The character used two-hand weapons with one hand while on horseback, so it reduces speed and damage.
My current secondary weapon is balanced sword, but sabre or broad sword is good too. I used to try SOW or great axe. They are good too, but I now prefer speed than damage.
So, here's possible secondary weapon choice.
1. Spear and pike: Awlpike and Pike give you good range, reasonable speed and damage as well as be able to do couching. Spears, when hit, always rare the horse up, so it saves your skin when you're de-horsed. It also keep enemy at bay if you used correctly. Drawback is it's rather hard to hit when you're attacking sideways and it's a bit slow compared with other weapons. If you're in to lancing (the deadliest of all attacks), that's your choice.
2. Two-handed weapon: Sword of War and Great Axe are the best choices in this type. They gave more damage and have long range which helps a lot if you're fighting against cavalry equipped with lance/spear (you can attack without the fear of getting couch-lanced) and while on foot. Drawback is since you can use only one hand while you're on horse back, it's sometimes too slow. If you want maximum damage with reasonable speed and range, they're the one.
3. One-handed weapons: Sword, Sabre, Nomad-Sabre and Broad Sword are the best choices. They're quick and have reasonable damage. Drawback is the range. Since they're much shorter than spear, lance and other two-handed weapons, you need to have some skills to get within their range and you're always in danger of getting hit first. When you're un-horsed, you're in a kind of trouble. It is because a horse can 'hit' you when its nose touches you, while you have to strike its chest to 'hit'. that is not a big problem with longer weapon but for one-hand weapon user, it's a kind of nightmare. If you want decent damage with fast attack speed and more than simple bashing, this is your choice.