## Materials related to “Some aspects of harmonic analysis on locally symmetric spaces related to real-form embeddings”

Mar 17th, 2011 by Eliot P. Brenner

This article, soon to be published by Springer in the *Serge Lang Memorial Volume* cites the following:

Section 7 of Preprint “Artin Formalism, for Kleinian Groups, via Heat Kernel Methods”:

This interactive graphic of the fundamental domain studied in Section 2 of the paper:

README for the graphics:

- To run: download onto a Windows computer (tested on Windows 7). Run .exe. Sorry, no binary yet for linux. Coming soon! (Write me if you want the source, which required openGL and GLUT libraries for C++ to compile).
- General description: axes x,y,z and real, imaginary, plus j (quaternionic) in upper-half-space model of the hyperbolic 3-space. The fundamental domain F(G) in the paper, is the region INSIDE the moccasin-colored triangular prism (going up to infinity, although it is only shown to a finite height of about 20) and OUTSIDE the red-colored sphere (centered at z=1 and of radius square root 2).
- Controls:

- Mouse: primary-key clicking with shift button depressed changes the viewpoint (the point the observer is looking at), without shift button the standpoint (the point the observer’s eye is located). Secondary key closes the graphics window (exits the app).
- Arrow keys: up-down arrows with shift button depressed moves the standpoint into/out of screen, without shift button moves the viewpoint into/out of screen. Left/right rotates the “up” vector.

Coming soon: text labels.