I have to look for a formula which gives the number of i-dimensional singular subspaces of a polar space of order $(s,t)$ and rank $n$. My idea was to prove this by induction on $i$. So I wanted to start with calculating the number of points. Here my idea was to use a double counting on the pairs (point, line through that point) but even this did not work. Anyone an idea please how to calculate the number of points in a polar space?
Further I would like to do induction on $i$, but I'm stuck how to calcuate het number of $(i+1)$-dimensional singular subspaces given the number of $i$-dimensional singular subspaces.Can someone help me?
Thanks in advance! I have been searching on it for a very long while so I would be very grateful.