Explore an advanced lecture on extending Lusztig's theory of character sheaves for connected reductive groups to positive-depth scenarios. Delve into a construction that produces generic character sheaves associated with positive-depth data, addressing a conjecture by Lusztig. Examine how this approach generates perverse sheaves linked to sufficiently generic multiplicative local systems, with trace-of-Frobenius functions matching parahoric Deligne-Lusztig induction. Gain insights into this cutting-edge development in representation theory, presented as joint work with R. Bezrukavnikov, in this 59-minute talk from the Hausdorff Center for Mathematics.