This is the atlas interaction from the seminar February 4, 2020. The script "bottom.at" is on the seminar web site. [Davids-MacBook-Pro-100:~/Dropbox/beamer/bottom] dav% atlas all bottom This is 'atlas' (version 1.0.7, axis language version 0.9.9), the Atlas of Lie Groups and Representations interpreter, compiled on Feb 1 2020 at 15:16:35. http://www.liegroups.org/ atlas> p5 Value: non-normal parameter(x=2,lambda=[3,5]/2,nu=[5,0]/2) {This is a representation of SO(4,1) with continuous parameter nu=5/2, whose unitarity we want to examine by comparison with SO(2,1). atlas> set L=SO(2,1) Variable L: RealForm atlas> {locally SL(2,R); K(R)=O(2)} atlas> set pL=trivial(L) Variable pL: Param atlas> pL Value: final parameter(x=1,lambda=[1]/2,nu=[1]/2) {This is the trivial of SO(2,1), with continuous parameter rho(G) = 1/2. atlas> print_sig_irr_long(pL,10) sig x lambda hw dim height 1 1 [ 1 ]/2 [ 0 ] 1 0 {signature of form on trivial, out to O(2) types height 8} {Not so interesting: trivial has only one K-type, dimension 1, where the form is positive.} atlas> print_sig_irr_long(pL*3,10) {signature of form on trivial*3, out to O(2) types height 8. Now the continuous parameter is (1/2)*3 = 3/2.} sig x lambda hw dim height 1 1 [ 1 ]/2 [ 0 ] 1 0 s 0 [ 1 ]/2 [ 1 ] 2 1 atlas> print_sig_irr_long(pL*7,10) {signature of form on trivial, out to O(2) types height 8} sig x lambda hw dim height 1 1 [ 1 ]/2 [ 0 ] 1 0 s 0 [ 1 ]/2 [ 1 ] 2 1 1 0 [ 3 ]/2 [ 2 ] 2 3 s 0 [ 5 ]/2 [ 3 ] 2 5 {Notice that the signature alternates from + to - as the highest weight of the O(2) type increases. atlas> G Value: disconnected real group with Lie algebra 'so(4,1)' atlas> p5 Value: non-normal parameter(x=2,lambda=[3,5]/2,nu=[5,0]/2) {these were defined in the script bottom.at read at the start.} atlas> print_sig_irr_long(pL*5,10) sig x lambda hw dim height 1 1 [ 1 ]/2 [ 0 ] 1 0 s 0 [ 1 ]/2 [ 1 ] 2 1 1 0 [ 3 ]/2 [ 2 ] 2 3 {signature of form on trivial*5, out to O(2) types height 8} atlas> pL*5 Value: final parameter(x=1,lambda=[1]/2,nu=[5]/2) {Notice the continuous parameter is 5/2, equal to that in p5.} atlas> print_sig_irr_long(p5,20) sig x lambda hw dim height 1 1 [ 5, 3 ]/2 [ 2, 0 ] 9 10 s 0 [ 5, 1 ]/2 [ 2, 1 ] 16 11 1 0 [ 5, 3 ]/2 [ 2, 2 ] 10 13 1 1 [ 7, 1 ]/2 [ 3, 0 ] 16 14 s 0 [ 7, 1 ]/2 [ 3, 1 ] 30 15 1 0 [ 7, 3 ]/2 [ 3, 2 ] 24 17 1 1 [ 9, 3 ]/2 [ 4, 0 ] 25 18 s 0 [ 9, 1 ]/2 [ 4, 1 ] 48 19 {sig of SO(4,1) form on p5, out to O(4) types height 20. We'll see that the "bottom layer" consists of the three representations of highest weight [ 2, j ]; and that the signatures 1, s, 1 there match the signatures for pL*5 computed above.} atlas> print_sig_irr_long(p5*(1/5),20) sig x lambda hw dim height 1 1 [ 5, 3 ]/2 [ 2, 0 ] 9 10 1 1 [ 7, 1 ]/2 [ 3, 0 ] 16 14 1 1 [ 9, 3 ]/2 [ 4, 0 ] 25 18 atlas> p5*(1/5) Value: non-dominant parameter(x=2,lambda=[3,5]/2,nu=[1,0]/2) {Now the continuous parameter for SO(4,1) is 1/2 instead of 5/2, and there is only a single bottom layer K-type (the first. There the signature matches that of the trivial on SO(2,1).} atlas> print_sig_irr_long(pL,10) sig x lambda hw dim height 1 1 [ 1 ]/2 [ 0 ] 1 0 atlas> pL Value: final parameter(x=1,lambda=[1]/2,nu=[1]/2) {idea: cohomological induction PRESERVES SIGNATURES on a few reps of K: this is the BOTTOM LAYER} atlas> set G=Sp(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> print_KGB(G) kgbsize: 11 Base grading: [11]. 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 7: 2 [C,n] 5 8 * 10 (0,0)#2 1x2^e 8: 2 [C,n] 6 7 * 10 (0,1)#2 1x2^e 9: 2 [n,C] 9 4 10 * (0,0)#1 2x1^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e {To begin studying bottom layer results, it's useful to look at parameters constructed with KGB elements x having no "complex descents": simple roots i where the entry in the third column is C, and the corresponding entry in column 3+i+1 is SMALLER than x. For example, root 0 is a complex descent for KGB elements 7 and 8, and root 1 is a complex descent for KGB element 9. This leaves a "no complex descent" list 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e These eight KGB elements can be used to construct parameters for the representations with lowest K-types in the eight regions of the blackboard picture (available on the web site, I hope!). #10 corresponds to the five lowest K-types near 0 (principal series lowest K-types). #0-#3 are the four discrete series, corresponding to the four octants (not very clear on the blackboard) covering "most" of K^. #4 is the northwest to southeast diagonal alternating singletons and pairs. #5 is the horizontal line of vertical triplets, and #6 is the vertical line of horizontal triplets. I'll talk 2/11 about how to make parameters for these lowest K-types using these x.} atlas> set q=parameter(KGB(G,5),[2,1],[0,0]) Variable q: Param {Here I've used x=5. The 2 in [2,1] says how far out in the horizontal row of triples to go; this turns out to have lowest K-type (3,1). The [0,0] says continuous parameter zero, a tempered (unitary) representation. atlas> set x2=KGB(G,2) Variable x2: KGBElt atlas> print_sig_irr_long(q,x2,15) sig x lambda hw dim height 1 5 [ 2, 1 ]/1 [ 3, 1 ] 3 6 1 2 [ 2, 1 ]/1 [ 3, 3 ] 1 7 1 0 [ 2, 1 ]/1 [ 3, -1 ] 5 7 1 2 [ 3, 0 ]/1 [ 4, 2 ] 3 9 1 0 [ 3, 0 ]/1 [ 4, 0 ] 5 9 1 4 [ 3, 2 ]/1 [ 3, -3 ] 7 10 1 0 [ 3, 2 ]/1 [ 4, -2 ] 7 11 2 5 [ 4, 1 ]/1 [ 5, 1 ] 5 12 2 2 [ 4, 1 ]/1 [ 5, 3 ] 3 13 2 0 [ 4, 1 ]/1 [ 5, -1 ] 7 13 1 4 [ 4, 3 ]/1 [ 4, -4 ] 9 14 2 2 [ 5, 0 ]/1 [ 6, 2 ] 5 15 1 2 [ 4, 3 ]/1 [ 5, 5 ] 1 15 1 1 [ 4, 3 ]/1 [ 3, -5 ] 9 15 2 0 [ 5, 0 ]/1 [ 6, 0 ] 7 15 2 0 [ 4, 3 ]/1 [ 5, -3 ] 9 15 {bottom layer is U(2)-reps with 1st coordinate of hwt = 3} {Corresponding Sp(2,R) rep is spherical, nu=0; has all _even_ SO(2) types 2m; shifted by 3,1 to get U(2) highest weights [ 3, 1+2m ].} {In this example everything is unitary, nothing to see about signature.} atlas> set q5=parameter(KGB(G,5),[2,1],[0,5]) Variable q5: Param {Now I've made the continuous parameter [0,5], so the signature on the bottom layer will match that of...} atlas> set p5=trivial(Sp(2,R))*5 Variable p5: Param (overriding previous instance, which had type Param) atlas> p5 Value: final parameter(x=2,lambda=[1]/1,nu=[5]/1) atlas> q5 Value: non-dominant parameter(x=5,lambda=[2,1]/1,nu=[0,5]/1) atlas> print_sig_irr_long(q5,x2,20) sig x lambda hw dim height 1 5 [ 2, 1 ]/1 [ 3, 1 ] 3 6 {***} s 2 [ 2, 1 ]/1 [ 3, 3 ] 1 7 {***} s 0 [ 2, 1 ]/1 [ 3, -1 ] 5 7 {***} s 2 [ 3, 0 ]/1 [ 4, 2 ] 3 9 1 0 [ 3, 0 ]/1 [ 4, 0 ] 5 9 1 4 [ 3, 2 ]/1 [ 3, -3 ] 7 10 {***} s 0 [ 3, 2 ]/1 [ 4, -2 ] 7 11 1+s 5 [ 4, 1 ]/1 [ 5, 1 ] 5 12 1+s 2 [ 4, 1 ]/1 [ 5, 3 ] 3 13 1+s 0 [ 4, 1 ]/1 [ 5, -1 ] 7 13 s 2 [ 5, 0 ]/1 [ 6, 2 ] 5 15 1 2 [ 4, 3 ]/1 [ 5, 5 ] 1 15 1 0 [ 5, 0 ]/1 [ 6, 0 ] 7 15 1 0 [ 4, 3 ]/1 [ 5, -3 ] 9 15 1 2 [ 5, 2 ]/1 [ 6, 4 ] 3 17 s 0 [ 5, 2 ]/1 [ 6, -2 ] 9 17 1+s 5 [ 6, 1 ]/1 [ 7, 1 ] 7 18 1+s 2 [ 6, 1 ]/1 [ 7, 3 ] 5 19 1+s 0 [ 6, 1 ]/1 [ 7, -1 ] 9 19 atlas> print_sig_irr_long(p5,100) sig x lambda hw dim height 1 2 [ 1 ]/1 [ 0 ] 1 0 s 1 [ 1 ]/1 [ -2 ] 1 1 s 0 [ 1 ]/1 [ 2 ] 1 1 1 1 [ 3 ]/1 [ -4 ] 1 3 1 0 [ 3 ]/1 [ 4 ] 1 3 {The Sp(4,R) signature on the U(2)-type (3,1+2m) matches the Sp(2,R) signature on the U(1)-type 2m. I added a *** after each bottom layer row.} {There _is no_ U(2)-type (3,-5), because that weight is not dominant!} {next week: we'll classify irr unitary of Sp(4,R) using known Sp(2,R) results and the blackboard PICTURE.}