# induction.at¶

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

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