Simple supercuspidal \(L\)-packets of quasi-split classical groups (Q6605389)

Let \(F\) denote a \(p\)-adic field and let \(G\) stand for a connected reductive group over \(F\). The problem of determining an explicit description of the local Langlands corresopndence for \(G\) consists of the following two tasks:\N\begin{itemize}\N\item[1.] Obtaining an explicit description of the endoscopic lifting from \(G\) to \(\mathrm{GL}_N\),\N\item[2.] Obtaining an explicit description of the local Langlands correspondence for \(\mathrm{GL}_N\).\N\end{itemize}\NIn the paper under the review, under the assumption \(p \neq 2\), the author studies previously mentioned tasks for simple supercuspidal representations of the following series of groups:\N\begin{itemize}\N\item the symplectic group \(\mathrm{Sp}_{2n}\) (then \(N = 2n+1\)),\N\item the quasi-split special orthogonal group \(\mathrm{SO}_{2n}^{\mu}\) corresponding to a ramified quadratic character \(\mu\) of \(F^{\times}\) (then \(N = 2n\)),\N\item the quasi-split special orthogonal group \(\mathrm{SO}_{2n+2}^{ur}\) corresponding to the non-trivial unramified quadratic character \(F^{\times}\) (then \(N = 2n+2\)).\N\end{itemize}\N\NWe note that the simple supercuspidal representations can be characterized as those having the minimal positive depth.\N\NThe author solves the first task, determining the structure of simple supercuspidal \(L\)-packets for mentioned series of groups. The main tool used is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation. Using known explicit descriptions of the \(L\)-parameters of simple supercuspidal representations for \(\mathrm{GL}_N\), the author also solves the second task for the lifted representations. As an application, the formal degree conjecture is also checked for the simple supercuspidal representations.