.. _parabolics.at: parabolics.at ===================================== See "Parabolic Subgroups and Induction", in dropbox, ultimately on atlas web site. Fix a subset S of the simple roots, defining the complex standard parabolic :math:`P_S` of type S. We define a set KGP(S) (a quotient of KGB) such that (roughly) KGP(S) <-> :math:`K\backslash G/P_S` . More precisely, for any :math:`x\in` KGB and :math:`p(\xi)=x` , KGP(S) is canonically in bijection with :math:`K_{\xi}\backslash G/P_S` ; i.e., :math:`K_{\xi}` conjugacy classes of parabolics of type S. K orbits on :math:`G/P_S` , equivalently K-conjugacy classes of parabolics of type S: Given a RealForm and a subset S of the simple roots, S -> partial order on KGB, generated by ascents in S -> equivalence relation generated by this KGB/equivalence <-> :math:`K\backslash G/P_S` Define KGP to be KGB modulo this equivalence. Data: ([int],KGBElt)=(S,x) where S lists the indices of a subset of the simple roots of root_datum(x) Equivalence: (S,x)=(S',y) if these correspond to the same K orbit on :math:`G/P_S` , which means: real_form(x)=real_form(y), S=S' (i.e. same complex parabolic), and x=y in the equivalence defined by S. In particular, given (S,x), taking x itself for the strong real form, (S,x) goes to the :math:`K_x` -conjugacy class of the standard parabolic :math:`P_S` . The data type is KGPElt or Parabolic (synonyms). Given (S,x), write :math:`[x_1,...,x_n]` for the S-equivalence class of :math:`x\subset` KGB. The last element :math:`x_n` is maximal, and is uniquely determined. This orbit of K on :math:`G/P_S` is closed <=> :math:`x_1` is closed in KGB. ComplexParabolic data type: (RootDatum rd,[int] S) <-> G-conjugacy class of standard parabolic with Levi factor given by subset S of simple roots More topics addressed in this file: parabolics with :math:`\theta` -stable Levi factor; :math:`\theta` -stable parabolics; real parabolics. **This script imports the following .at files:** | :ref:`misc.at` | :ref:`sort.at` | :ref:`Weylgroup.at` | :ref:`group_operations.at` | .. toctree:: :maxdepth: 1 parabolics_ref parabolics_index