## 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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