I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.

The meta question is, given a function on the unitary dual of $PGL(2, F)$ (or a more general reductive group $G$ over $F$), could its restriction to representations of fixed conductor introduce new points of discontinuity?

Let me precise the picture. Let $\phi$ be a function on the unitary dual of $PGL(2)$. Let $(M, \delta)$ be a Bernstein component of $PGL(2)$, that is a Levi $M$ and a discrete series $\delta$ on $M$. Assume the function (on unramified unitary characters of $M$) $$(\star) \qquad \phi_{M, \delta}: \chi \mapsto \phi\left(ind_M^G(\delta \otimes \chi)\right)$$

is continuous outside a set of measure zero (for the Plancherel measure).

**Question 1.** Does the restriction of $\phi$ to the generic dual (in order to have a notion of conductor) still satisfy $(\star)$? That is to say, if $\phi^{\mathrm{gen}}$ denotes the function defined by $\phi$ onthe generic dual and zero elsewhere on the unitary dual, does $\phi^{\mathrm{gen}}_{M, \delta}$ satisfy $(\star)$?

**Question 2.** Does the restriction of $\phi$ to the representations of conductor $c$ still satisfy $(\star)$? That is to say, if $\phi^{c}$ denotes function defined by $\phi$ on the representations of conductor $c$ and zero elsewhere on the unitary (or generic by Question 1) dual, does $\phi^{c}_{M, \delta}$ satisfy $(\star)$?

anysubspace canneverintroduce new points of discontinuity (although it may convert points of discontinuity to points of continuity). $\endgroup$ – LSpice Mar 5 '18 at 21:30