Invariant differential operators acting on quotient spaces and their index

Abstract: Let $G$ be a compact Lie group acting on a smooth manifold $M$ (without boundary), $E \to M$ be an equivariant bundle, and $P$ be a $G$-invariant pseudodifferential operator acting on the sections of $E$. Let $\alpha$ be an irreducible representation of $G$ and $\pi_\alpha(P)$ be the restriction of $P$ to the isotypical component corresponding to $\alpha$. We study the resulting algebra of symbols and we give a simple, necessary and sufficient criterion for $\pi_\alpha(P)$ to be Fredholm. We also provide a spectral sequence converging to the periodic cyclic homology of the corresponding algebra of symbols. This work was done in collaboration with A. Baldare, M. Benameur, R. Come, and M. Lesch.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

Noncommutative Geometry in NYC

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