Extension-group conjecture for principal series representations of ${\rm SL}_n(k)$

Let $k$ be the underlying nonarchimedean local field, let $\pi$ be an irreducible unitary principal series representation of ${\rm GL}_n(k)$ induced from a supercuspidal representation $\sigma$ of a Levi subgroup $M$, and define

$$S_\pi=\{\mu\mid \pi\otimes\mu\cong\pi\},\qquad S_\sigma=\{\mu\mid \sigma\otimes\mu\cong\sigma\},$$

where $\mu$ ranges over complex characters of $k^\times$, viewed as characters of ${\rm GL}_n(k)$ through the determinant. Let $SM=M\cap {\rm SL}_n(k)$, and let $Y$ be the character group of $SM$, regarded as a module for the Weyl group $W({\rm GL}_n(k),M)=N_{{\rm GL}_n(k)}(M)/M$ and hence for $S_\pi/S_\sigma$. Let $Q$ be the quotient of $k^\times$ parametrizing the relevant characters of $S_\pi$, and let $\pi_a$ and $\pi_b$ be the corresponding irreducible representations of ${\rm SL}_n(k)$ for $a,b\in Q$.

Extension-group conjecture. For $a,b\in Q$, one has

$$\operatorname{Ext}^r(\pi_a,\pi_b)\cong\Lambda^rY[ba^{-1}].$$

This generalizes the preceding computation for a special class of principal series representations of ${\rm SL}_n(k)$, where the Ext groups are described by isotypic components of exterior powers of the character group. The conjecture proposes that the same description holds for general principal series representations; the source supplies no resolution status.