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We define a representation of the unitary group $U(n)$ by metaplecticoperators acting on $L^2(\mathbb{R}^n)$ and consider the operator algebragenerated by the operators of the representation and pseudodifferentialoperators of Shubin class. Under suitable conditions, we prove the Fredholmproperty for elements in this algebra and obtain an index formula.

An Index Formula for Groups of Isometric Linear Canonical Transformations