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Let G be a connected, real, semisimple Lie group contained in its complexification G C , and let K be a maximal compact subgroup of G. We construct a K C -G double coset domain in G C , and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L ∞ bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maaß forms.

1989 04 15: Relative rates of shell dissolution and net sediment accumulationa commentary: can shell beds form by the gradual accumulation of biogenic debris on the sea floor? Lethaia, Vol. 22, pp. 207-212. Oslo. ISSN 0024-1164. Rates of shell production rarely exceed 500 g CaCOI.m-2yr-' in clastic sediments. Loss of shell carbonate by dissolution greatly exceeds loss by bioerosion and abrasion in most habitats. Rates of shell dissolution in modern sediments, estimated from rates of organic carbon degradation or measured directly, usually exceed loo0 g CaC0,.m-2yr-1. This taphonomic loss is concentrated at or just below the sediment-water interface in the taphonomically-active zone (TAZ). Consequently, except where rates of shell production are very high or rates of organic carbon degradation very low, shells cannot permanently accumulate on the sea floor. Preservation requires rapid burial, usually by physical 'event' processes, to slow down taphonomic loss. Only near the base of the TAZ does the long-term sedimentation rate become an effective mediator of shell preservation as sediment accumulation gradually removes buried shell material from the taphonomically-active zone. 0 Shell dissolution, shell accumulation, taphonomy, sedimentation rate, shell bed, time aoeraging. Daoid J. Daoies,

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IntroductionThe q-invariant of a self-adjoint elliptic differential operator on a compact manifold X was introduced by Atiyah, Patodi and Singer [A-P-S], in connection with the index theorem for manifolds with boundary. It is a spectral invariant which measures the asymmetry of the spectrum Spec(A) of such an operator A. To define it, one starts by setting, for Re(s)~>0, ~Spec(A)-(O~ 121 sThis is a holomorphic function which can be meromorphically continued to IE. Indeed, from the identityand the asymptotic behaviour of the heat operator at t=0, it follows that ~/(s, A) admits a meromorphic extension to the whole s-plane, with at most simple dim X -k poles at s= , (k=0, I, 2 .... ) and locally computable residues. The ord A remarkable, and considerably more difficult to establish, fact is that s=0 is not a pole, and this makes it possible to define the r/-invariant of A by setting where [7] runs over the nontrivial conjugacy classes in F=nl(X ), l (7) is the length of the (unique) closed geodesic c~ in the free homotopy class corresponding to [7], m (7) is the multiplicit~r of c~, Ph (7) is the restriction of the linear Poincar6 map P(7)=d4~l at (c~, d~)eTX to the directions normal to the geodesic flow 4~ and ~ is the parallel translation around c~ on A~ + = +i eigenspace of an(O~), with a~ denoting the principal symbol of B. He then proves that (0.6) Z (s) admits a meromorphic continuation to the entire complex plane;(0.7) log Z(0) = n i qx; and (0.8) Z(s) satisfies the functional equation Z(s) Z(-s) = e 2 ~ i.x.The appropriate class of Riemannian manifolds for which a result of this type can be expected is that of non-positively curved locally symmetric manifolds, while the class of self-adjoint operators whose eta invariants are interesting to compute is that of Dirac-type operators, eventually with additional coefficients in locally flat bundles. It is the purpose of this paper to formulate and prove such an extension of Millson's formula.We shall now present our main results. Let X denote a compact oriented odd-dimensional locally symmetric manifold, whose simply connected cover is a symmetric space of noncompact type. Let D denote a generalized Dirac operator associated to a locally homogeneous Clifford bundle over X. The fixed point set of the geodesic flow, acting on the unit sphere bundle T 1X, is a disjoint union of submanifolds Xr, parametrized by the nontrivial conjugacy classes [;~] ~ 1 in F= nl (X). Each X~ is itself a (possibly fiat) locally symmetric manifold of nonpositive sectional curvature. We denote by g1(F) the set of those conjugacy classes [7] for which X~ has the properly that the Euclidean