIiIjSSJlR0YmRidJImZHRiZGJyNGJ0YpRitGLCEiIg== QygtSSNoN0c2IjYjSSN6dEdGJSEiIi1JJXN1YnNHJSpwcm90ZWN0ZWRHNiRJJFN1YkdGJUkiJUdGJUYoPkkkaDduR0YlLUknZmFjdG9yRzYkRitJKF9zeXNsaWJHRiU2I0YuIiIi LCQqJiwqSSJiRzYiIiIiSSJkR0YmIiIjSSJlR0YmRidJImdHRiZGJyNGJ0YpRitGLCEiIg== These are the formulas given in section 5.6 of our paper. They show that for positive real parameters (a,b,c,d,e,f,g) each fundamental highest weight vector hj(z) takes a non-zero value at z=spt(a,b,c,d,e,f,g). This shows that condition (2) in Lemma 2.5 holds. It remains to check conditions (3) and (4) in Lemma 2.5. Condition (3) requires that limits of (generalized) spherical points exist if we take some variables to zero. First, we generate all possible ways the variables can go to zero. There are 126 proper non-empty subsets of {a=0, b=0, c=0, d=0, e=0, f=0, g=0}. with(combinat): zs:=[a=0,b=0,c=0,d=0,e=0,f=0,g=0]; ch:=[seq(choose(%)[i],i=2..127)]: :nops(%); NykvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm IiRFIg== The following code lists each possible setting of parameters a...g to zero and prints the limiting (generalized) spherical point. This verifies condition (3) in Lemma 3.5. In particular no singularities arise as we perfom these limits via setting parameters to zero in succession. for i from 1 to nops(ch) do: print(ch[i]); zf:=evalm(spt): for j from 1 to nops(ch[i]) do: zf:=subs(ch[i][j],evalm(zf)): end do: print(zf); end do: NyMvSSJhRzYiIiIh SSN6Zkc2Ig== NyMvSSJiRzYiIiIh SSN6Zkc2Ig== NyMvSSJjRzYiIiIh SSN6Zkc2Ig== NyMvSSJkRzYiIiIh SSN6Zkc2Ig== NyMvSSJlRzYiIiIh SSN6Zkc2Ig== NyMvSSJmRzYiIiIh SSN6Zkc2Ig== NyMvSSJnRzYiIiIh SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiYkdGJUYm SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiY0dGJUYm SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiZEdGJUYm SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiZUdGJUYm SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiZkdGJUYm SSN6Zkc2Ig== NyQvSSJhRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyQvSSJiRzYiIiIhL0kiY0dGJUYm SSN6Zkc2Ig== NyQvSSJiRzYiIiIhL0kiZEdGJUYm SSN6Zkc2Ig== NyQvSSJiRzYiIiIhL0kiZUdGJUYm SSN6Zkc2Ig== NyQvSSJiRzYiIiIhL0kiZkdGJUYm SSN6Zkc2Ig== NyQvSSJiRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyQvSSJjRzYiIiIhL0kiZEdGJUYm SSN6Zkc2Ig== NyQvSSJjRzYiIiIhL0kiZUdGJUYm SSN6Zkc2Ig== NyQvSSJjRzYiIiIhL0kiZkdGJUYm SSN6Zkc2Ig== NyQvSSJjRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyQvSSJkRzYiIiIhL0kiZUdGJUYm SSN6Zkc2Ig== NyQvSSJkRzYiIiIhL0kiZkdGJUYm SSN6Zkc2Ig== NyQvSSJkRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyQvSSJlRzYiIiIhL0kiZkdGJUYm SSN6Zkc2Ig== NyQvSSJlRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyQvSSJmRzYiIiIhL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJhRzYiIiIhL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJiRzYiIiIhL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJjRzYiIiIhL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJkRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyUvSSJkRzYiIiIhL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJkRzYiIiIhL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyUvSSJlRzYiIiIhL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJhRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJiRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NyYvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJjRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NyYvSSJkRzYiIiIhL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJhRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NycvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJiRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NycvSSJjRzYiIiIhL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZEdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiY0dGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiYkdGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJhRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== NygvSSJiRzYiIiIhL0kiY0dGJUYmL0kiZEdGJUYmL0kiZUdGJUYmL0kiZkdGJUYmL0kiZ0dGJUYm SSN6Zkc2Ig== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Finally we complete the verification that our action is well-behaved by showing that condition (4) in Lemma 3.5 applies. For each subset of the parameters (a,...,g) we obtained above a spherical point for the weight with those parameters set equal to zero. We must also check that the fundamental highest weight vectors associated with the complementary parameters are non-zero at this spherical point. For example at the spherical point Z for a weight of the sort aA1+0A2+cA3+0A4+0A5+fA6+gA7 (i.e. with b=d=e=0) we require that each of h1(Z), h3(Z), h6(Z) and h7(Z) be non-zero. The code below generates the following output for each of the 126 proper non-empty subsets of {a=0, b=0, c=0, d=0, e=0, f=0, g=0}: - A listing of the subset. These parameters are set to zero in succession to obtain a non-generic (generalized) spherical point as in the previous output. - A list of the fundamental highest weight vectors (h1...h7) associated with the complementary parameters. - A list of values for these hj's at the spherical point. The output shows that in all cases each fundamental highest weight vector for a complementary parameter takes a non-zero value at the limiting spherical point in question. Thus condition (4) from Lemma 3.5 does hold here, completing our analysis for this example. [1,2,3,4,5,6,7]: ch:=[seq(choose(%)[i],i=2..127)]: h:=[h1,h2,h3,h4,h5,h6,h7]: hn:=[h1n,h2n,h3n,h4n,h5n,h6n,h7n]: zs:=[a=0,b=0,c=0,d=0,e=0,f=0,g=0]: for i from 1 to nops(ch) do: s:=map(x->zs[x],ch[i]): print(s); zc:=[op({1,2,3,4,5,6,7}minus{op(ch[i])})]: hs:=map(x->hn[x],zc): for j from 1 to nops(s) do: hs:=subs(s[j],hs): end do: print(map(x->h[x],zc)); print(hs); end do: NyMvSSJhRzYiIiIh NyhJI2gyRzYiSSNoM0dGJEkjaDRHRiRJI2g1R0YkSSNoNkdGJEkjaDdHRiQ= NygsJCooLChJImJHNiIiIiJJImRHRidGKEkiZUdGJ0YoI0YoIiIjLCpGJkYoRilGLEYqRihJImdHRidGKEYrLChGJkYoRilGLEYqRigjISIiRixGMSoqLCZJImNHRidGKEYqRihGKywoRjRGLEYqRihJImZHRidGKEYoRjRGKywmRjRGLEYqRihGMSwkKipGJUYrRi1GKEYpRitGL0YxRjEqLkYzRitGNUYrRiVGK0YtRitGL0YwRjdGMCwkKiZGNUYrRjZGK0YxLCQqJkYtRitGLkYrRjE= NyMvSSJiRzYiIiIh NyhJI2gxRzYiSSNoM0dGJEkjaDRHRiRJI2g1R0YkSSNoNkdGJEkjaDdHRiQ= 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