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 $(\star)$?

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    $\begingroup$ "Continuity of the conductor" seems not to be what you want here. It seems rather that you mean "continuity on representations of fixed conductor", or suchlike. I'm confused about the question, but maybe just because I don't understand the topology. It seems to me that restricting a function from a big space to any subspace can never introduce new points of discontinuity (although it may convert points of discontinuity to points of continuity). $\endgroup$ – LSpice Mar 5 '18 at 21:30
  • $\begingroup$ @LSpice The term of "continuity" refers to an idea of "continuity along lines", in this case along twists by characters. Also, I am not really restricting, I am truncating by zero, thus there will be new discontinuity points (at the boundary of my sets). The question is: can these points lying on the boundary be of measure non-zero (knowing that before discontinuity points were of measure zero; this is important since for instance in SL(2, R) we always remove the trivial representation (measure 0) but not the Steinberg (measure nonzero, but already in the boundary, hence not a discontinuity) $\endgroup$ – Desiderius Severus Mar 6 '18 at 0:58
  • $\begingroup$ Oh, I see. Your title refers to restriction, but indeed the body explains that it is restriction-then-extension. $\endgroup$ – LSpice Mar 6 '18 at 1:37

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