The BPS decomposition theorem

For a proper map of algebraic varieties, the BBDG decomposition theorem gives a powerful description of the cohomological direct image when the source is smooth. I explain an analogue for good moduli space morphisms of smooth algebraic stacks, together with a description of the simple perverse sheaves appearing among the direct summands. Examples illustrate both the statement and the ideas that led to it. The talk draws in particular on joint work with Tasuki Kinjo.