Extending cuspidal representation to more bigger group.

Question

I am thinking of extending an irreducible cuspidal representation to more bigger group. My question is almost same with the earlier one posted by Neal Harris except the only one.

Let me first invoke his original question.

"Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermitian space over $E$.

Let $\tilde{G} := GU(V)$ and $G := U(V)$. Suppose that $(\pi, V_{\pi})$ is an irreducible cuspidal representation of $G.$

Is there an irreducible cuspidal representation $(\tilde{\pi}, V_{\tilde{\pi}})$ of $\tilde{G}$ such that $V_\pi \subset V_{\tilde{\pi}}|_{G}$? Note that here, the restriction is that of cusp forms, not of the representation itself."

The difference between mine and his is this; While his assumption G=U(2) and $\tilde{G}$=GU(2) hit the condition $\tilde{G}_{der}\subset G\subset\tilde{G}$, a core property after the Mattew's comment on the post, but my situation does not satisfy this.

Because, I am considering the case $G=E^{\times} , \tilde{G}=B^{\times}$ where E/F is quadratic extension of number fields and B is a quaternion algebra over F with a fixed embedding $E \hookrightarrow B$. Is there no hope in this case?

Since this question is very important to me, if you leave some comment or reference treating this, I will be very appreciate to you.

(Harris's original post.

extending cusp forms)

Answer 1

Looked at a slightly different way: the question of what happens when automorphic forms/repns are restricted to subgroups has complicated answers, in general. It can be treated as a problem in spectral decomposition, say in $L^2$. Another keyword is "period integral" for the integral expressions for the spectral components. Some examples are well-known in other terms: the "Mellin transform" integral that produces the standard L-function for GL(2) cuspforms can be construed (on the critical line!) as computing the spectral components of the restriction of the cuspform to the GL(1) imbedded in the upper left corner. The $GL(n-1)\times GL(n)$ integral formulas amount to computing the $GL(n-1)$ components of the restriction of a cuspform from $GL(n)$.

In other cases, sometimes it is feasible or elementary to compute periods of Eisenstein series, while the corresponding periods for cuspforms are much less elementary. (The Gross-Prasad conjectures were/are an example of this.)

In your specific situation, there are results of Waldspurger that may be approximately what you want.

Given that many of these periods/decomposition coefficients are values of L-functions at the center of the critical strip, except for trivial vanishing due to signs in functional equations, it is non-trivial to prove non-vanishing or vanishing. Thus, to ask that such a period be non-zero for exactly one cuspform would seem to be too much to ask, although perhaps very difficult to prove or disprove.