Parabolic principal-series and Satake conjecture

Let $\bmu$ be a parabolic character, let $L$ be the associated Levi subgroup, and let $\Pi^c$, $\sW^c$, and $\sH^c$ be the corresponding principal-series family, induced representation, and endomorphism algebra. Let $\hL$ be the relevant Langlands dual group. Parabolic principal-series and Satake conjecture.

$$\Pi^c \cong \sW^c,$$

and there exists a Satake-type isomorphism

$$\sH^c \cong \K_0(\operatorname{Rep}(\hL)).$$

These assertions relate the parabolic family to a compactly induced representation and its representation-theoretic Hecke algebra. The source records the conjecture as known in the case $G=\operatorname{GL}_N$, by work of Howe, while the general parabolic case is not resolved in the supplied text.