When an electron is excited from valence band to conduction band, it has to have a finite momentum in the case of indirect bandgap. Does that mean that the electron cannot be created at rest ? does it have a non zero "speed" ?
It means that to fulfill momentum conservation law, the momentum should be taken from somewhere: from phonons, impurities, etc.
The speed is a derivative of the energy (remember Hamilton's equations), thus the speed is zero at the minimum of the energy, wherever this point is located in k space