extended.at

references:
Parameters for Twisted Representations (Adams/Vogan)
Extended Parameters in Atlas (Adams)
Computing Twisted KLV Polynomials (Adams)
More on the 2i12, 2Ci, 1i cases (Adams)
Examples of extended parameters/braid relations (Adams)
all found at www.liegroups.org/papers/extended (no public link)

extended parameter:
  the type abbreviation ExtParam is defined in extended_misc.at

  ExtParam:(ic,delta,gamma,lambda,theta,g,l,omega,tau,t), is:

(InnerClass ic, {inner class}
 mat delta,     {twist}
 ratvec gamma,  {infinitesimal character, in X^*(H)\otimes Q}
 vec lambda,    {in X^*}
 mat theta,     {^theta_x = -theta_y}
 ratvec g,      {infinitesimal co-character, in X_*(H)\otimes Q}
 vec l,         {in X_*}
 mat omega,     {-^theta = ^theta_y = -theta_x}
 vec tau,       {in X^*}
 vec t          {in X_*}
)

 X^* = X^*(H) = X_*(H^v)
 X_* = X_*(H) = X^*(H^v)
 xi_0: distinguished involution of the inner class
 xi_0^v: distinguished involution of dual inner class
         (=-w_0\xi_0^t on level of root data)
 e(v)=exp(2i\pi v) (in \h=Lie(H) or \h^v=\h^*)

Reminders about what acts where
 delta, xi_0, theta, ^omega act on X^*,
                            which contains gamma,lambda,tau,rho,alpha
 ^delta, ^xi_0, omega, ^theta act on X_*,
                            which contains g,l,t,rho_check,alpha_check

 Warning: lambda is in X^*, NOT the same as lambda in (x,lambda,nu) in Fokko,
 (which is in X^*+\rho) rather it is the weight vector Fokko calls lambda-rho

 (g,l,theta) -> xi:
 g = infinitesimal co-character, in X_*\otimes Q, delta-fixed
 precondition: (1-^theta)g=(1-^theta)(l+rho^v)
 l in X_*
   -> torus element e((g-l)/2) in H
 theta: involution of X^*(H), theta=^theta_x, acting on X^*
   -> w such that theta=w*\xi_0
 (g,l,theta)
   -> xi=e((g-l)/2)\sigma_w\xi_0
   -> x=image of xi in KGB space X of G

 (gamma,lambda,omega) -> eta
 entirely dual to construction of xi
 precondition: gamma*(1-omega)=(lambda+rho)*(1-omega)
 also: omega=-^theta
   -> e((gamma-lambda)/2)\in H^v
   -> eta=e((gamma-lambda)/2)\sigma_{ww_0}\xi^v_0
   -> y=image of eta in KGB space X^v of G^v

 t: solution in X_* to (1-^delta)l=(1+^theta)t
   -> h=e(t/2)\in H(w)
   -> h\delta in the extended Cartan, i.e. \xi(h\delta)\xi^{-1}=h\delta
tau: solution to (1-delta)lambda=(1+^omega)tau
   -> e(tau/2)\in H^v(2)
   -> element of extended Cartan on dual side (don't give this element a name)

main construction of extended parameters
E: (ic,delta,gamma,x,g,y) -> (ic,delta,gamma,lambda,theta,g,l,omega,tau,t)
(ic inner class,
 delta is some distinguished involution of the based root datum,
 gamma is a logarithm of square(y)=square(real_form(y)),
 lambda = gamma-(torus_factor(y)+rho(ic))
 theta = involution(x) (!!!)
 g is a logarithm of square(x)=square(real_form(x)),
 l = g-(torus_factor(x)+rho_check(ic))
 omega = involution(y) = -^theta
 tau: solution of (1-delta)*lambda = (1+omega)*tau
 t:   solution of l*(1-delta)      = t*(1+theta)