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Minakshisundaram–Pleijel coefficients for non-compact higher rank symmetric spaces

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Abstract

We assign to a non-compact Riemannian symmetric space X a theta function and a zeta function. We compute all of the Minakshisundaram–Pleijel coefficients in a short-time asymptotic expansion of the theta function especially when X is of complex type, which we use to compute the one-loop effective potential—whose relevance for quantum field theory, for example, is briefly commented on. These coefficients, for X of general type, are also shown to be specifically related to residues and special values of the zeta function. The results presented extend previous ones in the literature where the rank of X is assumed to be one.

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Acknowledgements

Special thanks with appreciation is extended to Mrs. Yaping Yuan for her assistance and outstanding work in preparing the manuscript. The author also thanks the referee for a careful reading and review of the initial version of the manuscript and for suggested improvements thereof.

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  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Floyd L. Williams

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  1. Floyd L. Williams
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Correspondence to Floyd L. Williams.

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Williams, F.L. Minakshisundaram–Pleijel coefficients for non-compact higher rank symmetric spaces. Anal.Math.Phys. 10, 52 (2020). https://doi.org/10.1007/s13324-020-00396-x

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