Let $G = GL(n,\mathbb{R})$. Consider a Langlands data $(Q_F, \sigma, \lambda)$ with $F \subset \Delta$ (the set of simple roots), $Q_F$ the associated standard parabolic subgroup, $\sigma$ an irreducible tempered representation of $M_F$, and $\lambda$ an element of $(\mathfrak{a}_F)^*_\mathbb{C} = Hom(\mathfrak{a},\mathbb{C})$ satisfying $\langle Re \lambda, \alpha \rangle \ge 0 $ for $\alpha \in \Delta \setminus F$

Here $\mathfrak{a}_F$ is the Lie algbera of $A_F$ appearing in the Langlands decompostion $Q_F = M_F A_F N_F$.

Then the Langlands classification of irreducible admissible $(\mathfrak{g},K)$-modules is essentially given as the module $J(Q_F, \sigma, \lambda)$ occuring in the parabolically induced module $Ind_{Q_F}^G(\sigma \otimes \lambda \otimes 1)_K$ (the subscript $K$ denotes $K-$finite).

Is it the case that the infinitesimal character of Langland's quotient same as that of the $Ind_{Q_F}^G(\sigma \otimes \lambda \otimes 1)$ (with out $K$ subscript?)

(I am a novice in this subject.)