Limit multiplicity for unitary groups and the stable trace formula

We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups $U(a,b)$, conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of characters and discrete series on endoscopic groups, and the bounds are obtained using Arthur’s stabilization of the trace formula and the classification established by Mok, and Kaletha, Minguez, Shin and White.