Find the maximum of the valence band and the minimum of the conduction band.
They are not at the same k-vector.
Holes and conduction electrons will be found near the maximum/minimum. In order for a hole/electron pair to recombine they must "get rid" of the difference in k. An optical photon carries next to no momentum, so some other elementary excitation like a phonon has to be created.
In GaAs on the other hand, min and max occur at the same k. Holes and electrons can recombine and emit just a photon.
The same hold vice-versa for absorption of a photon.
The "stretched silicon" invented in the 90's reduced the degree of indirectness of the gap and allowed for more efficient silicon switching speeds. The idea was to grow silicon layers on mismatched substrates that were hot. When the wafer cooled, the silicon was stretched into a slightly different lattice constant.
I suppose there should be some qualitative argument using e.g. tight binding approximation on how the valence band and conduction band change at k=0 with k using kp-perturbation theory. As both bent downward, the band gap must be indirect.
The valence and conduction band are qualitatively due to the s and p orbitals of Si. The s orbitals are lower in energy than p. On the Gamma point, the total bonding-antibonding splitting is larger for s than for p orbitals. Taken together, the lowest valence band has s-character and is well separated from the highest valence band which has pure p character. The mainly anti-bonding p and s type conduction bands are nearly degenerate in energy. Once the Gamma point is left, the kp term will mix s and p bands. The p type valence band and s-type conduction band will repell whence the valence band has a maximum at k=0. On the other hand the s and p type conduction bands will repell even stronger as they are nearly degenerate. Hence the lowest conduction band will also have a maximum at k=0. If we believe in k=0 being the absolute maximum of the valence band then Si has to have an indirect band gap.