Index Formula for a Finite Group Г

Part of the Operator Theory: Advances and Applications book series (OT, volume 183)


Let Г be a finite group acting topologically freely on a smooth closed manifold M, and suppose that we are given a nonlocal elliptic operator1
$$ A = \sum\limits_{g \in \Gamma } {T\left( g \right)A\left( g \right):C^\infty \left( {M,\mathbb{C}^n } \right) \to C^\infty \left( {M,\mathbb{C}^n } \right).} $$
The nonlocal equation
$$ Au = f $$
can be reduced to a system of local equations. To this end, we replace the functions u and f by the vector functions
$$ U = \left\{ {U_g } \right\}_{g \in \Gamma } ,F = \left\{ {F_g } \right\}_{g \in \Gamma } $$
whose components
$$ U_g = T\left( g \right)u,F_g = T\left( g \right)f $$
are obtained as shifts of the original functions by the elements of Г.


Finite Group Cotangent Bundle Local Equation Chern Character Index Formula 
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© Birkhäuser Verlag AG 2008

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