dav% atlas all This is 'atlas' (version 1.0.8, axis language version 1.0), the Atlas of Lie Groups and Representations interpreter, compiled on Apr 23 2020 at 11:35:48. http://www.liegroups.org/ atlas> set G=Sp(4,R) Variable G: RealForm {want theta-stable parabolic, Levi U(1).Sp(2,R)} {use KGB element with a low number (from fundamental fiber: the CLOSED K orbits, which have length 0} {similarly, the OPEN KGB orbit, with number equal to #KGB(G)-1, can give real parabolic subalgebras.} atlas> set Q=parabolic_by_wt([1,0],KGB(G,0)) Variable Q: KGPElt (overriding previous instance, which had type mat) atlas> is_parabolic_theta_stable (Q) Value: true atlas> set L=Q.Levi Variable L: RealForm atlas> L Value: connected quasisplit real group with Lie algebra 'sl(2,R).u(1)' atlas> set p0=L.trivial Variable p0: Param atlas> p0 Value: final parameter(x=2,lambda=[0,1]/1,nu=[0,1]/1) atlas> p0.infinitesimal_character Value: [ 0, 1 ]/1 {Next, I want a parameter at the edge of the weakly fair range: "m=-2"} {in the notation from the slides. So I shift the lambda by [-2,0].} atlas> set q0=parameter(KGB(L,2),[-2,1],[0,1]) Variable q0: Param atlas> set Q0=theta_induce_irreducible(q0,G) Variable Q0: ParamPol atlas> Q0 Value: 1*parameter(x=7,lambda=[3,0]/1,nu=[1,0]/1) [0] 1*parameter(x=2,lambda=[1,0]/1,nu=[0,0]/1) [3] {Since we're in the weakly fair range, this is guaranteed to be a} {direct sum of unitary} atlas> for p in monomials(Q0) do is_unitary(p) od Value: [true,true] {Next, move lambda UP by [1,0], into the "fair range," for no reason} atlas> set q1=parameter(KGB(L,2),[-1,1],[0,1]) Variable q1: Param atlas> set Q1=theta_induce_irreducible(q1,G) Variable Q1: ParamPol atlas> Q1 Value: 1*parameter(x=5,lambda=[1,1]/1,nu=[0,1]/1) [3] atlas> Q1.infinitesimal_character Value: [ 1, 1 ]/1 atlas> Q0.infinitesimal_character Value: [ 1, 0 ]/1 {secretly the answer for Q0 is [0,1], but atlas made it dominant} {what happens beyond weakly fair? first reducibility?? Try m=-1} {I _think_ but do not know that the first reducibility for the} {generalized Verma here comes at m=-2. Start with m=-1.} atlas> set qm1=parameter(KGB(L,2),[-3,1],[0,1]) Variable qm1: Param atlas> set Qm1=theta_induce_irreducible(qm1,G) Variable Qm1: ParamPol atlas> Qm1 Value: 1*parameter(x=5,lambda=[1,1]/1,nu=[0,1]/1) [3] atlas> for p in monomials(Qm1) do is_unitary(p) od Value: [true] {still unitary! Now try m=-2.} atlas> set qm2=parameter(KGB(L,2),[-4,1],[0,1]) Variable qm2: Param atlas> set Qm2=theta_induce_irreducible(qm2,G) Variable Qm2: ParamPol atlas> Qm2 Value: -1*parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1) [0] 1*parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1) [6] atlas> for p in monomials(Qm2) do is_unitary(p) od Value: [true,true] atlas> Qm2.infinitesimal_character Value: [ 2, 1 ]/1 {minus sign in Qm2 means that cohom ind lives in more than one degree.} {Nevertheless all the irreducibles are unitary!} NEW SESSION for Sp(6,R) dav% atlas all This is 'atlas' (version 1.0.8, axis language version 1.0), the Atlas of Lie Groups and Representations interpreter, compiled on Apr 23 2020 at 11:35:48. http://www.liegroups.org/ atlas> set G=Sp(6,R) Variable G: RealForm atlas> set Q=parabolic_by_wt([1,0,0],KGB(G,0)) Variable Q: KGPElt (overriding previous instance, which had type mat) atlas> set L=Q.Levi Variable L: RealForm (overriding previous instance, which had type RealForm) atlas> set q3=L.trivial Variable q3: Param (overriding previous instance, which had type Param) atlas> q3 Value: final parameter(x=10,lambda=[0,2,1]/1,nu=[0,2,1]/1) {We want to go to the edge of the weakly fair range, which requires} {shifting lambda DOWN by rho(u) = [3,0,0]} atlas> set q0=parameter(KGB(L,10),[-3,2,1],[0,2,1]) Variable q0: Param (overriding previous instance, which had type Param) atlas> set Q0 = theta_induce_irreducible(q0,G) Variable Q0: ParamPol (overriding previous instance, which had type ParamPol) {should be unitary, since this is the edge of weakly fair} atlas> for p in monomials(Q0) do is_unitary(p) od Value: [true,true] {now we continue to NEGATIVE m. I think (don't know) that the first} {reducibility point for the generalized Verma is m=-3/. Start with m=-1.} atlas> set qm1=parameter(KGB(L,10),[-4,2,1],[0,2,1]) Variable qm1: Param atlas> set Qm1 = theta_induce_irreducible(qm1,G) Variable Qm1: ParamPol atlas> for p in monomials(Qm1) do is_unitary(p) od Value: [true] atlas> set qm2=parameter(KGB(L,10),[-5,2,1],[0,2,1]) Variable qm2: Param atlas> set Qm2 = theta_induce_irreducible(qm2,G) Variable Qm2: ParamPol atlas> for p in monomials(Qm2) do is_unitary(p) od Value: [true] atlas> set qm3=parameter(KGB(L,10),[-6,2,1],[0,2,1]) Variable qm3: Param atlas> set Qm3 = theta_induce_irreducible(qm3,G) Variable Qm3: ParamPol atlas> for p in monomials(Qm3) do is_unitary(p) od Value: [true,true] atlas> Qm3.infinitesimal_character Value: [ 3, 2, 1 ]/1 atlas> {secretly [-3,2,1]: three steps beyond weakly fair.} atlas> {first reducibility for Verma???} atlas> Qm3 Value: 1*parameter(x=44,lambda=[3,2,1]/1,nu=[3,2,1]/1) [0] 1*parameter(x=26,lambda=[3,2,1]/1,nu=[0,2,1]/1) [15] atlas> #KGB(G) Value: 45 {First constituent fo Qm3 is the trivial of G. To understand the second,} {let's set q3=trivial on L; Q3 = cohom ind(q3)} atlas> set q3=L.trivial Variable q3: Param (overriding previous instance, which had type Param) atlas> set Q3=theta_induce_irreducible(q3,G) Variable Q3: ParamPol (overriding previous instance, which had type ParamPol) atlas> Q3 Value: 1*parameter(x=26,lambda=[3,2,1]/1,nu=[0,2,1]/1) [15] {So the SECOND constituent of Qm3 is the irreducible Aq(lambda)} {representation Q3.} atlas> quit Bye.