What is the disk with three stops?

Disk with two stops, $k$ strands T-brane, has endomorphism algebra $NH_k$, with $q$ grading for crossing $q^{-2}$. Correspondingly, we have $(\pi: BB \to BG)_* \C_{BB}$, whose endomorphism involves $\pi^!$. Recall that for sheaves (not coherent sheaves), $\pi^! = \pi^* [\dim_\R fiber]$. This explains why we have those negative cohomological degrees.

Now we add another stop, say at the top. We will have $(k,0), (k-1,1), \cdots, (0,k)$ different types of $T$-branes.

We have a few observations:

1. $End(T_{(k_1, k_2)}) = NH_{k_1} \otimes NH_{k_2} =: NH_{k_1, k_2}. $. On the other side, we can consider $(BB \to BP_{k_1, k_2})_* \C$ whose endomorphism is also $NH_{k_1, k_2}$.