Frobenius-Schur indicator of a self-dual L-parameter

Question

Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC for $\mathrm{GL}_n$ we obtain a $n$-dimensional irreducible self-dual representation $\varphi_\pi$ of the Weil group $W_F$.

Is there an interpretation of the Frobenius-Schur indicator of $\varphi_\pi$ in terms of the representation $\pi$?

It is not naively the Frobenius-Schur indicator of the representation $\pi$, since Corollary 9 of Mishra "A note on sign of a self-dual representation" says the Frobenius-Schur indicator of $\pi$ is always $1$.

Answer 1

Prasad and Ramakrishnan (arXiv link) study how the signs of (discrete series) representations behave along the local Langlands correspondence, not just for GL($n$), but for all inner forms. Assume the base field has characteristic 0.

If $n$ is odd, $\pi$ and its Langlands parameter have the same sign.

If $n$ is even, you need to use the Jacquet-Langlands correspondence (at least to use Prasad-Ramakrishnan's result to get the sign of the parameter). Since $\pi$ is supercuspidal, then it corresponds to a representation $\pi_D$ on a division algebra $D$, which does not always have the same sign as $\pi$. Here the signs of $\pi_D$ and the Langlands parameter are opposite.