By Louis Boutet de Monvel, C. De Concini, C. Procesi

Lectures Given on the second consultation of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Venezia, Italy, June 12-20, 1992

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112 (1964), 229-247. Atiyah M. , Clifford modules, Topology (1964), no. 3, supplement, 3-83. Atiyah M. , Vector bundles and homogeneous spaces, Diff. Geometry, Proc. Syrup. , Amer. Math. , Providence (1961), 7-38. Atiyah M. , Segal G. , The index of elliptic operators II, Ann. Math. 87 (1968), 531-545. Atiyah M. , Singer I. M. [1], The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433. [2], The index of elliptic operators I, Ann. Math. 87 (1968), 484-530.

Yk) with rational coefficients, independent of m and such that: Ok = pk(r r , Ck) We then set ok(a) := pk(tr(a), t T ( a 2 ) , . . , tr(~k)). Next we can formally define for every element a in A and for every integer d a d th -characteristic polynomial: d Xd,~[t] := E ( - - 1 ) i a i ( a ) t d-i. i=0 This formal polynomial is useful in representation theory if we have the formal CayleyH a m i l t o n theorem: i) We say that an algebra R with trace satisfles the d - th [ormM CayleyHamilton theorem ifXd,a[a ] = 0 for every a G R.

1) F~ : U~ ----* T * X l'application qui prolonge F(Fe(~, h) = ~ + h) -Se : U~ ~ la projection T*Y Lorsque f est submersive, F~ est une immersion ferm6e et son image F ( H ) + SS U joue le r61e de SS(:Du-x). Remarquons que l'image directe f + ( M i u ) ne d@end fividemment que de supp M N U ( f + ( M i u ) = f + ( M i v ) si s u p p M N g = s u p p M N V). e. loealement un produit au sens analytique r~el (il suffira en fait que ceci soit vrai au voisinage de s u p p M N U), lorsque f e s t submersive, et que U est une varifit~ bord (ou s coins).