Representations of the \(p\)-adic \(\mathrm{GSpin}_4\) and \(\mathrm{GSpin}_6\) and the adjoint \(L\)-function (Q6613260)

In this paper, the authors study the local \(L\)-packets for the non-Archimedean split general spin groups \(\mathrm{GSpin}_{4}\) and \(\mathrm{GSpin}_{6}\), and prove the conjecture of Gross-Prasad for these groups, i.e. a local \(L\)-packet is generic if and only if the associated adjoint \(L\)-function is regular at \(s=1\).\N\NThey work in the larger generality for the split reductive group \(G_{m,n}^{r,s}:=\{(g,h)\in\mathrm{GL}_{m}\times\mathrm{GL}_{n}|\det(g)^{r}=\det(h)^{s}\}\), where \(\gcd(r,s)=1\), noting that \(\mathrm{GSpin}_{4}=G_{2,2}^{1,1}\) and \(\mathrm{GSpin}_{6}=G_{1,4}^{2,1}\). The strategy is to embed any irreducible admissible representation \(\pi\) of \(G_{m,n}^{r,s}\) to the restriction of some irreducible representation \(\pi_{m}\otimes\pi_{n}\) of \(\mathrm{GL}_{m}\times\mathrm{GL}_{n}\), then use the local Langlands correspondence (LLC) for \(G_{m,n}^{r,s}\) to write the adjoint \(L\)-function of \(\pi\) as the product of those for \(\pi_{m}\) and \(\pi_{n}\). The LLC for \(\mathrm{GSpin}_{4}\) or \(\mathrm{GSpin}_{6}\) is developed before by the same authors, thus the conjecture of Gross-Prasad follows from the known case for \(\mathrm{GL}_{n}\).\N\NMoreover, using the LLC for \(\mathrm{GL}_{n}\) and Zelevinsky's classification of generic representations for \(\mathrm{GL}_{n}\), the authors enumerate the \(L\)-packets of \(\mathrm{GSpin}_{n},\,n=4,6\), determine their genericity, and calculates explicitly the adjoint \(L\)-function for each \(L\)-packet.