:math:`K\backslash G/B` for other Cartan subgroups =================================================== Let us look at :math:`G=Sp(4,R)`. The :math:`K\backslash G/B` elements are:: atlas> G Value: connected split real group with Lie algebra 'sp(4,R)' 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 atlas> Recall that the first four form the fundamental fiber that go to the Cartan subgroup ``0``, the compact one. Elements ``5`` through ``8`` are attached to Cartan subgroup number ``2``, etc. The last collumn tells us that the fiber attached to the involution ``2^e`` consists of elements 5 and 6 and the fiber corresponding to the element ``1x2^e`` are elements ``7`` and ``8``. Here ``2^e`` is just Cayley transform by :math:`{\alpha}_2`, whereas ``1x2^e`` corresponds to conjugation by :math:`{\alpha}_1` composed with the Cayley transform by :math:`{\alpha}_2`. Let us recall which Cartan subgroups and Weyl groups correspond to each fiber:: atlas> set H=Cartan_class(G,0) Variable H: CartanClass (overriding previous instance, which had type string (constant)) atlas> print_Cartan_info (H) compact: 2, complex: 0, split: 0 canonical twisted involution: e twisted involution orbit size: 1; fiber size: 4; strong inv: 4 imaginary root system: C2 real root system: empty complex factor: empty atlas> As we know this is the Compact Cartan subgroup associated to the distinguished fiber:: atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is a Weyl group of type A1 W^R is trivial generators for W_ic: 2,1,2 atlas> This is a Weyl group of type ``A1``. So, the number of ``KGB`` orbits for this Cartan is ``8/4=2`` Now for one of the intermediate Cartan subgroups we have:: atlas> H:=Cartan_class(G,1) Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form atlas> atlas> print_Cartan_info (H) compact: 0, complex: 1, split: 0 canonical twisted involution: 2,1,2 twisted involution orbit size: 2; fiber size: 1; strong inv: 2 imaginary root system: A1 real root system: A1 complex factor: empty atlas> atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is an elementary abelian 2-group of rank 1 W_ic is trivial W^R is a Weyl group of type A1 generators for A 1 generators for W^R: 2,1,2 atlas> This is a copy of :math:`{\mathbb C}^\times` with Weyl group of order ``4``. So the number of ``KGB`` orbits is ``8/4=2`` Let us see what the :math:`W`-orbit of one element is, say:: atlas> set x=KGB(G,4) Variable x: KGBElt atlas> void: for w in W do prints(cross(w,x)) od KGB element #4 KGB element #4 KGB element #9 KGB element #9 KGB element #9 KGB element #9 KGB element #4 KGB element #4 atlas> Starting with element ``4`` the order of its stabilizer has four elements. And if we list all the elements of :math:`W`:: atlas> void: for (,w) in W do prints(w) od [] [0] [1] [1,0] [0,1] [0,1,0] [1,0,1] [1,0,1,0] atlas> We see that the elements ``[], [0], [1,0,1], and [1,0,1,0]`` all stabilize element ``4``. So the order of the stabilizer is ``4``. Similarly, for element ``9``. Now for the next Cartan subgroup:: atlas> H:=Cartan_class(G,2) Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms atlas> atlas> print_Cartan_info (H) compact: 1, complex: 0, split: 1 canonical twisted involution: 1,2,1 twisted involution orbit size: 2; fiber size: 2; strong inv: 4 imaginary root system: A1 real root system: A1 complex factor: empty atlas> This subgroup has order four. And its real Weyl group has order ``2``:: atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type A1 generators for W^R: 1,2,1 atlas> Then the number of ``KGB`` orbits is ``8/2=4`` and we can verify also that each stabilizer is order 2:: atlas> x:=KGB(G,5) Variable x: KGBElt atlas> atlas> void: for w in W do prints(cross(w,x)) od KGB element #5 KGB element #7 KGB element #5 KGB element #8 KGB element #7 KGB element #6 KGB element #8 KGB element #6 atlas> atlas> void: for (,w) in W do prints(w) od [] [0] [1] [1,0] [0,1] [0,1,0] [1,0,1] [1,0,1,0] atlas> Now for completeness, let us look at the split Cartan subgroup:: atlas> H:=Cartan_class(G,3) Value: Cartan class #3, occurring for 1 real form and for 3 dual real forms atlas> atlas> print_Cartan_info (H) compact: 0, complex: 0, split: 2 canonical twisted involution: 2,1,2,1 twisted involution orbit size: 1; fiber size: 1; strong inv: 1 imaginary root system: empty real root system: C2 complex factor: empty atlas> atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type B2 generators for W^R: 1 2 A Cartan Subgroup isomorphic to :math:`{\mathbb C}^\times \times {\mathbb C}^\times` and Weylgroup of type ``B2``. So the number of ``KGB`` orbits is ``8/8=1``:: atlas> set x=KGB(G,10) Variable x: KGBElt (overriding previous instance, which had type KGBElt) atlas> x:=KGB(G,10) Value: KGB element #10 atlas> atlas> void: for w in W do prints(cross(w,x)) od KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 atlas> This concludes this deiscussion on :math:`K\backslash G/B` orbits. In the next chapter we will discuss the representations associated to the intermediate Cartan subgroups. The parameter includes a discrete series of a Levi factor of a parabolic subgroup. So, to some extent it reduces to the case of discrete series. The idea is to look at the cuspidal data of an arbitrary parameter which gives a Levi factor :math:`M` and then applying what we learned about discrete series of M.