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In arXiv:1208.1945, Shin and Templier proved certain equidistribution boundson local components of certain families of automorphic representations. Weextend their weight-aspect results to families of automorphic representationswhere the Archimedean component is restricted to a single discrete-seriesrepresentation instead of an entire $L$-packet. We do this by using a so-called"hyperendoscopy" version of the stable trace formula developed by Ferrari.  The main technical difficulties are defining a version of hyperendoscopy thatworks for groups without simply connected derived subgroup and bounding thevalues of transfers of unramified functions. We also present an extension ofArthur's simple trace formula for test functions with Euler-Poincar\'ecomponent at infinity to non-cuspidal groups since it does not seem to appearelsewhere in the literature.

Sato-Tate Equidistribution for Families of Automorphic Representations through the Stable Trace Formula