  In this expository article, we consider first order elliptic differential
operators acting on smooth vector bundles over compact manifolds, and certain
invariants derived from the analysis of these operators, namely the eta
invariant} and the equivariant index. Many researchers have previously
considered these invariants before. What makes this work different is that we
are evaluating integer-valued indices corresponding to multiplicities of group
representations, and our eta invariant is a number dependent on the entire
group at once. Moreover, the techniques of proof and formulas obtained are new
and depend on equivariant heat asymptotics that may involve logarithmic terms.
For simplicity, we consider only elliptic differential operators, even though
the proofs outlined apply to transversally elliptic operators. In every case,
we outline the well-known proofs and theorems without Lie group actions first
and then show how these same ideas can be applied in the equivariant cases with
appropriate modifications. A more detailed and expanded article that applies to
transversally elliptic operators will appear in due time.
