Tame supercuspidal representations of 𝐺𝐿_{𝑛} distinguished by orthogonal involutions

For a p p -adic field F F of characteristic zero, the embeddings of a tame supercuspidal representation π \pi of G = GL n ( F ) G= \textrm {GL}_n (F) in the space of smooth functions on the set of symmetric matrices in G G are determined. It is shown that the space of such embeddings is nonzero precisely when − 1 -1 is in the kernel of π \pi and, in this case, this space has dimension four. In addition, the space of H H -invariant linear forms on the space of π \pi is determined whenever H H is an orthogonal group in n n variables contained in G G .