We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the $c$-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set $c$-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed $cge 1$ on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time ($n^{O(c)}$) algorithm for all fixed values of $c$, except $c=1$, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that $c$-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by $c$ and the length of the solution, as well as a tight ETH-based lower bound for both parameters.
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