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

Main functions:

\(A_q(\lambda)\) 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,wlambda,wlambda_q), where wlambda_q is weakly dominant.

Good/Fair conditions:

Condition on the roots of \(\mathfrak u\) :
For theta_induce(pi_L,G), gamma_L -> gamma_G=gamma_L+rho_u.
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
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.

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