Equivariant non-commutative residue and equivariant Weyl's theorem

by Dave, Shantanu.

Let M be a smooth, compact manifold acted upon smoothly by a group ?. The first objective of this thesis is to study the action of ? on the algebra of complete symbols ??(M)/???(M) and to determine the Hochschild and cyclic homology groups of the algebra B = ??(M)/???(M) ? ?. It turns out that these homology groups can be expressed in terms of the usual de-Rham cohomology of the fixed point manifolds S?M ? , where the elements ? run over any set of representatives of the conjugacy classes of ?. This is achieved using the spectral sequence associated with the natural filtration on B. In order to compute this spectral sequence, we also obtain an explicit identification for the Hochschild homology groups of the cross-product algebra C?(M)??. As usual, the 0-th Hochschild cohomology of our algebra B identifies with the space of traces on our algebra. In the second part of the thesis, we provide explicit determinations of the traces of the algebra B = ??(M)/???(M)?? as equivariant, delocalized noncommutative residues. We show that the delocalized noncommutative residue associated to a conjugacy class that acts as the identity on M exhibits the usual behavior of the noncommutative residue. However, the delocalized noncommutative residue associated to a conjugacy class that does not act as the identity, exhibits a behavior that is quite different from that of the usual residue. The relationship between the symplectic structures on T ?M and T ?M ? plays an improtant role in the background of all these results. For instance the description of the delocalized noncommutative residues in coordinate charts has the resemblance of “a fundamental form for symplectic submanifolds”. As an application of the equivariant, delocalized noncommutative residues, we obtain a Weyl type estimate for the occurrence of a ? representation ? in an invariant operator D, which might be the Laplacian of an invariant Riemannian metric. That is, the function N?,D(?) that counts multiplicity of ? in each eigensapace ker(D ? ?iI), ?i ? ?, grows asymptotically as N?,D(?) ? Cdim(?)? n m , where n = dim(M) and m = ord(D). iv