International Congress of Mathematicians

Heat Kernels and the Index Theorems • • • 367Since g is unitary, one verifies easily that the operator gP$M,>o9^1 is an orthogonalprojection on L 2 ((S(TM)(E)C N )\QM), and that gPdM,>o9^1 ^ PdM,>o is apseudodifferentialoperator of order less than zero. Moreover, the pair (D,gP$M,>o9^1)forms a self-adjoint elliptic boundary problem. We denote its associated ellipticself-adjoint operator by D gPgM >og -i-Yet L 2 p ! >0(S(TM )®C JV ) be the space of the direct sum of eigenspacesof nonnegative eigenvalues of D gPgM >og -i • Yet P g p BM > oS -\>o denote the orthogonalprojection from L 2 (S(TM) ® C k ) to L 2 gPgM >og_^>Q (S(TM) ® C*).Clearly, if « e L 2 (S(TM)® C*) verifies -POM,>O(«|OM) = 0, then gs verifiesgPdM,>og^ ((gs)\ 9 M) = 0.Definition 5.1 The Toeplitz operator T g is defined byT 9 = PgPeM.^og- 1 ,>o9 P PeM.>o,>0 '•L h M ,>_ 0 ,>o (S(TM) ® C») -+ L 2 gPgM>_ og^^0 (S(TM) ® C») -One verifies that T g is a Fredholm operator. The main result of this sectionevaluates the index of T g by more geometric quantities.5.2. An ^-invariant associated to gWe consider the cylinder [0,1] x dM. Clearly, the restriction of g on dMextends canonically to this cylinder.Let -D|[o,i]xOM be the restriction of D on [0,1] x dM. We equip the boundaryconditionPdM,>o at {0} x dM and the boundary condition Id — gPdM,>o9^1 at{1} x dM. Then (-D|[O,I]XOMJ PBM;>Q, Id — gP$M,>o9^1)forms a self-adjoint ellipticboundary problem. We denote the corresponding elliptic self-adjoint operator by-PeM,>o,S-PeM,>oS _1 'Let n(D PgM >0}9 p BM >og - 1 is ) be the ^-function of s £ C which, when Re(s) >>0, is defined by(n \ - Y^ sS n ( A )f l \ u PdM.xsigPaM.xsg- 1 1 s ) — / „ i x Is 'A#0 ' 'where A runs through the nonzero eigenvalues of D PgM >0 , g p BM >og - 1 -It is proved in [DZ2] that under our situation, n(D PgM >0}9 p BM >og -i,s) can beextended to a meromorphic function on C which is holomorphic at s = 0.Yet rj(Dp BM >Q}9 p BM >oS - 1 ) be the reduced ^-invariant defined by^ \ D PBM.yo^gPBM.yog- 1 )_ dfmker ^I>p eM ,> 0 , s p eM ,> oS -iJ + n (-Dp eM ,> 0 , s p eM ,> oS -i)