# Reference request: Principal series are equal in the Grothendieck group

In the usual setup, consider the category of Harish-Chandra $(\mathfrak{g},K)$-modules with given central character (if the central character is regular, this is equivalent to $K$-equivariant $D$-modules on the flag variety). Assume that everything is split.

I was said that the principal series modules in this category, all are equal in the Grothendieck group (and generically irreducible, so just isomorphic).

I would like a reference for this fact, but which is algebraic; I don't want it to use intertwining operators - one can make the problem meaningful over an algebraically closed field of characteristic $0$, and I want a proof that in principle will work in this purely algebraic context.

Thank you

I think the fact that you are referring to is the following, which holds in some generality. Suppose $M$ is a Levi factor in $G$ and $\pi$ is an irreducible representation of $M$. Suppose $P=MN$ is a parabolic subgroup containing $M$. Then the image of $Ind_{MN}^G(\pi\otimes 1)$ in the Grothendieck group is independent of the choice of $N$. The reason is that the character (which determines the image in the Grothendieck group) is given by the induced character formula, which only depends on $M$ (for example see Hecht-Schmid, Characters, asymptotics and n-homology of Harish-Chandra modules, MR0716371, Theorem 5.7). This holds over other fields as well.
The result you stated is not true. For example in $SL(2,\mathbb R)$ or $PGL(2,\mathbb R)$, $M=\mathbb R^\times$, $Ind_{MN}^G(\pi\otimes 1)$ and $Ind_{MN}^G((\pi\otimes sgn)\otimes 1)$ do not have the same image in the Grothendieck group, one is spherical and the other is not. For a good introduction see Chapter 1 of Representations of real reductive Lie groups, by David Vogan MR0632407.