Pumping lemma regarding {a^2k w | w ∈ {a, b}*, |w| = k}
In typical usage of the pumping lemma, you assume that the word you chose for pumping is written as $w=xyz$, with $x,y,z$ satisfying the conditions of the pumping lemma.

Then, in most cases, you reach a contradiction by saying that e.g., $xy^2z$ is not in the language. This is "pumping up", since you "add" copies of $y$.

In this case, however, you reach a contradiction by looking at the word $xz$ (i.e., $xy^0z$). So you are pumping "down", or subtracting copies of $y$.

The reason for this is that if you look at e.g., $xy^2z$, then you get a word of the form $a^{2p+k}b^p$, but this word might be written as $a^{2r} a^lb^p$ with $l+p=r$, so you would not reach a contradiction.