Here is a simple idea.
The origin of Langlands’ program is somewhere in trying to correlate the spectral data of Laplace operators on noncompact arithmetic spaces \(\Gamma \backslash X\) analogous to \(GL({\bf{Z}}^2) \backslash PGL({\bf{R}}^2)\) with spectra of various “dual” spaces. Compare J. Arthurs’ comment on Langlands and relative trace formulas (Arthur 2005, 2007). The dominant object is the spectrum of the Laplacian \(\Delta\) on the noncompact symmetric space \(\Gamma \backslash X\). Historically, the continuous part of the spectrum was discovered via Eisenstein series. Then the “discrete part” of the spectrum was defined by somehow “subtracting” the continuous part. Yet we find this procedure unconvincing. In a certain physical sense, a noncompact open space does not have any resonant frequencies because there is no boundary or reflection from which waves can “bounce back”. I.e. the noncompact space has no periodicity and no resonant frequencies.
The basic idea in our development is the maximal \(\bf{Q}\)-rational Borel-Serre bordification \(X[t]\) of \(X\). See (Borel and Serre 1973). Our construction of \(X[t]\) via excisions of rational horoballs at-infinity implies \(X[t] \subset X\) has a \(\Gamma\)-equivariant proper discontinuous boundary \(Y := \partial X[t]\). The key property of the rational bordification is that the reduced integral homology of \(Y\) when considered as a \(\bf{Z} \Gamma\)-module is the Bieri-Eckmann homological dualizing module.
Our main proposal is to replace the spectrum of the Laplacian on noncompact \(\Gamma \backslash X\) with the spectrum of the Steklov operator on the bordification \(\Gamma \backslash X[t]\). In this way all the technical difficulties of trying to separate the continuous and discrete parts of the spectra are avoided, and we have as foundation the discrete Steklov spectrum on \(X[t]\).
This idea naturally suggests a Steklov-Selberg trace formula on \(X[t]\) analogous to (Selberg 1956).
Claim: There exists a Poisson kernel \(P(x,y)\) and Poisson integral formula describing all harmonic extensions \(\hat{f}\) of bounded functions \(f\) defined on \(Y:=\partial X[t]\). The Steklov operator, also called the Dirichlet-to-Neumann map in the literature, is obtained by differentiation, and we have \[Sf(x):=\int_Y f(y) \frac{\partial P}{\partial \nu}(x,y) dy\] for all \(x\in X[t]\).
Claim: The Steklov operator \(S\) has discrete countable spectrum and the spectral trace defined by the sum of eigenvalues \(\sum \lambda\) converges absolutely.
Claim: The kernel \(k(x,y):= \frac{\partial P}{\partial \nu}(x,y)\) is absolutely integrable modulo \(\Gamma\) and the geometric trace defined by integration along the diagonal \(\int_Y k(y,y) dy\) converges absolutely modulo \(\Gamma\).
Claim: The spectral trace is equal to the geometric trace, and the geometric trace can be decomposed into a recursive sum of orbital-type integrals over conjugacy classes following Selberg.
[To Be Continued - JHM]