By L. Auslander, R. Tolimieri
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Extra resources for Abelian Harmonic Analysis Theta Functions and Function Algebras on a Nilmanifold
Ik are elements of bH 2k,p(M). It is clear that a given fp : P ---t Pn,o uniquely determines f M : M ---t Bn,o. The case G ~ O(N), dimM = 4k, is quite parallel. ik' i 1 + ... ik' i 1 + ... ik = k are elements of H 4k ,p(M). ik by the class of Hirzebruch genuss Lk then we get the notion of a (p, L k )-, (b, Lk)- or (b, p, Lk)-bundle (M, fM), respectively. 50 a) Suppose G c U(N), dim M = 2k. (M, g) satisfying (Bo), p 2 1. ik' i 1 + ... + i k = k. If additionally p = 1 and (M, g) is complete then even the corresponding characteristic numbers coincide.
We indicate (partial) answers given by Barbasch, Connes, Moscovici and Muller. Denote by R(k) the right regular representation R(k)f(g) = f(gk), TE : K ----t U(VE)' Then k ----t R(k) 0 TE(k) acts on COO(G) ® VE. We identify COO(E) with (COO(G) 0 VE)K, similarly L2(E) with (L2(G) ® VE)K. If ® is the Lie algebra of G, ®c its complexification, U(®) the universal enveloping algebra of ®, TE : K ----t U(VE), Tp : K ----t U(Vp) are unitary Relative Index Theory, Determinants and Torsion 52 representations then (U( ®) ® Hom (VE, VF))K shall denote the subspace of all elements in U(®) ®Hom (VE , VF ) which are fixed under k ------+ AdG(k) ® TE(k-1)t ® TF(k).
Under relatively restricting conditions concerning the geometry and topology at infinity the Fredholmness and an index theorem are proved in  and . 2) One could generalize the notion of Fredholmness (using other operator algebras) and then establish a meaningful index theory with applications. The discussion of these both approaches will be the content of this section. e. how much the analytical properties of D differ from those of D'. This approach will be discussed in detail in chapter V.