Abstract
Let $G_\infty$ be a semisimple real Lie group with unitary dual $\widehat{G}_{\infty}$. We produce new upper bounds for the multiplicities with which representations $\pi \in \widehat{G}_{\infty}$ of cohomological type appear in certain spaces of cusp forms on $G_\infty$. The main new idea is to apply noncommutative Iwasawa theory to certain $p$-adic completions of the cohomology of locally symmetric spaces.