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Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{i-1}=p_i\}$$ and, given a nilpotent matrix $N\in \mathcal{sl}_n,$ its associated $p$-partial Springer fibre $$\mathcal{B}_p^{N}=\{F\in \mathcal{B}_p \mid NF_i \subset F_{i-1}, \ i=1\dots n \}.$$ Define its natural extension $$ \widetilde{\mathcal{B}_p^{N}}:=\{F\in \mathcal{B}_p \mid NF_i \subset F_{i}, \ i=1\dots n \}.$$ The question: Is anything in general known about this variety, apart from (the obvious) that it is a projective variety?

In particular, analogously with known facts on ${\mathcal{B}_p^{N}},$ do we know anything about irreducible components of $\widetilde{\mathcal{B}_p^{N}}$, their dimensions, whether one can label them combinatorially etc?

NB When $p=(1,\dots,1)$ (ordinary case), these two varieties coincide $\widetilde{\mathcal{B}_p^{N}}=\mathcal{B}_p^{N},$ but not for general $p.$

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    $\begingroup$ One way I know how to combinatorially pave these varieties by affine spaces is as follows: we identify the partial flag variety as $G/P$ where $G=GL_n$ and $P$ is our standard block-wise upper triangular parabolic. We conjugate $N$ into some nice upper triangular form. Then we compute that every $B$-orbit in $G/P$ intersects the wanted closed subvariety ($\mathcal{B}_p^N$ or $\widehat{\mathcal{B}_p^N}$), and hopefully it is affine space of a combinatorially explicit dimension. $\endgroup$
    – Cheng-Chiang Tsai
    Commented Mar 29 at 4:14
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    $\begingroup$ (I think this method is in a paper of Tymoczko, though I don't know if that will be the earliest reference.) Though of course, affine paving might not be the same as knowing the irreducible components. $\endgroup$
    – Cheng-Chiang Tsai
    Commented Mar 29 at 4:15

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