Shin and Templier studied families of automorphic representations with
local restrictions: roughly, Archimedean components contained in a fixed
-packet of discrete series and
non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components
as either the weight of the
-packet
or level went to infinity. We extend their weight-aspect results to families where the
Archimedean component is restricted to a single discrete-series representation instead of an
entire
-packet.
We do this by using a so-called “hyperendoscopy” version of the stable trace
formula of Ferarri. The main technical difficulties are first, defining a version of
hyperendoscopy that works for groups without simply connected derived subgroup
and second, bounding the values of transfers of unramified functions. We also present
an extension to noncuspidal groups of Arthur’s simple trace formula since it does not
seem to appear elsewhere in the literature.
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