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In this thesis, we compute the equivariant cocycle of the Dirac operator D, when a finite group G acts by orientation preserving isometries, on a smooth compact Riemannian spin manifold M of dimension usd2n.usd We will work with the unital Banach algebra usd{\cal A} = C\sp{\infty}(G\times M),usd provided with the sup. norm, and usd{\cal H} = L\sp2({\cal E})usd where usd{\cal E}usd is the spinor bundle over M. Then usd(D, {\cal H})usd is a usd\thetausd-summable Fredholm module, and our goal is to compute the Chern character usdch(D, {\cal H})usd using the (JLO) formula. The first success of proving the Atiyah-Singer index theorem directly by heat kernel method was done in a magnificent paper by Patodi, who carried the "fantastic Cancellation" for the Laplace operators, and for the first time proved a local version of the Gauss-Bonnet-Chern Theorem. If we consider the asymptotic expansion of the heat kernel for the square of Dirac operator, i.e usdD\sp2,usd then it has the formusdusd{e\sp{-\rho\sp2/4t}\over (4\pi t)\sp{n}}\sum\sbsp{j=0}{N}t\sp{j}Uj(x,\xi)usdusd In the computation of D, we need to consider the supertrace and the limit as t tends to zero. The only term that survives both the supertrace and the limit is the one which contains exactly usd2nusd Clifford variables. Similar type of argument works when computing the cocycle or equivariant cocycle for the Dirac operator D. In our computation, we needed a way to keep track of the Clifford variables, and the powers of t. Yu introduced the usd\chiusd-degree map to keep track not only of the Clifford variables, but also of the partial derivative operators usd\partial/\partial x\sb{i}usd's, and the usdx\sb{k}usd's, which appears in the local expression of usdD\sp2.usd To keep track of powers of t, we use a device that was developed by Simon called The Canonical Order Calculus. Using the usd(\chi -no)usd degree map and the Canonical order calculus, we compute the index of the Dirac operator in chapter 1. Since our computation is local, working in a normal coordinates, we get a nice expression of usdD\sp2.usd Using the Duhamel's expansion combined with the above degree maps, we try to approximate the operator usdD\sp2usd by an operator which is easier to work with like the harmonic oscillator. In chapter 2, we extend the above method to the computation of the cocycle, thus presenting a different proof than the one in (4). In chapter 3, we compute the equivariant index of the Dirac operator. Finally, in chapter 4, we compute the equivariant cocycle, by combining what we did in chapters 2 and 3. It is a simple matter to extend these computations to twisted spinor bundle and twisted Dirac operator. We believe that these computations can be used for any geometric type operators. (Abstract shortened by UMI.)
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