Parabolic induction from real and \(\theta\) -stable parabolics; cuspidal and \(\theta\) -stable data of a parameter, and some functions related to \(\theta\) -stable parabolics.

Parabolic induction:

If L is a \(\theta\) -stable Levi subgroup of G, then KGB for L embeds into KGB for G.
For parabolic induction, a parameter p_L for the Levi L is assigned a parameter p_G for G:

p_L=(x_L,lambda,nu) -> p_G=(embed_KGB(x_L,G),lambda + appropriate rho-shift,nu).

For real parabolic induction, the rho-shift is: \(\rho_r(G)-\rho_r(L)+(1-\theta)(\rho_S(G)-\rho_S(L))\) .
(Here \(\rho_S\) is a certain half sum of complex roots.)
The Levi L must be the Levi factor of a REAL parabolic subgroup.

For \(\theta\) -stable (cohomological parabolic) induction, the rho-shift is:
\(\rho_i(G)-\rho_i(L)+\rho_{complex}(G)-\rho_{complex}(L) =\rho(G)-\rho_r(G)-\rho(L)+\rho_r(L)\) .
Since \(\mathfrak q\) is \(\theta\) -stable, \(\rho_r(G)-\rho_r(L)=0\) , so the shift is \(\rho(G)-\rho(L)=\rho(\mathfrak u)\) .
The group L must be the Levi factor of a THETA-STABLE parabolic subgroup of G.

Then \(\operatorname{Ind}_P^G I(p_L)=I(p_G)\) .

In the \(\theta\) -stable case, the shifted parameter p_G may be non-standard and needs to be standardized:
If p=(x,lambda,nu), and \(\langle \text{lambda},\alpha^{\vee}\rangle <0\) for some imaginary root \(\alpha\) (i.e. non-standard),
let i_root_system=imaginary roots for x(p). Find \(w\) so that \(w^{-1}\cdot\) lambda is dominant for
imaginary roots, set p_dom=parameter(x, \(w^{-1}\cdot\) lambda,nu) and return coherent continuation
action (wrt imaginary roots) of \(w\) on p_dom.

\(A_q(\lambda)\) 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
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;

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 weakly unipotent (e.g.,
one-dimensional), hence “weakly fair” is only defined if pi_L is one-dimensional.

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