  Let $G$ be a connected, simply connected, compact semisimple Lie group of
dimension $n$. It has been shown by Clerc \cite{Clerc1974} that, for any $f\in
L^1(G)$, the Bochner-Riesz mean $S_R^\delta(f)$ converges almost everywhere to
$f$, provided $\delta>(n-1)/2$. In this paper, we show that, at the critical
index $\delta=(n-1)/2$, there exists an $f\in L^1(G)$ such that
$$\limsup_{R\rightarrow\infty} \big|S_{R}^{(n-1)/2}(f)(x)\big|=\infty, \
\text{a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov
\cite{Kolmogoroff1923} for Fourier series on the circle, and a result of Stein
\cite{Stein1961} for Bochner-Riesz means on the tori $\mathbb T^{n}, n\geq 2$.
We also study localization properties of the Bochner-Riesz mean
$S_{R}^{(n-1)/2}(f)$ for $f\in L^1(G)$.

Classical Analysis and ODEs