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.
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