The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a ``rescaled'' spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.

58B34, 58J20

tangent groupoid, spinor bundle, local index theorem

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Department of Mathematics, Penn State University, University Park, PA 16802, USA

Chern Institute of Mathematics, Nankai University, Tianjin 300071, China