So, to build on my previous post about zeta functions and group actions, let me talk about a second application of linearly equivalent actions, this time in differential geometry.
So, pick your favorite compact Riemannian manifold , and consider the Laplacian . This is the negative of a symmetric elliptic differential operator, and thus has discrete spectrum, which each eigenvalue having finite multiplicity. Let be these eigenvalues (with multiplicity) in decreasing order. These eigenvalues are always negative (except for the constant function, an eigenvector for 0), since the Laplacian of a function is always negative at local maxima, and positive at local minima.
One way of packaging these eigenvalues is in the heat trace , which is the trace of the heat kernel $\eta(t)$ on . You can think of this as a bit like the zeta function of the manifold.
To me, the best way of thinking about these is as stationary solutions to the heat equation. Eigenvectors are ways of distributing heat such that over time, we just get an even decay to 0 (i.e., the distribution of heat over time is of the form ).
Now, if one takes a Galois cover of (in this context, “Galois” means that the group of deck transformations acts transitively on the pre-image of a point in ), then we have a finite group acting freely on with quotient $X$. Thus, if we take the Hilbert space , then acts on here and this action commutes with the Laplacian. Thus, each eigenspace for the Laplacian is invariant under this -action, and is the sum of its intersections with isotypic components for that action of .
That is, if ranges over the irreducible representationsof , then we get a bunch of commuting projections to the isotypic components , satisfying . Since , we get an identity for in terms of the components .
Of course, . Since each eigenspace intersected with an isotypic component is a direct sum of the same representation a number of times, applying invariants simply multiplies its contribution by . Applying this to each isotypic component, we get the formula
So, how do these multiplicities depend on the subgroup ? Well, Frobenius reciprocity tells us that , so these are precisely the multiplicities appearing in the permutation representation.
Thus, just as before, we get the
Theorem. (Sunada, 1986) The quotients of by two subgroups will be isospectral (i.e. have the same Laplacian spectrum) if .
Now, this may seem like a rather trivial observation, but it came after people had been thinking about isospectral manifolds for over 20 years. I think it’s a great example of a very simple observation in one field seeming very interesting in another.