# 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 $$P_S$$ of type S. We define a set KGP(S) (a quotient of KGB) such that (roughly) KGP(S) <-> $$K\backslash G/P_S$$ .

More precisely, for any $$x\in$$ KGB and $$p(\xi)=x$$ , KGP(S) is canonically in bijection with $$K_{\xi}\backslash G/P_S$$ ; i.e., $$K_{\xi}$$ conjugacy classes of parabolics of type S.

K orbits on $$G/P_S$$ , 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 <-> $$K\backslash G/P_S$$ 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 $$G/P_S$$ , 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 $$K_x$$ -conjugacy class of the standard parabolic $$P_S$$ .

The data type is KGPElt or Parabolic (synonyms).

Given (S,x), write $$[x_1,...,x_n]$$ for the S-equivalence class of $$x\subset$$ KGB.

The last element $$x_n$$ is maximal, and is uniquely determined. This orbit of K on $$G/P_S$$ is closed <=> $$x_1$$ 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 $$\theta$$ -stable Levi factor; $$\theta$$ -stable parabolics; real parabolics.

This script imports the following .at files: