Abstract
In this note, we extend the Hirzebruch proportionality principle to the coefficients in the asymptotic expansions for the Laplacians on differential forms with values in homogeneous vector bundles over symmetric spaces. The case zero-forms and the trivial line bundle is a proportionality principle for the trace of the heat kernel. For 2m-dimensional manifolds, the case of m-th order terms of some asymptotic expansions is a proportionality principle for the indices of certain elliptic complexes. These results have implications for the volumes of fundamental domains of discrete subgroups, and in refining these implications we also develop a proportionality principle for equivariant characteristic classes.
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