Let $\mathscr{C}_1$ and $\mathscr{C}_2$ denote classes 1 and 2 respectively. Since $X \sim \text{Ga}(\alpha, \beta)$ you have:
$$\begin{equation} \begin{aligned}
\mathbb{E}(X | X \in \mathscr{C}_1) = \mathbb{E}(X | X < T)
&= \frac{\mathbb{E}(X \cdot \mathbb{I}(X < T))}{\mathbb{P}(X < T)} \\[6pt]
&= \frac{ \int_0^T x \cdot \text{Ga}(x|\alpha, \beta) dx }{ \int_0^T \text{Ga}(x|\alpha, \beta) dx } \\[6pt]
&= \frac{ \int_0^T x^\alpha e^{-x/\beta} dx }{ \int_0^T x^{\alpha-1} e^{-x/\beta} dx } \\[6pt]
&= \beta \cdot \frac{ \int_0^T (x/\beta)^\alpha e^{-x/\beta} dx }{ \int_0^T (x/\beta)^{\alpha-1} e^{-x/\beta} dx } \\[6pt]
&= \beta \cdot\frac{ \int_0^T y^\alpha e^{-y} dy }{ \int_0^T y^{\alpha-1} e^{-y} dy } \\[6pt]
&= \beta \cdot\frac{ \gamma(\alpha+1,T) }{ \gamma(\alpha,T) }, \\[6pt]
\end{aligned} \end{equation}$$
where $\gamma$ is the lower incomplete beta function. Similarly, you have:
$$\begin{equation} \begin{aligned}
\mathbb{E}(X | X \in \mathscr{C}_2) = \mathbb{E}(X | X \geqslant T)
&= \frac{\mathbb{E}(X \cdot \mathbb{I}(X \geqslant T))}{\mathbb{P}(X \geqslant T)} \\[6pt]
&= \frac{ \int_T^\infty x \cdot \text{Ga}(x|\alpha, \beta) dx }{ \int_T^\infty \text{Ga}(x|\alpha, \beta) dx } \\[6pt]
&= \frac{ \int_T^\infty x^\alpha e^{-x/\beta} dx }{ \int_T^\infty x^{\alpha-1} e^{-x/\beta} dx } \\[6pt]
&= \beta \cdot \frac{ \int_T^\infty (x/\beta)^\alpha e^{-x/\beta} dx }{ \int_T^\infty (x/\beta)^{\alpha-1} e^{-x/\beta} dx } \\[6pt]
&= \beta \cdot\frac{ \int_T^\infty y^\alpha e^{-y} dy }{ \int_T^\infty y^{\alpha-1} e^{-y} dy } \\[6pt]
&= \beta \cdot\frac{ \Gamma(\alpha+1,T) }{ \Gamma(\alpha,T) }, \\[6pt]
\end{aligned} \end{equation}$$
where $\Gamma$ is the upper incomplete gamma function.