Suppose the prior for $μ$ is $Gamma(k,λ)$, so $fμ(m)=Cm^{k−1}exp(−λm)I(m≥0)$.
Further, suppose the data $X_1,…,X_n$ given $μ$ is iid Poisson with parameter $μ$, so for each $[Xj∣μ=m]∼[X∣μ=m]$, $f_X∣μ=m(i)=exp(−m)m^i/i!I(i∈{0,1,…})$,
How can I show that the posterior for μ is also Gamma? Would the parameters of the Gamma be a function of $k$ and $\lambda$?
The prior is
$$\pi(\mu)\propto \mu^{k-1}e^{-\lambda\mu}$$
The likelihood is
$$p(\mathbf{x}|\mu)\propto e^{-n\mu}\mu^{\Sigma_i x_i} $$
Thus the posterior is
$$\pi(\mu|\mathbf{x})\propto \pi(\mu)\cdot p(\mathbf{x}|\mu) =\mu^{(k+\Sigma_i x_i)-1}e^{-(\lambda+n)\mu}$$
Thus the posterior is still gamma with parameters
$$Gamma(k+\Sigma_i x_i;\lambda+n)$$
Hint: to simplify the calculations it is suggested to waste any quantity that doesn't depend on $\mu$