In the first part of the talk, I will outline a new approach to the theory of spectral zeta functions developed in the work of Jorgenson-Lang. Rather than precisely defining spectral zeta functions and the other objects involved, I will explain the methods by analogy with the classical derivation of the basic properties (meromorphic continuation, functional equation (f.e.)) of the Riemann zeta function from the Poisson Summation Formula.
The spectral zeta functions we consider are functions associated to an arithmetic locally symmetric spaces Gamma\G/K in the same way as the Riemann zeta function is associated with circle. Jorgenson-Lang turn around the classical approach to spectral zeta functions, in which the zeta function itself or its Euler product is viewed as the fundamental object. Instead they view the heat kernel of the Laplacian as the fundamental object. The first effect of this change of focus is to bring to prominence the spectral expansion of the heat kernel. We state the conjectured expansion in the case of
(1) G_n=SL(n,C), Gamma_n=SL(n,Z[i]), K_n=SU(n),
which is currently a theorem for the case n=2. In this case (n=2) there is a further theorem describing how to obtain the spectral zeta function from the expansion of the heatkernel.
In the second part of the talk, I will talk about ladders, which are structures similar to the sequence of locally symmetric spaces Gamma_n\G_n/K_n with their associated zeta functions, as in (1) above. I will give several diverse examples. Ladders enter in the program in two (related) ways. The first way is related to the appearance of smaller dimensional groups, roughly speaking of the same type as G, in the spectral expansion, in the form of the parabolic subgroups of G. Thus the ladder is the proper structural context in which to understand the spectral expansions of Gaussians such as the heat kernel. The second way is that the ladders (conjecturally) provide an alternate route of verifying the spectral expansions of the heat kernels. I will explain how this alternate route leads to a consideration of an analytic model of the space Gamma\G/K called a fundamental domain.
We must construct analytic models of the locally symmetric spaces that behave nicely with respect to normal projections. We propose for these models certain fundamental domains for the action of Gamma on G/K which we call Dirichlet domains at infinity. Instead of defining the Dirichlet domains at infinity rigorously, I will indicate how the axioms for the construction of the classical Dirichlet domain of a discrete subgroup of SL(2,R) acting on the upper half-plane can be modified to give the construction of the Dirichlet domain at infinity. The talk will conclude with a statement of my new results, which are explicit computations of the Dirichlet domains at infinity for SO(3,Z[i]) and (SO(2,1))(Z) acting on the upper half-space and upper half-plane, respectively. I will display pictures of the (infinite but finite volume) convex polytopes in question.