# Perverse Sheaves and the Geometric Satake Equivalence

The idea behind “perverse sheaves” originally had its roots in the work of Mark Goresky and Robert MacPherson on “intersection homology”, but has since taken a life of its own after the foundational work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne and has found many applications in mathematics. In this post, we will describe what perverse sheaves are, and state an important result in representation theory called the geometric Satake equivalence, which makes use of this language.

A perverse sheaf is a certain object of the “derived category of sheaves with constructible cohomology”, satisfying certain conditions. This is quite a lot of new words, but we shall be defining them in this post, starting with “constructible”.

Let $X$ be an algebraic variety with a stratification, i.e. a decomposition

$\displaystyle X=\coprod_{\lambda\in\Lambda}X_{\lambda}$

of $X$ into a finite disjoint union of connected, locally closed, smooth subsets $X_{\lambda}$ called strata, such that the closure of any stratum is a union of strata.

A sheaf $\mathcal{F}$ on $X$ is constructible if its restriction $\mathcal{F}\vert_{X_{\lambda}}$ to any stratum $X_{\lambda}$ is locally constant (for every point $x$ of $X_{\lambda}$ there is some open set $V$ containing $x$ on which the restriction $\mathcal{F}\vert_{V}$ to $U$ is a constant sheaf). A locally constant sheaf which is finitely generated (its stalks are finitely generated modules over some ring of coefficients) is also called a local system. Local systems are quite important in arithmetic geometry – for instance, local sheaves on $X$ correspond to representations of the etale fundamental group $\pi_{1}(X)$. The character sheaves discussed at the end of The Global Langlands Correspondence for Function Fields over a Finite Field are also examples of local systems (in fact, perverse sheaves, which we shall define later in this post, can be viewed as a generalization of local systems and are also important in the geometric Langlands program).

Now let us describe roughly what a derived category is. Given an abelian category (for example the category of abelian groups, or sheaves of abelian groups on some space $X$) $A$, we can think of the derived category $D(A)$ as the category whose objects are the cochain complexes in $A$, but whose morphisms are not quite the morphisms of cochain complexes in $A$, but instead something “looser” that only reflects information about their cohomology.

Let us explain what we mean by this. Two morphisms between cochain complexes in $A$ may be “chain homotopic”, in which case they induce the same morphisms of the corresponding cohomology groups. Therefore, as an intermediate step in constructing the derived category $D(A)$, we first create a category $K(A)$ where the objects are the cochain complexes in $A$, but where the morphisms are the equivalence classes of morphisms of cochain complexes in $A$ where the equivalence relation is that of chain homotopy. The category $K(A)$ is called the homotopy category of cochain complexes (in $A$).

Finally, a morphism of chain complexes in $A$ is called a quasi-isomorphism if it induces an isomorphism of the corresponding cohomology groups. Therefore, since we want the morphisms of $D(A)$ to reflect the information about the cohomology, we want the quasi-isomorphisms of chain complexes in $A$ to actually become isomorphisms in the category $D(A)$. So as our final step, to obtain $D(A)$ from $K(A)$, we “formally invert” the quasi-isomorphisms.

We do not yet have everything we need to define what a perverse sheaf is, but we have mentioned previously that they are an object of the derived category of sheaves on an algebraic variety $X$ with constructible cohomology. We denote this latter category $D_{c}^{b}(X)$ (this is used if there is some stratification of $X$ for which we have this category; if the stratification $\Lambda$ is specified, we say $\Lambda$-constructible instead of constructible, and we denote the corresponding category by $D_{\Lambda}^{b}(X)$).

Let us say a few things about the category $D_{c}^{b}(X)$. Having “constructible cohomology” means that the cohomology sheaves of $D_{c}^{b}(X)$ are complexes of sheaves, we can take their cohomology, and this cohomology is valued in sheaves (these sheaves are what we call cohomology sheaves) which are constructible, i.e. on each stratum $X_{\lambda}$ they are local systems. The category $D_{c}^{b}(X)$ is also equipped with a very useful extra structure (which we will also later need to define perverse sheaves) called the six-functor formalism.

These six functors are $R\mathrm{Hom}$, $\otimes^{\mathbb{L}}$, $Rf_{*}$, $Rf^{*}$, $Rf_{!}$, and $Rf^{!}$, the first four being the derived functors corresponding to the usual operations of Hom, tensor product, pushforward, and pullback, respectively, and the last two are the derived “shriek” functors (see also The Hom and Tensor Functors and Direct Images and Inverse Images of Sheaves). The functor $\otimes^{\mathbb{L}}$ makes $D_{c}^{b}(X)$ into a symmetric monoidal category, and $R\mathrm{Hom}$ is its right adjoint. The functor $Rf_{*}$ is right adjoint to $Rf^{*}$, and similarly $Rf_{!}$ is right adjoint to $Rf^{!}$. In the case that $f$ is proper, $Rf_{!}$ is the same as $Rf_{*}$, and in the case that $f$ is etale, $Rf^{!}$ is the same as $Rf^{*}$. We note that it is quite common in the literature to omit the $R$ from the notation, and to let the reader infer that the functor is “derived” from the context (i.e. it is a functor between derived categories).

A derived category is but a specific instance of the even more abstract concept of a triangulated category, which we have defined already, together with the related concepts of a t-structure and the heart of a t-structure, in The Theory of Motives.

In fact we will need the concept of a t-structure to define perverse sheaves. Let us now define this t-structure on the derived category of constructible sheaves. Let $X=\coprod_{\lambda\in\Lambda} X_{\lambda}$ be an algebraic variety with its stratification, and for every stratum $X_{\lambda}$ let $d_{\lambda}$ denote its dimension. We write $D_{\mathrm{const}}^{b}$ for the subcategory of $D_{\Lambda}^{b}$ whose cohomology sheaves are locally constant, and for any object $\mathfrak{F}$ of some derived category we write $\mathcal{H}^{i}(\mathfrak{F})$ for its $i$-th cohomology sheaf. We define

$\displaystyle ^{p}D_{\lambda}^{\leq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i> d_{\lambda}\rbrace$

$\displaystyle ^{p}D_{\lambda}^{\geq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i< -d_{\lambda}\rbrace$

Now let $i_{\lambda}:X_{\lambda}\to X$ be the inclusion of a stratum $X_{\lambda}$ into $X$. We further define

$\displaystyle ^{p}D^{\leq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}^{b}:Ri_{\lambda}^{*}\frak{F}\in ^{p}D_{\lambda}^{\leq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

$\displaystyle ^{p}D^{\geq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}:Ri_{\lambda}^{!}\frak{F}\in ^{p}D_{\lambda}^{\geq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

This defines a t-structure, and we define the category of perverse sheaves on $X$, denoted $\mathrm{Perv}(X)$, as the heart of this t-structure.

With the definition of perverse sheaves in hand we can now state the geometric version of the Satake correspondence (see also The Unramified Local Langlands Correspondence and the Satake Isomorphism). Let $k$ be either $\mathbb{C}$ or $\mathbb{F}_{q}$, and let $K=k((t))$, and let $\mathcal{O}=k[[t]]$. Let $G$ be a reductive group. The loop group $LG$ is defined to be the scheme whose $k$-points are $G(K)$ and the positive loop group $L^{+}G$ is defined to be the scheme whose $k$-points are $G(\mathcal{O})$. The affine Grassmannian is then defined to be the quotient $LG/L^{+}(G)$.

The geometric Satake equivalence states that there is equivalence between the category of perverse sheaves $\mathrm{Perv}(\mathrm{Gr}_{G})$ on the affine Grassmannian $\mathrm{Gr}_{G}$ and the category $\mathrm{Rep}(^{L}G)$ of representations of the Langlands dual group $^{L}G$ of $G$. It was proven by Ivan Mirkovic and Kari Vilonen using the Tannakian formalism (see also The Theory of Motives) but we will not discuss the details of the proof further here, and leave it to the references or future posts.

As we have seen in The Global Langlands Correspondence for Function Fields over a Finite Field, the geometric Satake equivalence is important in being able to define the excursion operators in Vincent Lafforgue’s approach to the global Langlands correspondence for function fields over a finite field. It has (in possibly different variants) also found applications in other parts of arithmetic geometry, for example in certain approaches to the local Langlands correspondence, as well as the study of Shimura varieties. We shall discuss more in future posts on this blog.

References:

Perverse sheaf on Wikipedia

Constructible sheaf on Wikipedia

Derived category on Wikipedia

Satake isomorphism on Wikipedia

An illustrated guide to perverse sheaves by Geordie Williamson

Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson and Anna Romanov

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Perverse sheaves and fundamental lemmas (notes by Chao Li from a course by Wei Zhang)

Perverse sheaves in representation theory (notes by Chao Li from a course by Carl Mautner)

Geometric Langlands duality and representations of algebraic groups over commutative rings by Ivan Mirkovic and Kari Vilonen