Treumann–Venkatesh admissibility and functoriality conjectures for Tate cohomology

Let $G$ and $H$ be the groups occurring in the Local Langlands correspondence, let $\boldsymbol{Z}[[\sigma]$ be the group ring of $\langle \sigma\rangle\cong \boldsymbol{Z}/\ell\boldsymbol{Z}$, and let $\mathrm{T}^j(\Pi)$ denote the Tate cohomology groups of a $G(E)\rtimes\sigma$-representation $\Pi$, viewed as representations of $H(E)$. Let $\Pi$ be an irreducible smooth representation of $G(E)$ whose isomorphism class is fixed by $\sigma$, so that its $G(E)$-action uniquely extends to a $G(E)\rtimes\sigma$-action. For each $j\in\mathbf{Z}/2\mathbf{Z}$, two assertions are made. Treumann–Venkatesh admissibility and functoriality conjectures. First, $\mathrm{T}^j(\Pi)$ is admissible as a representation of $H(E)$. Second, the $L$-parameter of every irreducible $H(E)$-subquotient of $\mathrm{T}^j(\Pi)$ is sent by the $\sigma$-dual homomorphism $\check{\psi}$ to the $L$-parameter for $\Pi^{(\ell)}:=\Pi\otimes_{k,\operatorname{Frob}}\ell$, the Frobenius twist of $\Pi$. These conjectures propose that Tate cohomology realizes the expected representation-theoretic and $L$-parameter-level compatibility in the Local Langlands Correspondence, but the supplied text gives no evidence resolving either assertion.