Average multiplicities

This note summarizes my talk at the ICERM workshop on Murmurations and Arithmetic, 6 June 2023 to 8 June 2023. The talk was based on the paper [14], to which the reader is referred for details and proofs. The two addenda at the end appeared neither in the talk nor in [14] but are complementary to both. It is a pleasure to thank ICERM and the organizers of the workshop for inviting me to participate in a meeting that was both enjoyable and productive.