I edited out a lot of the missteps; I hope that what remains here may make the main points a bit clearer, and still allow you to repeat such experiments. [Davids-MBP-100:~] 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 4 2020 at 08:30:24. http://www.liegroups.org/ atlas> set G=Sp(6,R) Variable G: RealForm atlas> {K=U(3); rational infl char is three rational numbers, up to permutation, sign change} atlas> {Look at cohom ind from q=l+u; need to specify q} atlas> set Q=parabolic_by_wt([1,1,1],KGB(G,0)) Variable Q: KGPElt (overriding previous instance, which had type mat) {reason for 1 1 1: centralizer is GL(3) (real form) ---> Levi L} atlas> set L=Q.Levi Variable L: RealForm atlas> L Value: connected quasisplit real group with Lie algebra 'su(2,1).u(1)' {this is U(2,1)} {Want to cohomologically induce from L to G} atlas> set pL = L.trivial Variable pL: Param atlas> pL Value: final parameter(x=5,lambda=[1,0,-1]/1,nu=[1,0,-1]/1) {Want to tensor trivial of U(2,1) with a positive 1-diml character} atlas> set pL2=parameter(KGB(L,5),[1,0,-1]+2*[1,1,1] ,[1,0,-1]) Variable pL2: Param atlas> pL2.infinitesimal_character Value: [ 3, 2, 1 ]/1 {We've added 2*[1,1,1]} atlas> set pL0=pL Variable pL0: Param atlas> pL0.infinitesimal_character Value: [ 1, 0, -1 ]/1 atlas> set PL2=theta_induce_irreducible(pL2,G) Variable PL2: ParamPol atlas> PL2 Value: 1*parameter(x=17,lambda=[5,4,3]/1,nu=[1,0,-1]/1) [36] atlas> PL2.infinitesimal_character Value: [ 5, 4, 3 ]/1 atlas> {"shifted" in [1,1,1] direction} atlas> is_unitary(first_param(PL2)) Value: true atlas> is_unitary(pL2) Value: true {This is cohomological induction in the good range; takes unitary to unitary} atlas> set PL0=theta_induce_irreducible(pL0,G) Variable PL0: ParamPol atlas> PL0 Value: 1*parameter(x=17,lambda=[3,2,1]/1,nu=[1,0,-1]/1) [18] atlas> PL0.infinitesimal_character Value: [ 3, 2, 1 ]/1 atlas> {shifted down [2,2,2], but still in the good range} atlas> is_unitary(first_param(PL0)) Value: true atlas> set pLminus1=parameter(KGB(L,5),[1,0,-1]+(-1)*[1,1,1] ,[1,0,-1]) Variable pLminus1: Param atlas> pLminus1 Value: final parameter(x=5,lambda=[0,-1,-2]/1,nu=[1,0,-1]/1) atlas> pL0 Value: final parameter(x=5,lambda=[1,0,-1]/1,nu=[1,0,-1]/1) atlas> set PLminus1=theta_induce_irreducible(pLminus1,G) Variable PLminus1: ParamPol atlas> PLminus1.infinitesimal_character Value: [ 2, 1, 0 ]/1 atlas> is_unitary(first_param(PLminus1)) Value: true atlas> set pLminus2=parameter(KGB(L,5),[1,0,-1]+(-2)*[1,1,1] ,[1,0,-1]) Variable pLminus2: Param atlas> set PLminus2=theta_induce_irreducible(pLminus2,G) Variable PLminus2: ParamPol atlas> is_unitary(first_param(PLminus2)) Value: true atlas> PLminus2.infinitesimal_character Value: [ 1, 1, 0 ]/1 at shifts of 0, -1, -2, found infl chars [3,2,1], [2,1,0], [1,0,-1] But atlas makes infinitesimal characters DOMINANT; so [1,0,-1] becomes [1,1,0]. atlas> set pLminus3=parameter(KGB(L,5),[1,0,-1]+(-3)*[1,1,1] ,[1,0,-1]) Variable pLminus3: Param atlas> set PLminus3=theta_induce_irreducible(pLminus3,G) Variable PLminus3: ParamPol atlas> PLminus3 Value: 1*parameter(x=41,lambda=[3,2,0]/1,nu=[2,1,0]/1) [0] 1*parameter(x=30,lambda=[4,1,0]/1,nu=[2,0,0]/1) [5] 1*parameter(x=17,lambda=[2,1,0]/1,nu=[1,0,-1]/1) [9] atlas> for q in monomials(PLminus3) do is_unitary(q) od Value: [true,true,true] atlas> set pLminus4=parameter(KGB(L,5),[1,0,-1]+(-4)*[1,1,1] ,[1,0,-1]) Variable pLminus4: Param atlas> set PLminus4=theta_induce_irreducible(pLminus4,G) Variable PLminus4: ParamPol atlas> PLminus4 Value: 1*parameter(x=44,lambda=[3,2,1]/1,nu=[3,2,1]/1) [0] 1*parameter(x=41,lambda=[4,3,1]/1,nu=[3,2,0]/1) [5] 1*parameter(x=30,lambda=[3,2,1]/1,nu=[3,0,0]/1) [13] 1*parameter(x=17,lambda=[3,2,1]/1,nu=[1,0,-1]/1) [18] atlas> for q in monomials(PLminus4) do is_unitary(q) od Value: [true,false,false,true] pLx = trivial of U(2,1) shifted by det^x: ALWAYS UNITARY PLx = cohom ind(pLx) to Sp(6,R): KNOWN UNITARY IN FAIR RANGE x \ge m2 (the "m2" means "minus two") PL2 1*parameter(x=17,lambda=[5,4,3]/1,nu=[1,0,-1]/1) [height = 36] PL1 1*parameter(x=17,lambda=[4,3,2]/1,nu=[1,0,-1]/1) [height = 27] PL0 1*parameter(x=17,lambda=[3,2,1]/1,nu=[1,0,-1]/1) [height = 18] PLm1 1*parameter(x=17,lambda=[2,1,0]/1,nu=[1,0,-1]/1) [height = 9] PLm2 1*parameter(x=38,lambda=[3,3,0]/1,nu=[1,1,0]/1) [height = 0] PLm3 1*parameter(x=41,lambda=[3,2,0]/1,nu=[2,1,0]/1) [height = 0] (unitary) 1*parameter(x=30,lambda=[4,1,0]/1,nu=[2,0,0]/1) [5] (unitary) 1*parameter(x=17,lambda=[2,1,0]/1,nu=[1,0,-1]/1) [9] (unitary) PLminus4 1*parameter(x=44,lambda=[3,2,1]/1,nu=[3,2,1]/1) [height = 0] (unitary: TRIVIAL of Sp(6,R)) 1*parameter(x=41,lambda=[4,3,1]/1,nu=[3,2,0]/1) [height = 5] (not unitary) 1*parameter(x=30,lambda=[3,2,1]/1,nu=[3,0,0]/1) [height = 13] (not unitary) 1*parameter(x=17,lambda=[3,2,1]/1,nu=[1,0,-1]/1) [height = 18] (unitary) SUMMARY {Timothy lectures: parameters pLx are in GOOD RANGE if x ge 0} {pLm1: weakly good} {pLm2:weakly fair -- still know cohom ind from unitary is unitary; proof uses that some generalized Verma module is irreducible} {below -2: NOT weakly fair; eventually (namely at -3) the generalized Verma becomes reducible, cohom ind becomes complicated} {Example above suggests: at first reducibility for Verma (pLminus3 in this case) still get unitary composition factors???} {Proving this (or finding counterexamples, and figuring out what _is_ true) seems like a NICE MATH PROBLEM; easy to test a bunch of exs.} atlas> set Qprime=parabolic_by_wt([1,1,1],KGB(G,5)) Variable Qprime: KGPElt {choice of 5 is because I looked at print_KGB... it showed me that this would make Levi subgroup U(3)} atlas> set Lprime = Qprime.Levi Variable Lprime: RealForm atlas> Lprime Value: compact connected real group with Lie algebra 'su(3).u(1)' {U(3) = K} atlas> set pLprime0=parameter(KGB(Lprime,0),[1,0,-1]+(0)*[1,1,1] ,[0,0,0]) {Again I want to start with the trivial pLprime0 and twist it by det^x to get pLprimex, then cohomologically induce to PLprimex. I'll just print answers...} PLprime3 1*parameter(x=5,lambda=[6,5,4]/1,nu=[0,0,0]/1) [49] PLprime0 1*parameter(x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1) [22] PLprimem1 1*parameter(x=5,lambda=[2,1,0]/1,nu=[0,0,0]/1) [13] PLprimem2 1*parameter(x=21,lambda=[1,3,0]/1,nu=[0,1,0]/1) [5] {end of weakly fair} PLprimem3 1*parameter(x=40,lambda=[4,3,0]/1,nu=[2,1,0]/1) [0] (unitary) 1*parameter(x=5,lambda=[2,1,0]/1,nu=[0,0,0]/1) [13] (unitary) PLprimem4 1*parameter(x=44,lambda=[3,2,1]/1,nu=[3,2,1]/1) [0] (unitary TRIVIAL of Sp(6,R)) 1*parameter(x=40,lambda=[3,2,1]/1,nu=[3,2,0]/1) [5] (not unitary) 1*parameter(x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1) [22] (unitary) Based on this example, the proposed problem "are all comp factors still unitary at "first reducible" Verma point"? still looks good. Next week: how does this tell you about unipotent reps? End with the K-types of... atlas> PLprime3 Value: 1*parameter(x=5,lambda=[6,5,4]/1,nu=[0,0,0]/1) [49] This is a holomorphic discrete series with one-dimensional LKT. Fun to see its K-types, which were computed by Schmid in a paper in the 1960s: atlas> print_branch_irr_long(PLprime3,KGB(G,5),80) m x lambda hw dim height 1 5 [ 6, 5, 4 ]/1 [ 7, 7, 7 ] 1 49 1 5 [ 8, 5, 4 ]/1 [ 9, 7, 7 ] 6 59 1 5 [ 8, 7, 4 ]/1 [ 9, 9, 7 ] 6 65 1 5 [ 8, 7, 6 ]/1 [ 9, 9, 9 ] 1 67 1 5 [ 10, 5, 4 ]/1 [ 11, 7, 7 ] 15 69 1 5 [ 10, 7, 4 ]/1 [ 11, 9, 7 ] 27 75 1 5 [ 10, 7, 6 ]/1 [ 11, 9, 9 ] 6 77 1 5 [ 12, 5, 4 ]/1 [ 13, 7, 7 ] 28 79