we have $(1-\xi^{-ip^s}g^{p^s})e_i=\frac{1}{t}[1-(\xi^{-ip^s}g^{p^s})^t]=0$, that is, $g^{p^s}e_i=\xi^{ip^s}e_i$. Hence $\{g^{i_1}e_i|0\leqslant
i_1\leqslant p^s-1\}$ is a basis of $kGe_i$ and dim$kGe_i=p^s$. Under this basis, the matrix of the action of $g$ on $kGe_i$ is $$\left(\begin{array}{ccccc}
0&0&\cdots&0&\xi^{ip^s}\\
1&0&\cdots&0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&\cdots&0&0\\
0&0&\cdots&1&0
\end{array}\right)_{p^s\times p^s}.$$ The characteristic polynomial of $g$ is $p(x)=x^{p^s}-\xi^{ip^s}=(x-\xi^i)^{p^s}$. Acting on $kGe_i$, $g$ has a unique eigenvalue $\xi^i$ with multiplicity $p^s$. By Lemma \[2.2\], $g^p\in Z(H(\lambda,\mu))$, the center of $H(\lambda,\mu)$. Hence $\{e_0, e_1, \cdot, e_{t-1}\}$ is a set of central orthogonal idempotents of $H(\lambda, \mu)$. It follows that $H(\lambda,\mu)=\bigoplus_{0\leqslant i\leqslant
t-1}H(\lambda,\mu)e_i$ is a decomposition of the left regular module $H(\lambda,\mu)$, which is also a composition of $H(\lambda, \mu)$ as two-sided ideals. Thus, the action of $g$ on $H(\lambda,\mu)e_i$ has the unique eigenvalue $\xi^i$ (with multiplicity of $p^{s+2}$). So $g$ has the unique eigenvalue $\xi^i$ when it acts on every principal projective module occurring in $H(\lambda,\mu)e_i$.

Note that dim$H(\lambda, \mu)$=dim$(\mathcal{B}(V)\# kG)=p^2n=p^{s+2}t$ and $$H(\lambda, \mu)e_i={\rm span}\{g^{i_1}a^{i_2}b^{i_3}e_i|
0\leqslant i_1\leqslant p^s-1, 0\leqslant i_2, i_3\leqslant
p-1\}.$$ Hence dim$H(\lambda, \mu)e_i=p^{s+2}$ and $\{g^{i_1}a^{i_2}b^{i_3}e_i|
0\leqslant i_1\leqslant p^s-1, 0\leqslant i_2, i_3\leqslant p-1\}$ is a basis of $H(\lambda,\mu)e_i$.

Now we can prove the main results of this section.

\[3.8\] Let $\{T_0, T_1, \cdots, T_{t-1}\}$ be the complete set of non-isomorphic simple $H(0,\mu)$-modules given in 