I think I understand formula 3.71: paying cash dividend $\kappa_j$ at time $T_j$ will cause the stock price to drop by the same amount immediately, as is reflected in the subtraction from the firm capital value $G_t$ by $D_t$ in the stock price. (I understand the capital value as the stock price that would have been if no past cash dividends had been issued.)
But what does the second formula (3.72) mean? To me, $\tilde D_t$ means the present value (at $t$) of all future dividends to be paid after $t$. But what does the sum $S_t=G_t+\tilde D_t$ mean? How can the stock price exceed the capital value of the firm? Also, as opposed to $G_0=S_0$ implied by the first formula, the second formula implies that $G_T=S_T$, how is it reasonable? It looks as if by the expiry, the dividends paid will have had no impact on the stock price at all.