Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms

Let G(infinity) be a semisimple real Lie group with unitary dual (G) over cap (infinity). We produce new upper bounds for the multiplicities with which representations pi is an element of (G) over cap (infinity) of cohomological type appear in certain spaces of cusp forms on G(infinity). The main new idea is to apply noncommutative Iwasawa theory to certain p-adic completions of the cohomology of locally symmetric spaces.