I am trying to read the lecture notes of Joseph Bernstein on the representations of $p$-adic groups and struggling to understand a certain claim regarding the Bernstein center.
Suppose $G$ is a reductive $p$-adic group and $D$ is a cuspidal component of a Levi subgroup $M$ with a projective generator $F\bigotimes\rho$, where $\rho$ is some representation in $D$, and $F$ the algebra of regular functions on the variety of unramified characters of $M$.
Due to the existence of a right adjoint to the parabolic induction functor $i_{G,M}:Rep(M)\rightarrow Rep(G)$, we get that $i_{G,M}(F\bigotimes\rho)$ is a projective generator in the subcategory $Rep_{(M,D)}(G)$ of $Rep(G)$ consisting of representations belonging to the cuspidal component associated with the pair $(M,D)$. From this we get that $Rep_{(M,D)}(G)$ is equivalent to the category of right modules of the ring $\Lambda=Hom_{G}(i_{G,M}(F\bigotimes\rho),i_{G,M}(F\bigotimes\rho))$, which is in natural bijection with $Hom_{M}(r_{M,G}\circ i_{G,M}(F\bigotimes\rho),F\bigotimes\rho)$, since by Frobenius reciprocity $i_{G,M}$ has a left adjoint, $r_{M,G}:Rep(G)\rightarrow Rep(M)$. And from this we get an imbedding through post composition $\Lambda(D)\hookrightarrow\Lambda$, where $\Lambda(D)=Hom_{M}(F\bigotimes\rho,F\bigotimes\rho)$.
My question is why should the center of $\Lambda$ be contained in the imbedded copy of $\Lambda(D)$?
The paper I'm reading on the subject claims that its easy to see this but I'm not sure how to approach this claim.
