solid state physics - Why does AC Conductivty in Drude Theory have an Imaginary and a Real Part?

In the Drude Model the direct current (DC) conductivity is given by the following formula:

$$\sigma_0=\frac{ne^2 \tau}{m}$$ where $\tau$ is the relaxation time.

Furthermore, the AC conductivity in the Drude Model is given by this formula:

$$\sigma(\omega)=\frac{\sigma_0}{1-i\omega \tau}$$

one can see that for zero frequency the AC conductivity yields the DC conductivity.

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In this paper by Scheffler et al., the frequency-dependent microwave conductivity of the heavy fermion metal $UPd_2Al_3$ is examied. The following graph is taken from their paper:

Why are there two different conductivities? The authors seem to have separated the solution into an imaginary and a real part but I don't understand why the conductivity should have an imaginary part in the first place. Could someone explain to me why there are two curves and not just one?