# Equivariant non-commutative residue and equivariant Weyl's theorem

Abstract (Summary)

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
Bibliographical Information:

Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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