# Do $W$-conjugate characters induce the same modules under parabolic induction?

Suppose $G$ is an algebraic group, with maximal rational torus $T$, corresponding character group $X(T)$, and Weyl group $W$. Then $W$ acts on $X(T)$ via $(w\cdot \chi)(t)=\chi(t^w)$, where $t^w=w^{-1}tw$.

If $\chi_1$ and $\chi_2$ are irreducible and conjugate under $W$ in $X(T)$, and $R_T^G$ denotes the parabolic induction functor, is it true that $R_T^G(\chi_1)$ and $R_T^G(\chi_2)$ are isomorphic as modules? (I'm blurring the distinction between characters and representations.)

• It is too early in the day for me to work this out myself: Is this the same action as the one we "usually" consider on the characters, i.e. the one that comes from the action on the root system? Also, I am not particularly familiar with how parabolic induction works, but should you not start with a representation of a parabolic subgroup for that? – Tobias Kildetoft Apr 22 '17 at 7:11
• @TobiasKildetoft Yes, I think the action is the usual one we consider on characters. Parabolic induction is independent of the parabolic subgroup you choose, and is the same as inflating a representation of the Levi to a representation of a parabolic containing it, and then inducing to $G$ in the usual way. So I think we can start with the representation of $T$, and inflate and induct through any parabolic containing it as its Levi. – Denise Gi Apr 22 '17 at 7:28
• Ok, so we are really just doing induction from the Borel? But which induction? Is it the left- or the right adjoint of restriction? – Tobias Kildetoft Apr 22 '17 at 7:40
• @TobiasKildetoft Yes, inducting through the Borel. Left adjoint is fine. I'm sorry, I'm a little confused by your question since parabolic induction and restriction are biadjoint. – Denise Gi Apr 22 '17 at 8:32
• Restriction does not admit a biadjoint in the category of algebraic representations, so this suggests that this is not the category in which you work, which means it is something I don't know anything about. – Tobias Kildetoft Apr 22 '17 at 8:46