The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists
θ
=
θ
(
G
)
(
3
⩽
θ
>
4
)
\theta = \theta (G)(3 \leqslant \theta > 4)
such that, if
χ
1
,
…
,
χ
N
{\chi _1}, \ldots ,{\chi _N}
are N distinct characters of G,
d
1
,
…
,
d
N
{d_1}, \ldots ,{d_N}
their dimensions,
c
1
,
…
,
c
N
{c_1}, \ldots ,{c_N}
complex numbers of modulus greater than or equal to one, then, for all
p
>
θ
,
|
|
|
Σ
j
=
1
N
c
j
d
j
χ
j
|
|
|
p
⩾
const
p
N
α
p
p > \theta ,|||\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}||{|_p} \geqslant \text {const}_p N^{\alpha _p}
where
|
|
|
⋅
|
|
|
p
||| \cdot ||{|_p}
denotes the
L
p
(
G
)
{L^p}(G)
convolutor norm and
const
p
\text {const}_p
and
α
p
=
α
p
(
G
)
{\alpha _p} = {\alpha _p}(G)
are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.