# Who was Peter?

The fundamental fact about the representation theory of a compact topological group $G$ is the Peter-Weyl Theorem, which can be described as follows: the regular representation of $G$ on $L^2(G)$ (defined using the probability Haar-measure on $G$) decomposes as an orthogonal Hilbert direct sum of the spaces of matrix coefficients of the finite-dimensional irreducible representations of $G$. (Books, for instance the one of Knapp on semisimple Lie groups, often include more in what is called Peter-Weyl Theory, but this is the statement that is proved in the original paper.)

As I am currently preparing to write down a proof of the Peter-Weyl theorem for my notes on representation theory, I had a look at this paper. Although I probably won’t follow it quite completely, I found it very interesting — it is quite subtly different from all modern treatments I have seen, in an interesting way, and without being much more complicated than what is found, e.g., in Knapp’s book. (For a short masterful online account, this post of Terry Tao is very good; from a search in the same Göttingen archive web site, it seems that — maybe — the modern treatment dates back to Pontryagin, in 1936.)

In any case, the question for today is: Who was Peter? Only the initial “F.” and the affiliation “in Karlsruhe” identifies this coauthor on the original paper (even the “F.” is misread as “P.” on the PDF cover page…) It seem he was a student of Weyl, but note that there is no Peter on the math genealogy page for Weyl. (A joke here at ETH, is that when the lecture room known as the Hermann Weyl Zimmer is renovated this summer, some unfortunate skeletons will be found in the closets and under the floor, or behind the blackboards…)