.. _thetastable.at:

thetastable.at
=====================================

Some functions related to theta-stable parabolics. The basic functions
are in parabolics.at and induction.at.

Main functions:

:math:`A_q(\lambda)`  construction:
============================

| Note: theta_induce_irreducible(pi_L,G) has infinitesimal character:
|infinitesimal character(pi_L)+rho(u).
|Aq(x,lambda,lambda_q) is defined as follows:
if lambda_q is weakly dominant set q=q(x,lambda_q),
apply derived functor to the one-dimensional lambda-rho(u) of L.
|
|  REQUIRE: lambda-rho(u) must be in X^*.
|
|Aq(x,lambda,lambda_q) has infinitesimal character lambda+rho_L,
thus the one-dimensional with weight lambda has infinitesimal character lambda+rho_L
for L, and goes to a representation with
infinitesimal character lambda+rho_L for G; i.e.,
Aq takes infinitesimal character gamma_L to SAME infinitesimal character
for G.
| If lambda_q is not weakly dominant, define
| Aq(x,lambda,lambda_q)=Aq(wx,w\lambda,w\lambda_q),
where w\lambda_q is weakly dominant.

Good/Fair conditions:
=====================

|  Condition on the roots of :math:`\mathfrak u` :
|  For theta_induce(pi_L,G), gamma_L -> gamma_G=gamma_L+rho_u.
|  Then:
|  GOOD:  <gamma_L+rho_u,alpha^vee> > 0;
|  WEAKLY GOOD:  <gamma_L+rho_u,alpha^vee> \ge 0;
|  FAIR: <gamma_L-rho_L+rho_u,alpha^vee> > 0.
|
| For  Aq(x,lambda,lambda_q): gamma_L=lambda+rho_L;
| gamma_L -> gamma_G=gamma_L = lambda+rho_L
| Aq(x,lambda)=theta_induce(x,lambda-rho_u)
| Then:
| GOOD: <lambda+rho_L,alpha^vee> > 0;
| WEAKLY GOOD: <lambda+rho_L,alpha^vee> >= 0;
| FAIR: <lambda,alpha^vee> > 0;
| WEAKLY FAIR: <lambda,alpha^vee> \ge 0.
|
| theta_induce(pi_L,G) = Euler-Poincare characteristic of the
cohomological induction functor.
|
| fair => vanishing outside middle degree => honest representation
| weakly fair: same implication.
| NB: <gamma_L-rho_L_rho_u,alpha^vee> >= 0 does NOT imply vanishing
(in general) if pi_L is not one-dimensional,
hence "weakly fair" is only defined if pi_L is one-dimensional.


**This script imports the following .at files:**

| :ref:`induction.at<induction.at>`
| :ref:`finite_dimensional.at<finite_dimensional.at>`
|


.. toctree::
   :maxdepth: 1

   thetastable_index
   thetastable_ref