Reference request: Principal series are equal in the Grothendieck group

Question

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

Answer 1

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.

By the way what you refer to as "central character" is usually called the "infinitesimal character", i.e. character of the center of the universal enveloping algebra.