K.at¶
Embedding of the complex group K defined by compact imaginary root system
\(T_{K_0}=(H^{\delta})_0 =\) Cartan subgroup of \(K_0\) \(T_{K_0}\subset T_K=H^{\delta} \subset H\) \(T_{K_0} is a torus in \) abelian (possibly disconnected) ` X^*(T_{K_0})= X^*/(X^*)^{-delta} :math:` ` X^*(T_K) = X^*/(1-delta)X^* twoheadrightarrow X^*(T_{K_0}) :math:` (restriction map is surjective) see W_K.at
K_0=identity component of K, with Cartan subgroup T_K0 B=basis of X_*(T_K0) (as columns) = cocharacter lattice for (K_0,T_K0) returns a matrix B with rank(K_0) columns, rank(ic) rows columns are a basis of the +1 left-eigenspace delta
This matrix ` B :math:` satisfies ` ^delta*B=B :math:` left multiplication by ` ^B :math:` is projection ` X^*(H) -> X^*(T_{K_0}) = X^*(H)/X^*(H)^{-delta} :math:` left multiplication by ` B :math:` is injection ` X_*(T_{K_0})-> X_*(H) :math:` [` ^delta*v=v :math:` for ` v$ in image]
This script imports the following .at files: