# small maps, extension of IC sheaves and BM homology

This is a question very much related to the one here extensions of IC sheaves I didn't know if I should ask a new question or edit the old one which wasn't very precise...

Homology means Borel-Moore homology.

Let $\pi:X'\to X$ be a proper small map (i.e. $X'\times_X X'$ is of dimension $n=\dim(X)$ and every irreducible component of $X'\times_X X'$ of dim $n$ maps to a dense subset of $X$) between smooth algebraic varieties. The sheaf $\mathcal{F} = R\pi_*\mathbb{C}_X$ is perverse on $X$. I'm interested in the following algebra $A=Ext^*(\mathcal{F},\mathcal{F})$. In the book of Chriss-Ginzburg it is proved that there is an isomorphism $A\simeq H_*(X'\times_X X')$ (see also the answer of David Ben-Zvi in the link above).

Let us make an additional assumption on the map $\pi$, namely that it is a Galois cover over some open subset, say $U$, of $X$ with Galois group $W$ (by the smallness it is always a covering map but not necessarily Galois).

Denote by $L$ the local system which is the restriction of $\mathcal{F}$ to this open subset $U$. Since $\pi$ is small we have $\mathcal{F} = IC(X,L)$

The zero degree part of $A$ is $End(\mathcal{F}) = End(L) = \mathbb{C}[W]$.

We also have an algebra map $B:=H^*(X') = Ext^* (\mathbb{C}_{X'},\mathbb{C}_{X'})\to A$ by functoriality.

Question1: Is it true that the image of $B$ together with $\mathbb{C}[W]$ generate the algebra $A$? (we could ask the same question but equivariantly - supposing that a (reductive) group $G$ acts on both $X$ and $X'$). I've never seen adressed this question so I wonder if the answer is trivially no.

Question2: If/when Question1 is true can we write the algebra $A$ as some sort of smash product between $W$ and $B$?

If $p:X''\to X$ is another small map which is a Galois covering with group $W$ over some (other) open subset of $X$ then the sheaf $Rp_*(\mathbb{C}_{X''})$ is isomorphic to $IC(L)$ and hence the Ext-algebras are the same. This implies that the BM-homologies $H_*(X''\times_X X'')$ and $H_*(X'\times_X X')$ are isomorphic...

Question 3: I would like to understand (maybe a more elementary reason?) why this is true. Does this imply some sort of rigidity for small maps? (I don't dare say uniqueness but something close to it)

I must say that I don't know any example of two (different, non-trivial i.e. not already coverings) small maps $X',X''\to X$ which are Galois coverings over some open subset of $X$ with the same Galois group $W$... If you know an example I would be glad to hear about it.

Some examples (of positive answers) regarding the Questions 1,2:

1. Let $\mathfrak{g}$ be a finite dimension simple Lie algebra. Take the Grothendieck resolution $\pi:\mathfrak{g}'\to\mathfrak{g}$, where $\mathfrak{g}' = G\times_B \mathfrak{b}$. It is a small map and the $Ext$ algebra, or equivalently $H_*(\mathfrak{g}'\times_{\mathfrak{g}}\mathfrak{g}')$, is isomorphic to the smash product of $\mathbb{C}[W]$ and $H_*(\mathfrak{g}') = H_*(\mathcal{B})$ where $\mathcal{B}$ is the flag variety. (some references: Douglass, J. Matthew, Röhrle, Gerhard "Homology of the Steinberg variety and Weyl group coinvariants" ; Kwon, Namhee -"Borel-Moore homology and K-theory on the Steinberg variety"). These are written for the nilpotent cone but (if I understood correctly) a Fourier transform argument says that the Ext algebra of the Springer resolution is the same as for the Grothendieck resolution. The same thing but equivariantly should(?) give the degenerate affine Hecke algebra (see answer of David Ben-Zvi) - couldn't find a reference for this result.

2. In Lusztig's cuspidal local systems I - he proves results of the same type as Question 1 asks (much more precise actually) not for the trivial local system but for some cuspidal local systems (which are local systems on nilpotent orbits that do not appear in the permutation representation on the Springer fiber of the component group of the centralizer of a nilpotent element in this orbit... I think...)

3. In Varagnolo-Vasserot the question 1 has an affirmative answer (in equivariant homology). (the maps are not small though, but they resemble the map $\mathfrak{g}'\to\mathfrak{g}$).

(All the examples I know of are around the Steinberg variety and quiver varieties[the example 3] and there are some variations with (equivariant) K-theory instead of homology. See Ch7 in Chriss-Ginzburg)

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Hi Dragos, it seems that your definition in the second paragraph is for a semi-small map rather than a small map? –  Geordie Williamson Jan 3 '13 at 20:59
Hi Geordie, you're right. I'll fix it. –  Dragos Fratila Jan 4 '13 at 9:46
Hello again! With regards to your question 3, the Atiyah flop (see wikipedia) gives two inequivalent resolutions of the singularity $xw = yz$ in $C^4$. (This singularity occurs in a Schubert variety in the Grassmannian of 2-planes in 4 space.) –  Geordie Williamson Jan 4 '13 at 10:31
That's a cute example. A question though: I have the impression that the two varieties are isomorphic - not through the natural map but rather through the one that interchanges $x$ with $y$ and $w$ with $z$. Or in other words we can obtain one of the resolutions from the other by using the automorphism of the singularity $xw=yz$ that sends $x$ to $y$ and $w$ to $z$. Is this correct? –  Dragos Fratila Jan 4 '13 at 13:37
I'm not sure what you mean in item 3 (the VV paper). In what sense does question 1 have an affirmative answer? –  Ben Webster Jan 4 '13 at 15:51
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