When a character is mounted, his/her combat speed and hence number of action points per combat turn may be restricted.
If a mounted character's Riding (including relevant specialisms) is lower than his/her Combat (including combat speed specialism), his/her Riding (including relevant specialisms) is used instead of his/her Combat (including combat speed specialism) in the dice roll to determine how many action points he/she has for that combat turn.
For example, Cogan the barbarian has a Combat skill score of 6 1/2 with a +1 specialism bonus in combat speed, a Riding skill score of 2 with a +1 specialism in riding horses, and a CO score of +1/2. On foot, Cogan can choose to have a guaranteed number of action points of 8 (= 6 1/2 + 1 + 1/2); when riding a horse (in this case, a warhorse (+2 bonus to Riding dice rolls) equipped with saddle, harness, reins and stirrups (+1 bonus to Riding dice rolls)), Cogan is much less comfortable, and his guaranteed action points are only 6 1/2 (= 2 +1 + 1/2 + 2 +1).
Note that while both rider and mount are considered one figure for the purposes of position on the combat board, they are treated separately for their actions and phases. Hence, both mount and rider have separate action points for the combat turn. The mount's action points are used for movement actions and possibly for the mount's attacks (e.g. a warhorse kicking). The mounted character's action points are used for actions to control a mount plus attacks, blocks/parries and dodges.