Cohomological Parabolic Induction ==================================== Defining :math:`\theta`-stable parabolic subalgebras for a given real group :math:`G` is a little trickier than defining real parabolic subgroups because there can be more that one :math:`K` conjugacy class of such structures attached to a given (type of) complex parabolic subgroup. Defining a :math:`\theta`-Stable Parabolic Subalgebra --------------------------------------------------------------- In the section on Real Parabolic Induction we discussed two different ways of defining a real parabolic subgroup: by giving the complex parabolic subgroup type (i.e., a subset of the simple roots), and by giving a weight :math:`\lambda`. Let's focus here on the second technique, which may seem more natural and familiar. Since a :math:`\theta`-stable parabolic subalgebra may be thought of as :math:`\mathfrak q(\lambda)=\mathfrak l(\lambda)+\mathfrak u(\lambda)` for a weight :math:`\lambda` in the compact part of the fundamental Cartan subalgebra, the first step is to choose a ``KGB`` element ``x`` attached to the fundamental Cartan. It also needs to be in the distinguished fiber (which is automatic in the equal rank case). In order to obtain a :math:`\theta`-stable parabolic subalgebra, you need to choose a weight :math:`\lambda` that is fixed by the involution :math:`\theta_x`. In the equal rank case, this is, of course, also automatic for this choice of ``x``. Let's look at an example: Let :math:`G=Sp(4,\mathbb R)` once again, and let's choose ``x`` to be ``KGB`` element 2. This is the element attached to the holomorphic (or antiholomorphic) discrete series of :math:`G`, so that the simple root #0, :math:`e_1-e_2`, is compact. Then the unique :math:`\theta`-stable parabolic subalgebra with Levi factor :math:`U(1,1)` is the one attached to, for example, the weight :math:`\lambda=(1,-1)`:: atlas> set G=Sp(4,R) Variable G: RealForm atlas> set x=KGB(G,2) Variable x: KGBElt atlas> set P=parabolic([1,-1],x) Parabolic is theta-stable. Variable P: ([int],KGBElt) atlas> P Value: ([0],KGB element #0) atlas> set L=Levi(P) Variable L: RealForm atlas> L Value: connected quasisplit real group with Lie algebra 'sl(2,R).u(1)' Notice that ``atlas`` prints a message that the parabolic is indeed :math:`\theta`-stable. Notice also that ``atlas`` conjugates the weight :math:`\lambda` and ``x`` simultaneously to make the weight dominant before calculating ``P``; we could have defined this parabolic using the dominant weight :math:`(1,1)` and the ``KGB`` element #0:: atlas> set Q=parabolic([1,1],KGB(G,0)) Parabolic is theta-stable. Variable Q: ([int],KGBElt) atlas> P=Q Value: true If we use the weight :math:`(1,1)` with our original ``x`` to construct a parabolic, we get one with compact Levi factor:: atlas> set P2=parabolic([1,1],x) Parabolic is theta-stable. Variable P2: ([int],KGBElt) atlas> P2 Value: ([0],KGB element #2) atlas> Levi(P2) Value: compact connected real group with Lie algebra 'su(2).u(1)' and :math:`\lambda=(-1,-1)` would give the opposite parabolic. You can get a list of all :math:`\theta`-stable parabolics for :math:`G` using the command ``theta_stable_parabolics``:: atlas> set tsp=theta_stable_parabolics(G) Variable tsp: [([int],KGBElt)] atlas> void: for P@i in tsp do prints(i," ",P) od 0 ([],KGB element #0) 1 ([],KGB element #1) 2 ([],KGB element #2) 3 ([],KGB element #3) 4 ([0],KGB element #2) 5 ([0],KGB element #3) 6 ([0],KGB element #4) 7 ([1],KGB element #5) 8 ([1],KGB element #6) 9 ([0,1],KGB element #10) Notice that there are four :math:`\theta`-stable parabolics corresponding to the empty set of simple roots, i.e., to the Borel, one for each discrete series. You can also get a list of the parabolics of a certain type only:: atlas> set tsp_0=theta_stable_parabolics_type(G,[0]) Variable tsp_0: [([int],KGBElt)] atlas> void: for P@i in tsp_0 do prints(i," ",P) od 0 ([0],KGB element #2) 1 ([0],KGB element #3) 2 ([0],KGB element #4) In this list, the ``KGB`` element given is the maximal element in the equivalence class on ``KGB(G)`` defined by the set of simple roots ``[0]``; our parabolic ``P`` defined earlier is #2 in this list:: atlas> P=tsp_0[2] Value: true atlas> equivalence_class_of(P) Value: [KGB element #0,KGB element #1,KGB element #4] The equivalence class of ``P`` is the set of ``KGB`` elements obtained from ``x`` by cross actions and Cayley transforms through simple root #0 (in general, through any of the simple roots listed). Each of these ``KGB`` elements will define the same parabolic. (See the summary for the script file ``parabolics.at`` on the ``atlas Library`` page for more details.) Theta-Stable Induction -------------------------- The commands for theta-stable, or cohomological parabolic, induction work in a similar fashion to the corresponding commands in the real parabolic case. We can theta-induce standard modules (``theta_induce_standard``) or irreducibles, and the answers need to be understood in those terms. Let's focus here on the second type of induction: inducing an irreducible representation of :math:`L` to get the composition series of the resulting representation of :math:`G`. Let's stay with :math:`G=Sp(4,\mathbb R)`, and :math:`P` the :math:`\theta`-stable parabolic with Levi factor :math:`L=U(1,1)`. First take the trivial representation of :math:`L`:: atlas> t:=trivial(L) Value: final parameter (x=2,lambda=[1,-1]/2,nu=[1,-1]/2) atlas> set p=theta_induce_irreducible(t,G) Variable p: ParamPol atlas> p Value: 1*final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2)Variable p: ParamPol atlas> goodness(t,G) Good This is of course an :math:`A_{\mathfrak q}(\lambda)` module in the good range, and therefore, as expected, irreducible. Theta-induction takes representations of infinitesimal character :math:`\gamma` to representations of infinitesimal character :math:`\gamma+\rho(\mathfrak u)`:: atlas> infinitesimal_character (t) Value: [ 1, -1 ]/2 atlas> infinitesimal_character (p) Value: [ 2, 1 ]/1 atlas> rho_u(P) Value: [ 3, 3 ]/2 The output is of type ``ParamPol``. Next, let's induce the one-dimensional :math:`det^{-1}` of :math:`L`:: atlas> set p1=parameter(L,2,[-1,-3]/2,[-1,-3]/2) Variable p1: Param atlas> goodness(p1,G) Weakly good atlas> theta_induce_irreducible(p1,G) Value: 1*final parameter (x=4,lambda=[1,0]/1,nu=[1,-1]/2) Of course, we can choose any irreducible representation on :math:`L` at all. For a non-unitary example, here is a finite dimensional representation:: atlas> set p=parameter(L,2,[1,-5]/2,[1,-5]/2) Variable p: Param (overriding previous instance, which had type Param) atlas> dimension(p) Value: 3 atlas> goodness (p,G) None atlas> theta_induce_irreducible(p,G) Value: 1*final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2) 1*final parameter (x=9,lambda=[2,1]/1,nu=[3,3]/2) This parameter is outside the fair range, and the induced representation is reducible. The calculation involves wall crossings and coherent continuation action. (See the summary for the script file ``induction.at`` on the ``atlas Library`` page for more details.) Notice that the induction functions will accept only parameters on Levi factors of the right kind of parabolics; entering a parameter on a Levi subgroup that does not come from a real parabolic subgroup will result in an error message:: atlas> real_induce_irreducible(t,G) Runtime error: L is not Levi of real parabolic (in call at basic.at:8:57-71 of error@string, built-in) [b=false, message="L is not Levi of real parabolic"] ...(output truncated) Similarly, the function ``theta_induce_irreducible`` requires the input of a parameter on a Levi subgroup coming from a :math:`\theta`-stable parabolic subalgebra. Indeed, a Levi subgroup of :math:`G` uniquely defines the parabolic it came from. The command ``make_parabolic(L,G)`` reverses the function ``Levi(P)``.