{Timothy talked about GL(n,C), and mentioned an example in GL(6,C)} set G=GL(6,C) Variable G: RealForm (overriding previous instance, which had type RealForm) {Since some things are more interesting for other groups, I'll keep also a second example} atlas> set G1=Sp(6,R) Variable G1: RealForm {He talked about theta-stable parabolics. Atlas will list all of those for you.} atlas> set Q1=theta_stable_parabolics(G1) Variable Q1: [KGPElt] atlas> set Q=theta_stable_parabolics(G) Variable Q: [KGPElt] {You can ask how many there are.} atlas> #Q Value: 32 atlas> #Q1 Value: 34 {You can print a list, with their Levi subgroups.} atlas> void:for q in P do prints(q,", Levi = ",Levi(q)) od ([],KGB element #0), Levi = connected quasisplit real group with Lie algebra 'gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,5],KGB element #5), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([1,6],KGB element #4), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,5,6],KGB element #48), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([2,7],KGB element #3), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,2,5,7],KGB element #18), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([1,2,6,7],KGB element #35), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,2,5,6,7],KGB element #258), Levi = connected quasisplit real group with Lie algebra 'sl(4,C).gl(1,C).gl(1,C).gl(1,C)' ([3,8],KGB element #2), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,3,5,8],KGB element #17), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([1,3,6,8],KGB element #13), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,3,5,6,8],KGB element #96), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C)' ([2,3,7,8],KGB element #26), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,2,3,5,7,8],KGB element #85), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C)' ([1,2,3,6,7,8],KGB element #200), Levi = connected quasisplit real group with Lie algebra 'sl(4,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,2,3,5,6,7,8],KGB element #621), Levi = connected quasisplit real group with Lie algebra 'sl(5,C).gl(1,C).gl(1,C)' ([4,9],KGB element #1), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,4,5,9],KGB element #16), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([1,4,6,9],KGB element #12), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,4,5,6,9],KGB element #95), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C)' ([2,4,7,9],KGB element #9), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,2,4,5,7,9],KGB element #42), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C)' ([1,2,4,6,7,9],KGB element #70), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).sl(2,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,2,4,5,6,7,9],KGB element #358), Levi = connected quasisplit real group with Lie algebra 'sl(4,C).sl(2,C).gl(1,C).gl(1,C)' ([3,4,8,9],KGB element #21), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' ([0,3,4,5,8,9],KGB element #80), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C)' ([1,3,4,6,8,9],KGB element #64), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C)' ([0,1,3,4,5,6,8,9],KGB element #250), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).sl(3,C).gl(1,C).gl(1,C)' ([2,3,4,7,8,9],KGB element #175), Levi = connected quasisplit real group with Lie algebra 'sl(4,C).gl(1,C).gl(1,C).gl(1,C)' ([0,2,3,4,5,7,8,9],KGB element #317), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(4,C).gl(1,C).gl(1,C)' ([1,2,3,4,6,7,8,9],KGB element #561), Levi = connected quasisplit real group with Lie algebra 'sl(5,C).gl(1,C).gl(1,C)' ([0,1,2,3,4,5,6,7,8,9],KGB element #719), Levi = connected quasisplit real group with Lie algebra 'sl(6,C).gl(1,C)' {Each q=l+u gives rise to a particular LONGEST KGB element xq: as a Borel subgroup, it's the Iwasawa Borel in l, plus the nil radical u of u. This element xq is the second thing listed. First thing is the (atlas numbers of) the simple roots in the Levi. Dynkin diagram of GL(6,C) is 0 -- 1 -- 2 -- 3 -- 4 5 -- 6 -- 7 -- 8 -- 9 These two pieces are interchanged by the distinguished involution, so a theta-stable L is given by (any) subset of {0,1,2,3,4} together with the corresponding subset of {5,6,7,8,9}: 32 altogether. The Levi is described as a real group next: a product of various gl(m,C), written here as a product of sl(m,C) x gl(1,C).} {Here's Sp(6,R)} atlas> void:for q in Q1 do prints(q,", Levi = ",Levi(q)) od ([],KGB element #0), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #1), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #2), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #3), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #4), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #5), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #6), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([],KGB element #7), Levi = compact connected quasisplit real group with Lie algebra 'u(1).u(1).u(1)' ([0],KGB element #2), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([0],KGB element #5), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([0],KGB element #6), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([0],KGB element #7), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([0],KGB element #8), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([0],KGB element #9), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([1],KGB element #1), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([1],KGB element #3), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([1],KGB element #5), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([1],KGB element #7), Levi = compact connected real group with Lie algebra 'su(2).u(1).u(1)' ([1],KGB element #10), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([1],KGB element #11), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([0,1],KGB element #5), Levi = compact connected real group with Lie algebra 'su(3).u(1)' ([0,1],KGB element #7), Levi = compact connected real group with Lie algebra 'su(3).u(1)' ([0,1],KGB element #17), Levi = connected quasisplit real group with Lie algebra 'su(2,1).u(1)' ([0,1],KGB element #18), Levi = connected quasisplit real group with Lie algebra 'su(2,1).u(1)' ([2],KGB element #12), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([2],KGB element #13), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([2],KGB element #14), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([2],KGB element #15), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).u(1).u(1)' ([0,2],KGB element #14), Levi = connected real group with Lie algebra 'su(2).sl(2,R).u(1)' ([0,2],KGB element #15), Levi = connected real group with Lie algebra 'su(2).sl(2,R).u(1)' ([0,2],KGB element #16), Levi = connected quasisplit real group with Lie algebra 'sl(2,R).sl(2,R).u(1)' ([1,2],KGB element #26), Levi = connected quasisplit real group with Lie algebra 'sp(4,R).u(1)' ([1,2],KGB element #27), Levi = connected quasisplit real group with Lie algebra 'sp(4,R).u(1)' ([0,1,2],KGB element #44), Levi = connected split real group with Lie algebra 'sp(6,R)' {Here I'll just mention that the first eight terms are Borel subalgebras, the eight closed KGB orbits (corresponding to the eight kinds of discrete series rep for Sp(6,R)} {I want to look harder at the parabolic in GL(6,C) that Timothy discussed, with Levi GL(2,C)xGL(1,C)xGL(3,C). It's not so hard to decide that it's the one with simple roots [0,3,4,5,8,9] (so KGB element 80). I'd like to pick that out of my long list. So I should have typed instead} atlas> void:for q@i in Q do prints("i = ",i,", q = ",q,", Levi = ",Levi(q)) od {I won't rewrite all the output, but part is} i = 24, q = ([3,4,8,9],KGB element #21), Levi = connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C).gl(1,C).gl(1,C)' i = 25, q = ([0,3,4,5,8,9],KGB element #80), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C)' i = 26, q = ([1,3,4,6,8,9],KGB element #64), Levi = connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C)' {So the parabolic I want is q=Q[25]:} atlas> set q = Q[25] Variable q: KGPElt (overriding previous instance, which had type KGPElt) atlas> set L=Levi(q) Variable L: RealForm (overriding previous instance, which had type RealForm) atlas> L Value: connected quasisplit real group with Lie algebra 'sl(2,C).sl(3,C).gl(1,C).gl(1,C).gl(1,C) {We're interested in cohomological induction from L, which atlas computes as (for example)} atlas> theta_induce_irreducible(L.trivial,G) Value: 1*parameter(x=80,lambda=[5,3,1,-1,-3,-5,5,3,1,-1,-3,-5]/2,nu=[1,-1,0,2,0,-2,1,-1,0,2,0,-2]/2) [60] {On L, the lowest L\cap K=U(2) x U(1) x U(3)-type is the trivial. In the induced parameter, the highest weight of the lowest K-type is this SHIFTED by 2rho(u\cap p) = (since this is a complex group) rho(u). Atlas tells us atlas> rho_u(q) Value: [ 4, 4, 1, -3, -3, -3, 4, 4, 1, -3, -3, -3 ]/2 {To put this in human coordinates, add the first 6 coordinates to the last six, getting LKT [4,4,1,-3,-3,-3]. We can check, first getting the parameter on G, then restricting the irreducible to K:} atlas> set r = first_param (theta_induce_irreducible(L.trivial,G)) Variable r: Param (overriding previous instance, which had type Param) atlas> r Value: final parameter(x=80,lambda=[5,3,1,-1,-3,-5,5,3,1,-1,-3,-5]/2,nu=[1,-1,0,2,0,-2,1,-1,0,2,0,-2]/2) atlas> print_branch_irr(r,70) m x lambda 1 0 [ 5, 5, 1, -3, -3, -3, 3, 3, 1, -3, -3, -3 ]/2 1 0 [ 5, 5, 1, -3, -3, -3, 5, 3, -1, -3, -3, -3 ]/2 1 0 [ 5, 5, 3, -3, -3, -3, 3, 3, 1, -3, -3, -5 ]/2 1 0 [ 7, 5, -1, -3, -3, -3, 5, 3, -1, -3, -3, -3 ]/2 2 0 [ 5, 5, 1, -3, -3, -3, 5, 3, 1, -3, -3, -5 ]/2 {We get the highest weights of these five U(6) reps by adding the first six coordinates to the last 6: [4,4,1,-3,-3,-3] [5,4,0,-3,-3,-3] [4,4,2,-3,-3,-4] [6,4,-1,-3,-3,-3] [5,4,1,-3,-3,-4]} {Timothy wanted to get LKT mu = (3,3,1,-1,-1,-1). To do that, you must start with mu_L = mu - 2rho(u\cap p) = mu - rho(u) = [3,3,1,-1,-1,-1]-[4,4,1,-3,-3,-3] = [-1,-1,0,2,2,2] To get an L parameter with this LKT, we must add mu_L to the first six coords of lambda for L.trivial:} atlas> L.trivial Value: final parameter(x=11,lambda=[1,-1,0,2,0,-2,1,-1,0,2,0,-2]/2,nu=[1,-1,0,2,0,-2,1,-1,0,2,0,-2]/2) atlas> set rL= parameter(KGB(L,11),[-1,-3,0,6,4,2, 1,-1,0,2,0,-2]/2, [0,0,0,0,0,0,0,0,0,0,0,0]) Variable rL: Param {Now cohomologically induce to G, and look at the K-types:} atlas> set r=first_param (theta_induce_irreducible(rL,G)) Variable r: Param (overriding previous instance, which had type Param) atlas> r Value: final parameter(x=0,lambda=[3,3,1,-1,-1,-1,3,3,1,-1,-1,-1]/2,nu=[0,0,0,0,0,0,0,0,0,0,0,0]/1) atlas> print_branch_irr(r,38) m x lambda 1 0 [ 3, 3, 1, -1, -1, -1, 3, 3, 1, -1, -1, -1 ]/2 1 0 [ 5, 3, 1, -1, -1, -1, 3, 1, 1, -1, -1, -1 ]/2 1 0 [ 5, 1, 1, -1, -1, -1, 5, 1, 1, -1, -1, -1 ]/2 2 0 [ 5, 3, 1, -1, -1, -1, 3, 3, -1, -1, -1, -1 ]/2 2 0 [ 3, 3, 1, 1, -1, -1, 3, 3, 1, -1, -1, -3 ]/2 {These five U(6) types have highest weights (1st 6 + last 6) [3,3,1,-1,-1,-1] {LKT; bottom layer} [4,2,1,-1,-1,-1] {bottom layer} [5,2,1,-1,-1,-1] {not bottom layer} [4,3,0,-1,-1,-1] {bottom layer} [3,3,1,0,-1,-2] {bottom layer} ***************************************** Now let's look at some hermitian forms. atlas> set rL= parameter(KGB(L,11),[-1,-3,0,6,4,2, 1,-1,0,2,0,-2]/2, [3/2,-3/2,0,3/2,0,-3/2, 0,0,0,0,0,0]) Variable rL: Param (overriding previous instance, which had type Param) {This is an L rep with LKT mu_L, but now given a nonzero nu to make it nonunitary. After loading bottom.at, we can do} atlas> print_sig_irr_long (rL,5) sig x lambda hw dim height s 0 [ -1, -1, 0, 2, 2, 2, -1, -1, 0, 2, 2, 2 ]/2 [ -1, 0, 0, 0, 1, 2, 0, -1, 0, 2, 1, 0 ] 1 0 1 0 [ 1, -1, 0, 2, 2, 2, -1, -3, 0, 2, 2, 2 ]/2 [ 0, 0, 0, 0, 1, 2, 0, -2, 0, 2, 1, 0 ] 3 2 1 0 [ 1, -3, 0, 2, 2, 2, 1, -3, 0, 2, 2, 2 ]/2 [ 0, -1, 0, 0, 1, 2, 1, -2, 0, 2, 1, 0 ] 5 4 1+s 0 [ -1, -1, 0, 4, 2, 2, -1, -1, 0, 2, 2, 0 ]/2 [ -1, 0, 0, 1, 1, 2, 0, -1, 0, 2, 1, -1 ] 8 4 {These four L\cap K types have highest weights (add first six coords to last six; I keep the signature attached) s.[-1,-1,0,2,2,2] 1.[0,-2,0,2,2,2] 1.[1,-3,0,2,2,2] (1+s)[-1,-1,0,3,2,1] {Adding 2rho(u\cap p)=(4,4,1,-3,-3,-3) gives some predictions about the signature of the cohomologically induced rep:} {s.[3,3,1,-1,-1,-1] 1.[4,2,1,-1,-1,-1] 1.[5,1,1,-1,-1,-1] (1+s).[3,3,1,0,-1,-2]} {So let's cohomologically induce and look at the signature} atlas> set r = first_param(theta_induce_irreducible(rL,G)) Variable r: Param (overriding previous instance, which had type Param) atlas> r Value: final parameter(x=80,lambda=[5,3,1,1,-1,-3,3,1,1,1,-1,-3]/2,nu=[3,-3,0,3,0,-3,3,-3,0,3,0,-3]/4) atlas> print_sig_irr_long (r,40) sig x lambda hw dim height s 0 [ 3, 3, 1, -1, -1, -1, 3, 3, 1, -1, -1, -1 ]/2 [ -1, 0, 0, 0, 1, 2, 4, 3, 1, -1, -2, -3 ] 7056 34 1 0 [ 5, 3, 1, -1, -1, -1, 3, 1, 1, -1, -1, -1 ]/2 [ 0, 0, 0, 0, 1, 2, 4, 2, 1, -1, -2, -3 ] 15750 36 1 0 [ 5, 1, 1, -1, -1, -1, 5, 1, 1, -1, -1, -1 ]/2 [ 0, -1, 0, 0, 1, 2, 5, 2, 1, -1, -2, -3 ] 12375 38 1+s 0 [ 5, 3, 1, -1, -1, -1, 3, 3, -1, -1, -1, -1 ]/2 [ 0, 0, 0, 0, 1, 2, 4, 3, 0, -1, -2, -3 ] 16128 38 1+s 0 [ 3, 3, 1, 1, -1, -1, 3, 3, 1, -1, -1, -3 ]/2 [ -1, 0, 0, 1, 1, 2, 4, 3, 1, -1, -2, -4 ] 40320 38 highest weights are s.[3,3,1,-1,-1,-1] 1.[4,2,1,-1,-1,-1] 1.[5,1,1,-1,-1,-1] (1+s).[4,3,0,-1,-1,-1] (not bottom layer) (1+s).[3,3,1,0,-1,-2] PERFECTLY matching the signature on the (L\cap K)-types above.