I'm trying to read a paper on Random walks on compact lie groups here. On Pg. 28, one of the key Lemmas that I'm interested in makes use of a specific subset $\mathcal{H}_r \subset L^2(G)$ and then proceeds to construct an identity on it. For the purposes of my research, it is important that I understand what this space is, its properties, more examples and preferably a thorough study of it (if any). I was able to find any reference about these functions. Could any one please provide me any good references on $\mathcal{H}_r$ spaces?
The definition of $\mathcal{H}_r$ from the paper is the following:
If $G$ is a compact connected semi-simple Lie group, the space $L^2(G)$ can be decomposed as an orthogonal sum of finite dimensional irreducible representations. We write $\mathcal{H}_r\subset L^2(G)$ for the sum of those constituents which have highest weight $v$ with $|v|\le r$.
Thanks in advance!