# Components of Springer fibers, category O, and Khovanov’s “functor valued invariant of tangles”

I’ll just note at the beginning, this post is a bit of an experiment. At this point, it is about a semi-finished research thought of mine (which I’m not 100% sure is original, but I’m putting it out on the internet at least in part hoping that the internet will be able to tell me whether it’s original or not), and will consequently probably be a bit more technical than the average post on this blog, but hopefully, at least a few of you will be able to follow me.

As many of you know, my co-blogger Joel recently posted a preprint (with Sabin Cautis), which constructs a knot homology theory using the geometry of coherent s heaves and Fourier-Mukai transforms on convolutions of minuscule orbits in the affine Grasmannian of $SL_2$.

On the other hand, last year, Catharina Stroppel published a couple of papers on the relationship between Khovanov’s original construction of “a functor valued invariant of tangles” and various flavors of category O. From what I understand, underlying this is a philosophy that the $\mathfrak{sl}_n$ version of Khovanov-Rozansky will be related a block to category O that lies on a dimension $n-1$-dimensional wall of the Weyl chamber of $\mathfrak {sl}_d$ (where $d$ is a number relating to the number of strands in your tangle diagram).

One natural question leaps to mind: how are these related?

Underlying each of these categorifications of knot invariants is a categorification of the dth tensor power of the standard representation of $SL_2$. C&K use the category of coherent sheaves on the aforementioned convolution product of d copies of the minuscule stratum in the affine Grassmannian of $SL_2$, and Stroppel uses a direct sum of the principal blocks of parabolic category $\mathcal O$ for all maximal parabolics of $\mathfrak{sl}_d$.

Now, one obvious way these ideas could relate to each other is that these could be “the same” categorification, that is, we could find an equivalence between these categories (or their derived categories, more likely). Unfortunately, this is just not the case. By definition, Stroppel’s category has a block for each weight space of the representation, and C&K’s category is irreducible (as a rule, the categories of coherent sheaves on varieties tend to be), so they can’t possibly be equivalent.

But what they could be is independent generalizations of Khovanov’s original invariant, since in both constructions, since that only depended on the 0 weight space of the $\mathfrak{sl}_2$ representation. Perhaps a smaller part of these categories is equivalent. I believe this to be the case, at least when $d=2m$ is even; allow me to explain why.

Now, when I talked to Joel about this a few weeks ago, he suggested the connection would go through a certain subvariety of the convolution variety under consideration, which I’ll call “the Springer fiber” to spare myself trouble. This variety is indeed a Springer fiber, for a nilpotent of Jordan type $(m,m)$, and in fact an open subset of the convolution variety is the unique crepant resolution to the Slodowy slice transverse to that nilpotent orbit (if that means anything to you).

This is a reducible projective variety, its components are all smooth and have smooth pairwise intersection (which is not true of all Springer fibers), and they are indexed by Temperley-Lieb diagrams (though people are more likely to say “standard Young tableau of shape $(m,m)$“). A more remarkable fact is the following:

Let $E\cong \oplus_{C}\mathcal{O}_C$ be the direct sum of the structure sheaves of these components.

Theorem. The Ext-algebra $\mathrm{Ext}^\bullet(E,E)$ is naturally isomorphic to Khovanov’s algebra $H^n$.

I don’t think this theorem is in the literature (yet?), and I’m actually not 100% sure who to ascribe it to. It might be in the paper that Khovanov claimed 5 years ago in this paper that he and Paul Seidel were writing, and never published (hey, they’re busy guys), and it does seem the important ideas for the theorem above are due to Khovanov, even if he never wrote them down quite like that.

But that’s not the only place that $H^n$ has shown up. Let $\mathfrak{S}$ be the set of permutations on 2m letters whose insertion symbol (i.e. Q-symbol) is the standard tableau of shape $(m,m)$ whose lower row is $1,\ldots, m$ (these are also in bijection with standard tableau of that shape by Robinson-Schensted). Let $\lambda=\sum_{i=1}^m-\omega_i$ be minus the sum of the first m fundamental weights of $\mathfrak{sl}_{2m}$, and let $L(\mu)$ be the unique irreducible representation of $\mathfrak{sl}_{2m}$ with highest weight $\mu$. Let $F\cong\oplus_{\sigma\in\mathfrak{S}}L(\sigma(\lambda+\rho)-\rho)$.

Theorem. (Stroppel, Khovanov) The Ext-algebra $\mathrm{Ext}^\bullet(F,F)$ is naturally isomorphic to $H^n$.

Well, this is a pretty interesting statement. It certainly suggests that something is going on here, but as many people will attest, there is more information in a category than the Ext algebras in question. You also need to know about the $A_\infty$ structures on these algebras. If the natural $A_\infty$-algebras on these algebras coincide, THEN you are cooking with gas.

Theorem? The triangulated subcategories of the derived category of coherent sheaves on the convolution variety and of category $\mathcal{O}$ generated by the collections $\{\mathcal{O}_C\}$ (where $C$ ranges over components of the Springer fiber) and by $\{L(\sigma(\lambda+\rho)-\rho)\}$, where $\sigma$ ranges over $\mathfrak{S}$ are isomorphic, with an isomorphism preserving the t-structure, and taking one collection to the other.

A suggestion of a proof in a moment. Though first I’ll ask a question: are Koszul rings intrinsically formal? This would allow us to conclude instantly that don’t these structures are formal, and we would be done.

We already know (by work of I. Frenkel, Khovanov and Stroppel on the algebraic side, I’m not so sure about the geometric) that after applying Grothendieck group, these collections will generate the $\mathfrak{sl}_2$ invariants of our representation (the 2m-fold tensor product of the standard), and in fact, their classes are exactly the canonical basis (or maybe dual canonical; I’ve applied dualities too many times to feel sure), so this would be a not so surprising equivalence, but still pretty cool.

Proof? So, what we have to show is that these Ext-algebras have the same $A_\infty$ structure. I claim they are both formal (higher products all vanish). This is easy for category $\mathcal{O}$. By work of Soergel, any block of category $\mathcal{O}$ has a graded version which is equivalent to the category of graded modules over a certain finite dimensional Koszul algebra, and simple objects correspond to pure ones under this correspondence. Thus, if we take a minimal projective resolution of the image of $F$ in this category, all its differentials will be of positive degree, and will be killed by applying $Hom$ to this same object. Thus, $\mathrm{Ext}^\bullet(F,F)$ is indeed formal.

The geometric side of things is harder. The gist is as follows: the components of the Springer fiber aren’t completely random varieties. Indeed, they’re 1) holomorphic Lagrangian and 2) complete intersections, i.e. for each there’s a vector bundle of rank m (actually, the sum of m line bundles) and a section of said bundle which the component is the vanishing set of.

Thus, there is a vector bundle $V_\pi$ for each standard tableau of the right shape, and a locally free resolution of $\mathcal{O}_{C_\pi}$ of the form

$\cdots\to\wedge^2 V_\pi\to V_\pi\to\mathcal{O}_X$.

Calculating the Ext’s between these, we can take $\mathrm{Hom}$ of the sum of these resolutions to $E$, and get summands of the form

$\cdots\leftarrow\wedge^2 V_\pi|_{C_\pi'}\leftarrow V_\pi|_{C_{\pi'}}\leftarrow\mathcal{O}_{C_{\pi'}}$.

Now, on $D=C_{\pi}\cap C_{\pi'}$,the bundle $V_\pi$ is isomorphic to $T_{C_{\pi}}^*$ and thus, a splitting as $T^*_{D}\oplus N^*_{D/C_{\pi'}}$, with the differential vanishing on the first summand, and on the latter giving a locally free resolution of $\mathcal{O}_D$ as a $\mathcal{O}_{C_{\pi'}}$-module. Thus, we have a surjection of sheaves of dg-algebras from a locally free model for the sheaf Ext algebra $\mathcal{E}xt^\bullet(E,E)$ to sheaf of dg-algebras with no differential, which is a direct sum of sheaves of the form $\oplus_{i=1}^m\wedge^i T^*D_{\pi,\pi'}$, which have a flabby resolution by the Doulbeaut complex. Hodge theory shows that these summands are themselves formal (each $D_{\pi,\pi'}$ is projective and thus Kähler). Q. E. D.?

Well, congrats to those of you who made it to the bottom. I hope some part of the above was illuminating.

## 11 thoughts on “Components of Springer fibers, category O, and Khovanov’s “functor valued invariant of tangles””

1. Thanks. That’s an excellent summary.

I was confused about Koszul duality for a moment, but I see that you have cleverly switched on the category O side to working with the singular blocks, and gone from projectives to irreducibles.

I think that most of what you wrote is “known to experts” (where experts = me, Sabin, Stroppel, Khovanov, Seidel, Roman, etc.). Perhaps other people already know of your last argument, though I haven’t seen it expressed exactly that way before. Roman once gave me a little different (more local) argument for this same fact. On the symplectic side, Denis once outlined for me an argument showing that the higher products vanished in HF(C_1, C_2). I don’t believe that there are any more literature references than the ones you give — but you should consult Roman to see if what he and Rina are doing which contains some of this.

One question for you, since I was thinking about it today: what can you say about Ext(O_{C_1}, O_{C_2}) if the intersection is not smooth?

2. Allen Knutson says:

Not to sidetrack too much, but what’s the relation between the Springer fiber with its lovely component structure you describe, and the orbital scheme of {M^2=0}, whose components are also indexed by TL-diagrams?

3. There’s always a bijection between the components of the Springer fiber and the orbital variety (in type A, I think it’s more complicated in other types), which can be realized as followed. Look at the closure of your favorite nilpotent orbit (let’s pretend we’re Brian Rothbach, and say that’s {M^2=0}), and take its preimage under the Springer resolution. Since we’re in type A, the components of this preimage are in bijection with components of the Springer fiber (if we were in other type, we would have to worry about monodromy). On the other hand, this variety has a map to the flag variety (which must be surjective and a fiber bundle, since it’s a G-invariant subvariety of $T^*G/B$), whose fiber is the orbital variety for that nilpotent orbit. Our funny variety also must have components in bijection with the orbital variety, since G/B is simply connected.

That’s about all I know. Melnikov seems to have done some stuff about the exact connection between Springer fibers and orbital varieties (understanding intersections of components, and stuff like that).

4. Here is a more relevant version of Allen’s question:

We have bijections (in this case)
(component of springer fibre)
TL diagrams
a subset of the Schubert varieties in G/P

Can you geometrically describe this composition? (Roman asked me this question once).

I’m thinking of this subset of Schubert varieties from the “parabolic categories” version of things. These Schubert varieties are those such that the corresponding projective modules are PIMS.

5. Anonymous says:

What’s {M^2 = 0}?

6. Of course, it depends a bit on what you mean by “geometrically describe,” but I would say yes.

1) The bijection between orbital varieties for M^2=0 and projective-injective perverse sheaves is as follows.

Consider the lowest coset representatives for $W/W_P$ as a subset of G. Let $\mathfrak{n}_x$ be the stabilizer of $xP\in G/P$ in $\mathfrak{n}$, and $\mathfrak{m}_x$ be the stabilizer of $Bx\in B\backslash G$. Then:

1) the indecomposable projective perverse sheaf whose head is $\mathbf{IC}(BxP)$ is also injective if and only if $B\cdot \mathfrak{m}_x=B\cdot \mathfrak{m}_{w_0}$, and the map sending such a sheaf to the orbital variety $\overline{B\cdot \mathfrak{n}_x}$ is a bijection between P-I perverse sheaves constant on the Schubert stratification and orbital varieties of {M^2=0}.

This is just combining classical results of Irving and (I think) Springer.

To get a component of a Springer fiber, consider said orbital variety as a subset of $T^*_{eB}G/B$, and take it’s $G$-orbit. This should intersect any Springer fiber over a nilpotent of $(m,m)$ type in a single component.

The correspondence to TL diagrams is less geometric. One way to think of it is as the $P$-symbol of the lowest coset representative of $W/W_P$, using the bijection between standard tableau of shape $(m,m)$ and crossingless matchings (the bottom row is the left end of the arcs, and the bottom row is the right end).

Alternatively, using work of Fung, the components of the Springer fiber can be explicitly in terms of inclusion relations on flags. I’m not sure if you can really make it more geometric than that.

7. Anonymous-

{M^2=0} is a convenient shorthand for the set of matrices which square to 0, though we’ve been talking about orbital varieties, so we’ve mostly meant the set of upper triangular matrices which square to 0. This is a reducible algebraic variety, and it’s components have a natural bijection with TL diagrams.

8. Allen-

I just realized I was being stupid, and answered your question for the transpose of the Young diagram you asked it about. I have no idea what transposition means for orbital varieties, except for some funny business about commuting unipotents.

9. One question for you, since I was thinking about it today: what can you say about Ext(O_{C_1}, O_{C_2}) if the intersection is not smooth?

My general suspicion is that it should be a fucking disaster. There is a sensible conjecture about which Ext between simples in category $\mathcal{O}$ it should be, but I have no idea how one would go about proving that.

10. Matt says:

Hi,
I do not believe that Koszul rings are automatically intrinsically formal. Consider $\bigwedge V$. Then $HH^d(\bigwedge V,\bigwedge V) \cong \text{Sym}^d V^* \otimes \bigwedge V$. The piece corresponding to $HH^{d,2-d}$ is then $\text{Sym}^d V^* \otimes \bigwedge^2 V$.