irreducible representations of the Weyl algebras

By translating the irreducible representation $\Phi$ in the previous section, we obtain a family of irreducible representations.

LEMMA 7.5   Let $k$ be a field of characteristic $p\neq 0$ . Let $n$ be a positive integer. Let $c\in k^{2 n}$ . Then there is a finite dimensional representation $\Phi_{c}$ of $A_n(k)$ on $k[x_1,x_2,\dots,x_n]/(x_1^p,x_2^p,\dots,x_n^p)$ defined as follows.

$$\Phi(\gamma_i) f=(x_i+c_i) f, \qquad \Phi(\gamma_{n+i}) f=(\partial_i+c_{n+i}) \cdot f \qquad (i=1,2,\dots,n)$$

Then from what we have shown in the previous section, we obtain the following results.

LEMMA 7.6   Let $k,c, \Phi_{c}$ as in Lemma above. Then:

1. The kernel of $\Phi_c$ is given as
   $$\operatorname{Ker}(\Phi_{c})=\mathfrak{a}_{c}= (\gamma_1^p-c_1^p,\gamma_2^p-c_2^p,\gamma_3^p-c_3^p,\dots, \gamma_{2 n}^p-c_{2 n}^p).$$

2. $\Phi_{c}$ gives rise to a $k$ -algebra isomorphism
   $$\overline{\Phi_{c}}: A_n(k)/\mathfrak{a}_{c} \cong M_{p^n}(k).$$

3. $\Phi_{c}$ is an irreducible representation of $A_n(k)$ .