Endoscopic classification conjecture for unitary similitude groups

Let $F$ be the underlying nonarchimedean local field, and let ${\mathrm{GU}}(J_{2n,\varepsilon_1})(F)$ and ${\mathrm{GU}}(J_{2n,\varepsilon_2})(F)$ be the two relevant unitary similitude groups. Write

$$\Pi_{irr,temp}({\mathrm{GU}}_{2n})=\Pi_{irr,temp}({\mathrm{GU}}(J_{2n,\varepsilon_1}))\cup \Pi_{irr,temp}({\mathrm{GU}}(J_{2n,\varepsilon_2}))$$

for the set of their irreducible admissible tempered representations. For $g_i\in {\mathrm{GU}}(J_{2n,\varepsilon_i})(F)$, write $g_1\leftrightarrow g_2$ when they have the same characteristic polynomial, and let $\theta_\pi$ denote the distribution character of a representation $\pi$.

Endoscopic classification conjecture. The set $\Pi_{irr,temp}({\mathrm{GU}}_{2n})$ is a disjoint union of finite local tempered Vogan $L$-packets:

$$\Pi_{irr,temp}({\mathrm{GU}}_{2n})=\bigcup_{\phi}\Pi_\phi,$$

where $\phi$ runs over all tempered $L$-parameters of ${\mathrm{GU}}_{2n}(F)$ and

$$\Pi_\phi=\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_1}))\cup\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_2})).$$

Each packet consists of finitely many tempered representations and satisfies: for $i=1,2$,

$$\theta_{\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_i}))}:=\sum_{\pi_{\varepsilon_i}\in\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_i}))}\theta_{\pi_{\varepsilon_i}}$$

is stable; $\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_1}))$ contains a unique generic representation; and, whenever $g_i\in {\mathrm{GU}}(J_{2n,\varepsilon_i})(F)_{reg}$ satisfy $g_1\leftrightarrow g_2$,

$$\theta_{\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_1}))}(g_1)=-\theta_{\Pi_\phi({\mathrm{GU}}(J_{2n,\varepsilon_2}))}(g_2).$$

Here ${\mathrm{GU}}(J_{2n,\varepsilon_2})(F)$ is not quasi-split. The conjecture supplies the expected endoscopic classification of tempered representations for the two unitary similitude groups, extending the endoscopic classification for unitary groups and the proposed reduction from similitude groups to classical groups. It is assumed in the paper's main theorems and its resolution status is not specified in the supplied text.