Archimedean Langlands classification

Question

I am trying to clarify how the archimedean admissible dual is classified in the $G=GL(n, \mathbf{R})$ case.

Fix a semistandard Levi subgroup $M$ in $G$, $\delta$ a square-integrable representation of $M^1$, and $\nu \in \mathfrak{a}_\mathbf{C}^\star / W$, where $\mathfrak{a}_\mathbf{C}^\star$ is the dual of the complexification of the Cartan subalgebra of $\mathfrak{g}$ and $W$ the Weyl group. There is only one $\nu$ in its class modulo $W$ such that the unitary induction $Ind_P^G (\delta \otimes e^\nu)$ admits an irreducible subquotient, which we denote by $\pi(\delta, \nu)$. The Langlands classification states that those are admissible representations, and that the whole admissible dual arises in this way. So that the "continuous parameters" of the admissible dual are parametrized by $\mathfrak{a}_\mathbf{C}^\star/W$.

How is that $\nu$ related to the infinitesimal character $\lambda_\pi = \mathfrak{a}_\mathbf{C}^W \to \mathbf{C}$ of $\pi$, which also lies in $\mathfrak{a}_\mathbf{C}^\star/W$?

Answer 1

It is not necessarily the case that $\mathfrak a_\mathbb C$ is a Cartan subalgebra of $\mathfrak g$. In general a Cartan subalgebra of $\mathfrak g$ has the form $\mathfrak h_\mathbb C=\mathfrak t_\mathbb C\oplus \mathfrak a_\mathbb C$. The infinitesimal character is an element of $\mathfrak h_\mathbb C^*$, and $\nu$ is the restriction of this to $\mathfrak a_\mathbb C$. There is also a contribution from the (relative) discrete series on $M$. (Also $W$ is the "relative" Weyl group.)

For example in $GL(2n,\mathbb R)$, $M=GL(2,\mathbb R)^n$, $\dim(\mathfrak t_\mathbb C)=\dim(\mathfrak a_\mathbb C)=n$.

See Knapp's book "Representation Theory of Semisimple Groups, and Overview Based on Examples", Section VIII.6.