parameters.at

unnormalized_torus_factor

torus_factor, aka "normalized torus factor", is $\theta$-fixed,
$\xi=\exp_1(torus_factor/2)\exp_1(\rho^\vee/2)\sigma_w\xi_0$ where
$\exp_1(t)=\exp(2\pi i t)$ is the "base 1 exponential function"; note
"over 2" in those $\exp_1$ terms, Marc writes $\exp_{-1}(x)=\exp(\pi ix)$
instead, for which $\xi=\exp_{-1}(torus_factor+\rho^\vee)\sigma_w\xi_0$.
Jeff prefers $\xi=exp_1(unnormalized_torus_factor)\sigma_w\xi_0$
Therefore:
 |unnormalized_torus_factor=(torus_factor+rho^\vee)/2|
 |torus_factor=2*unnormalized_torus_factor-rho^\vee|
NOTE: result of |unnormalized_torus_factor| is not $\theta-fixed$
probably better not to use |unnormalized_torus_factor|

also defined in synthetic.at

square_unnormalized

compute xi^2 and test if it is central
this is formula when v= unnormalized_torus_factor
xi=exp(2\pi iv)\sigma_w\xi_0
xi^2=exp(2\pi i[(1+^theta)*v+(1-^theta)*rho^v/2]
    =v*(1+theta)+rho^v*(1-theta)/2
    =rho^v + (2*v-rho^v)*(1+theta)/2

square_normalized

square

for valid arguments one should have
square(ic,theta,tf)=square(real_form(ic,theta,tf))

also in this file at line 42, line 43; also defined in synthetic.at

square2

this should always be twice |square|

square 2 overloads

the following is the same as central_cocharacter@RealForm in Vogan-dual.at

also in this file at line 37; also defined in synthetic.at

square_is_central

tf=normalized torus_factor: xi=exp(\pi i(tf\rho^v))\sigma_w\xi_0
so square_is_central(ic,theta(x),torus_factor(x)) should always be true

also defined in synthetic.at, extended_misc.at

parity

parity condition

dual_inner_class

to compute y values over a block of parameters, precompute dual inner class

lambda 2 overloads

also defined in basic.at

nu 2 overloads

allow unused y

also defined in basic.at, extended.at

make_regular

sometimes gamma returned by y^2 is not regular, need to make it regular

also defined in stable.at

choose_g 3 overloads

choose infinitesimal character appropriate for dual parameter

tits

contragredient of a parameter

KGB_elt

also defined in basic.at, synthetic.at

inverse

also defined in basic.at, combinatorics.at, ratmat.at, tits.at, affine.at, cyclotomic.at, Gaussian_elim.at, cyclotomic_Gaussian_elim.at, lazy_lists.at

contragredient

contragredient: (x,lambda,nu) -> (w_0\times x,-w_0\lambda,-w_0\nu)
sketch: the contragredient is given by the Chevalley involution (of the real group)
which acts by -1 on lambda,nu, but keeps the same x (it doesn't change the "group")
-lambda,-nu are in the wrong chamber (for our fixed parabolic), so use w_0 (the long
element of the Weyl group)

parameter 2 overloads

gamma=infinitesimal character to combine with x, found from real_form(y)

final argument gamma would work

the next function used to be wrong, with choose_g(x,y) instead

computes gamma from real_form(y), x-dominant

also defined in basic.at, L_packet.at, K.at, extended_misc.at

is_subset

A is a subset of B

also defined in basic.at

is_proper_subset

A is a proper subset of B

also defined in misc.at

complement

complement of A in B

also defined in basic.at, misc.at

param_pol 2 overloads

turn [Param] into a ParamPol

turn [Param] into a ParamPol; error if #params=0

also defined in basic.at

remove_duplicates

replace [Param] with list with multiplicity one
uses properties of ParamPols

also defined in restricted_roots.at, nilpotent_orbits.at, disconnected.at

compare

compare two lists of parameters

compare_long