process

A Schwartz type algebra for the tangent groupoid 195Lemma 4.11. PQc W S c . U V / ! S c. U 0V/ is well defined.Proof. The first observation is that the integral in the definition of PQc is always welldefined. Indeed, this is a consequence of the following two facts:• for t D 0, 7! F .x; ; ; 0/ 2 S.R q /,• for t ¤ 0, 7! F.x;;;t/ 2 Cc 1.Rq /.Taking derivatives under the integral sign, we obtain that PQc .F / 2 C 1 . U 0V/. Then,we just have to show that PQc .F / verifies the two conditions of Definition 4.1. For thefirst, if K U Œ0; 1 is the compact conic support of F , then it is enough to putK 0 D .P Id Œ0;1 /.K/in order to obtain a conic compact subset of U 0 Œ0; 1 relative to V and to check thatK 0 is the compact conic support of PQc .F / . Let us now verify the condition .s 1 /. Letk; m 2 N, l 2 N p and ˇ 2 N q . We want to find C .k;m;l;ˇ/ >0such that.1 Ckk 2 / k k@ l x @ˇ @m Qt P c .F /.x; ; t/k C .k;m;l;˛/ :For k 0 k C q 2 and ˛ D .0; ˇ/ 2 Rq R q we have by hypothesis that there exists>0such thatC 0.k 0 ;m;l;˛/Then we also have thatwithk@ l x @ˇ @m tk@ l x @ˇ @m t F.x;;;t/kC 0 1.1 Ck.; /k 2 / k0 :ZPQc .F /.x; ; t/k C 0 1df2R q W.x;;;t/2 U V g .1 Ck.; /k 2 / k0 C 0 111d C.1 Ckk 2 / q 2Zf2R k0 q g .1 Ckk 2 / k0 .1 Ckk 2 / kZC D C 0 1d:f2R q g .1 Ckk 2 / k0We can now give the proof of Proposition 4.8.Proof of Proposition 4.8. Let us first fix some notation. We suppose dim G D p C qand dim G .0/ D p, in particular this implies that dim G .2/ D p C q C q. Let .U;/and .U 0 ; 0 / be G .0/ -slices in G .2/ and G , respectively, such that the following diagramcommutes:U m U 0 0 U PU 0 ,