Tame supercuspidal representations of \(\mathrm{GL}_{n}\) distinguished by orthogonal involutions (Q2836479)

Let \(\pi\) be an irreducible tame supercuspidal representation of \(GL_n\) over a \(p-\)adic field of characteristic zero. This paper shows that \(\pi\) is distinguished with respect to some orthogonal group if and only if its central character \(\varpi\) is trivial at \(-1\). Moreover the paper determines for each orthogonal group \(H\), the dimension \(\ell_H\) of \(H-\)invariant linear forms on \(\pi\). Thus the paper gives a complete picture for the distinction of \(\pi\) with respect to an orthogonal group.NEWLINENEWLINEAssume \(\varpi(-1)=1\). When \(n\) is odd, \(\pi\) is distinguished by \(H\) if and only if \(H\) is split, with \(\ell_H=1\). (This result was previously established in a work of the author and J. Lansky). The \(n\) even case is more complicated. When \(H\) is split, \(\ell_H\) is either \(2\) or \(3\) depending on \(\pi\). When \(\ell_H=2\), \(\pi\) is also distinguished by one other (conjugacy class of) orthogonal group \(H'\) which is quasi-split, with \(\ell_{H'}=1\). When \(\ell_H=3\), \(\pi\) is distinguished by the unique (conjugacy class of) non-quasisplit group \(H''\) with \(\ell_{H''}=1\); it is not distinguished by any quasi-split group.NEWLINENEWLINEAn interesting observation is that there is a uniform description for the above result. Let \(C^\infty(S)\) be the space of smooth functions on the set of symmetric matrices \(S\) in \(GL_n\), then the dimension of \(GL_n\)-equivariant embeddings of \(\pi\) in \(C^\infty(S)\) equals 4.NEWLINENEWLINEThis well-written paper also includes a nice exposition of the background information on tame supercuspidal representations and forms of orthogonal groups.