process

A Schwartz type algebra for the tangent groupoid 183consider the Schwartz algebra S.A G /. Indeed, the Fourier transform shows that thislast algebra is stable under holomorphic calculus on C 0 .A G / and so it has the “good”K-theory, meaning that K 0 .A G / D K 0 .S.A G //. None of the inclusions in (1) isstable under holomorphic calculus, but that is precisely what we wanted because ouralgebra S c .G T / has the remarkable property that its evaluation at zero is stable underholomorphic calculus, while its evaluation at one (for example) is not.The algebra S c .G T / is, as a vector space, a particular case of a more generalconstruction that we do for “deformation to the normal cone manifolds” from which thetangent groupoid is a special case (see [4] and [13]). A deformation to the normal conemanifold (DNC for short) is a manifold associated to an injective immersion X,! Mthat is considered as a sort of blow up in differential geometry. The constructionof a DNC manifold has very nice functorial properties (Section 3) which we exploitto achieve our construction. We think that our construction could be used also forother purposes, for example, it seems that it could help to give more understanding inquantization theory (see again [4]).The article is organized as follows. In the second section we recall the basic factsabout Lie groupoids. We explain very briefly how to define the convolution algebraCc1 .G /. In the third section we explain the “deformation to the normal cone” constructionassociated to an injective immersion. Even if this could be considered as classicalmaterial, we do it in some detail since we will use in the sequel very explicit descriptionsthat we could not find elsewhere. We also review some functorial properties associatedto these deformations. A particular case of this construction is the tangent groupoid associatedto a Lie groupoid. In the fourth section we start by constructing a vector spaceS c .DX M / for any deformation to the normal cone manifold DM X; this space alreadyexhibits the characteristic of being a field of vector spaces over the closed interval Œ0; 1whose fiber at zero is a Schwartz space while for values different from zero it is equalto Cc1.M /. We then define the algebra S c.G T /; the main result is that its product iswell-defined and associative. The last section is devoted to motivate the constructionof our algebra by explaining shortly some further developments that will immediatelyfollow from this work. All the results of the present work are part of the author’s PhDthesis.I want to thank my PhD advisor, Georges Skandalis, for all the ideas that heshared with me, and for all the comments and remarks he made on the present work.I would also like to thank the referee for the useful comments he made for improvingthis paper.2 Lie groupoidsLet us recall what a groupoid is.Definition 2.1. A groupoid consists of the following data: two sets G and G .0/ , andmaps• s; r W G ! G .0/ , called the source and target map respectively,