.. _induction.at: induction.at ===================================== Parabolic induction from real and :math:`\theta` -stable parabolics; cuspidal and :math:`\theta` -stable data of a parameter, and some functions related to :math:`\theta` -stable parabolics. Parabolic induction: _________________________ | If L is a :math:`\theta` -stable Levi subgroup of G, then KGB for L embeds into KGB for G. | For parabolic induction, a parameter p_L for the Levi L is assigned a parameter p_G for G: | | p_L=(x_L,lambda,nu) -> p_G=(embed_KGB(x_L,G),lambda + appropriate rho-shift,nu). | | For real parabolic induction, the rho-shift is: :math:`\rho_r(G)-\rho_r(L)+(1-\theta)(\rho_S(G)-\rho_S(L))` . | (Here :math:`\rho_S` is a certain half sum of complex roots.) | The Levi L must be the Levi factor of a REAL parabolic subgroup. | | For :math:`\theta` -stable (cohomological parabolic) induction, the rho-shift is: | :math:`\rho_i(G)-\rho_i(L)+\rho_{complex}(G)-\rho_{complex}(L) =\rho(G)-\rho_r(G)-\rho(L)+\rho_r(L)` . | Since :math:`\mathfrak q` is :math:`\theta` -stable, :math:`\rho_r(G)-\rho_r(L)=0` , so the shift is :math:`\rho(G)-\rho(L)=\rho(\mathfrak u)` . | The group L must be the Levi factor of a THETA-STABLE parabolic subgroup of G. | | Then :math:`\operatorname{Ind}_P^G I(p_L)=I(p_G)` . | | In the :math:`\theta` -stable case, the shifted parameter p_G may be non-standard and needs to be standardized: | If p=(x,lambda,nu), and :math:`\langle \text{lambda},\alpha^{\vee}\rangle <0` for some imaginary root :math:`\alpha` (i.e. non-standard), | let i_root_system=imaginary roots for x(p). Find :math:`w` so that :math:`w^{-1}\cdot` lambda is dominant for | imaginary roots, set p_dom=parameter(x, :math:`w^{-1}\cdot` lambda,nu) and return coherent continuation | action (wrt imaginary roots) of :math:`w` on p_dom. | :math:`A_q(\lambda)` construction: ______________________________________________ | Note: theta_induce_irreducible(pi_L,G) has infinitesimal character: | infinitesimal character(pi_L)+rho(u). | Aq(x,lambda,lambda_q) is defined as follows: | if lambda_q is weakly dominant set q=q(x,lambda_q), | apply derived functor to the one-dimensional lambda-rho(u) of L. | | REQUIRE: lambda-rho(u) must be in X^*. | | Aq(x,lambda,lambda_q) has infinitesimal character lambda+rho_L, | thus the one-dimensional with weight lambda has infinitesimal character | lambda+rho_L for L, and goes to a representation with | infinitesimal character lambda+rho_L for G; i.e., Aq takes infinitesimal | character gamma_L to SAME infinitesimal character for G. | If lambda_q is not weakly dominant, define | Aq(x,lambda,lambda_q)=Aq(wx,w\lambda,w\lambda_q), | where w\lambda_q is weakly dominant. | Good/Fair conditions: ________________________ | Condition on the roots of :math:`\mathfrak u` : | For theta_induce(pi_L,G), gamma_L -> gamma_G=gamma_L+rho_u. | Then: | GOOD: > 0; | WEAKLY GOOD: \ge 0; | | For Aq(x,lambda,lambda_q): gamma_L=lambda+rho_L; | gamma_L -> gamma_G=gamma_L = lambda+rho_L | Aq(x,lambda)=theta_induce(x,lambda-rho_u) | Then: | GOOD: > 0; | WEAKLY GOOD: >= 0; | FAIR: > 0; | WEAKLY FAIR: \ge 0. | | theta_induce(pi_L,G) = Euler-Poincare characteristic of the | cohomological induction functor. | | fair => vanishing outside middle degree => honest representation | weakly fair: same implication. | NB: >= 0 does NOT imply vanishing (in general) if pi_L is not weakly unipotent (e.g., | one-dimensional), hence "weakly fair" is only defined if pi_L is one-dimensional. | **This script imports the following .at files:** | :ref:`misc.at` | :ref:`parabolics.at` | :ref:`kl.at` | :ref:`coherent.at` | :ref:`synthetic.at` | :ref:`K.at` | :ref:`finite_dimensional.at` | .. toctree:: :maxdepth: 1 induction_ref induction_index