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Spectral invariant of the zeta function of the Laplacian on \(Sp(r+1)/Sp(1)\times{}Sp(r)\). (English) Zbl 0765.58032

Let \(A\) be a self-adjoint elliptic pseudo-differential operator of order \(m>0\) acting on a compact \(n\)-dimensional manifold \(X\). Its zeta function \(\zeta(A,s)\) can be defined as \(\zeta(A,s)=\text{Trace} A^{- s}=\sum_{\lambda>0}\lambda^{-s}\) (the eigenvalues of \(A\) are \(\lambda>0)\). This series gives (for \(\text{Re} (s)>n/m)\) a holomorphic function of the complex variables. The author has given a general method of analytic continuation to compute the spectral invariant of the zeta function \(\zeta(\Delta,s)\) at \(s=0\) of the Laplace-Beltrami operator \(\Delta\) acting on 2-forms on the sphere \(S^{4r-1}\). The main purpose of the present paper is to compute the spectral invariant of the zeta function \(\zeta(\Delta,s)\) at \(s=0\) of \(\Delta\) acting on forms of degree 1 on the space \(Sp(r+1)/Sp(1)\times Sp(r)\). The main theorem contains an explicit form of the mentioned invariant.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
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