Bernstein's New Approach to Representation Theory

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MSc Mathematics

Master Thesis

Bernstein’s New Approach to

Representation Theory

Author: Supervisor:

Robert Christian Subroto

dr. A.L. Kret

Examination date:

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Abstract

Joseph Bernstein proposed in his paper Representation Theory and Stacks [2] a new language for representation theory. He presented an alternative construction using stacks to obtain the category Rep(G) of smooth G-representations, giving a more geometric approach to representation theory. In this thesis we will discuss his new language of representation theory for the case of smooth representations.

Title: Bernstein’s New Approach to Representation Theory

Author: Robert Christian Subroto, bobby-robert@hotmail.com, 11570571 Supervisor: dr. A.L. Kret

Second Examiner: prof. dr. L.D.J. Taelman Examination date: July 17, 2019

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

The Langlands program is considered to be one of (if not) the most interesting topics in modern mathematical research. It involves a series of conjectures connecting number theory and geometry. As nicely described on the Wikipedia page, it seeks to relate Galois groups in algebraic number theory to representation theory of algebraic groups over local fields and adeles.

The local Langlands program is part of the Langlands program where one restricts to the representations of algebraic groups over local fields. Local fields are topological fields which are locally compact. For simplicity, we only consider finite extensions of the p-adic numbers Qp which we will denote by F (these are the non-Archimedean local

fields of characteristic 0). We also restrict ourselves to the general linear groups GLn

over F .

Some cases of the Langlands program have already been proven. This is especially true when restricting to local fields. For example, consider the one-dimensional case GL1(F ) where F is a local field. The local Langlands conjectures in this case is a result

of local class field theory. The case for GL2(F ) has also been proven. A full proof of

the Langlands correspondence for G = GL2(F ) can be found in the book The Local

Langlands Conjecture for GL(2) [4] by Colin J. Bushnell and Guy Henniart. However, the Langlands program hasn’t been proven in full generality. Mathematicians are trying to find new ways of studying representations of GLn(F ) which might help them prove

the Langlands conjectures.

Let G = GLn over a local field F as above. In the standard approach, we study

representations of G over F by studying the category of smooth representations of the F -points G = G(F ) of G. Roughly speaking, smooth representations of a topological group G are representations taking into account the topological structure of the group G. More about smooth representations will be explained in Chapter 1.

In 2016, a mathematician named J. Bernstein known for his book Representations of the group GLn(F ), where F is a non-Archimedean local field [3] released an article titled

Stacks in Representation Theory [2] suggesting to use the concept of stacks in the study of smooth representations. He claims that his new approach of representation theory is more natural than the standard approach.

A stack X is defined over a category S with endowed Grothendieck topology (which is a generalization of the concept of coverings of topological spaces to categories) and can be viewed as a sheaf on S in categorical sense that takes values in categories instead of sets or vector spaces. A category with a Grothendieck topology is called a site. There is a well-defined notion of sheaves on a site S which in a way generalizes the notion of a sheaf on a topological space. The category of sheaves on S is denoted by Sh(S). A stack X over a site S naturally inherits a Grothendieck topology from S making X a

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site, thus we also have the category Sh(X ). This category is the main point of interest of Bernstein’s approach to representation theory.

To put Bernstein’s idea into perspective, let F be a finite extension of Qp and let

G = GL_n over F (see Section 1.3). The idea of using stacks is to put the F -points G = GLn(F ) of GLn inside a ”proper geometric environment” which in this case means

a suitable category endowed with a proper Grothendieck topology. A good candidate for G is the site S consisting of locally profinite spaces also known as l-spaces, together with the standard Grothendieck topology which is basically the categorical variant of the standard topological covering of l-spaces. Using the basic stack BG, which is the stack consisting of principal G-bundles over the corresponding l-space, we can (and we will) prove that the category Sh(BG) is equivalent to Rep(G). The construction of Rep(G) using stacks can be formulated in a diagram as follows:

GLn−→ G = GLn(F ) −→ stack BG over S −→ Sh(BG).

Hence using stacks, Bernstein also managed to construct the standard category Rep(G). Inspired by this construction, he proposed a slightly different approach to study repre-sentations of GLn. Instead of looking at the F -points, he proposed to put GLn in the

site of schemes of finite type over F denoted by SchF (the geometric environment of

GLn viewed as a scheme over F ) and to consider the basic stack BG which consists of

principal G-bundles over schemes with respect to ´etale topology (see Section 5.4 of [2] for more details). The stack BG is the analogue of BG in terms of schemes. In diagram form, we get the construction:

G −→ stack BG over Sch_F −→ Sh(BG).

We might wonder whether Sh(BG) is categorically equivalent to Sh(BG). This is in general not the case and this is what Bernstein’s main argument of using stacks in representation theory is based upon. He proposed to study the category Sh(BG) instead since in general, the category Sh(BG) is larger and contains more information than Sh(BG). For example, Sh(BG) can be written as a product of categories Sh(BGi)

where Gi is a pure inner form of G (see [2]).

The main goal of this thesis is to study how we can derive the standard approach of representation theory using stacks and to prove that Sh(BG) is indeed equivalent to the category Rep(G) consisting of smooth G-representations.

Structure of the thesis

Here is an overview of what will be discussed in this thesis.

• In the first part of Chapter 1, we will briefly discus some basic results of l-groups and l-spaces needed for the remainder of this thesis. This includes basic definitions and theorems. We will also briefly discuss the general linear group GLn and its

relation to l-groups. In the second part, we will discuss some basic notions of the theory of smooth representation theory

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• In Chapter 2, we will discuss the classical theory of sheaves on topological spaces. There will be a small emphasize on the construction of sheaves from a presheaf (sheafification) which will prove to be useful in later chapters. We will only consider sheaves where we assign complex vector spaces.

• Chapter 3, we will make the link between sheaves and representation theory. We will discuss the notion of G-equivariant sheaves and provide proof for two important results which links representation theory and sheaves. This chapter serves as a bridge between the standard approach of smooth representations in Chapter 3 and the approach by using stacks which will be discussed in the remaining chapters. • In Chapter 4 we will discuss the necessary material from category theory needed to

understand and define stacks. There will be some emphasis on fibered categories (a special type of category) and sites (a category together with a Grothendieck topology). Also, we will discuss the notion of a sheaf on a site.

• In Chapter 5 we will discuss some examples of useful stacks like the basic stack and quotient stack. This is a semi-technical chapter where I will refer to external literature for the proof.

• In Chapter 6 we will explicitly show the categorical equivalence between smooth representation theory and sheaves on stacks. This equivalence is the main result of Bernstein’s article in which he proposed to use stacks as a new approach to representation theory.

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1 l-groups and Smooth Representations

In this chapter, we will cover some basic theory about locally profinite groups also known as l-groups which are the main type of groups of interest in this thesis. We will also discuss the notion of a smooth representation regarding these l-groups.

The first part of this chapter where we discuss l-groups is based on the article G-Equivariant l-Sheaves of Jakub Witaszek [10] and the book Representations of the group GLn(F ), where F is a non-archimedean local field of J. Bernstein and A. Zelevinsky [3].

The second part where we discuss smooth representations is based on Chapter 1 of the book The Local Langlands Conjecture for GL(2) [4].

1.1 Topological Groups

Although group theory and topology seems to be quite different in nature, there are many cases where a space is endowed with both group theoretic and topological properties. Profinite groups are one of them.

Definition 1.1. A topological group is a group G which is also a topological space such that the multiplication law m : G × G → G and the inverse map i : G → G are continuous maps.

A homomorphism between topological groups is a homomorphism of the underlying groups which is continuous. An isomorphism of topological groups is a homomorphism of the underlying groups which is a homeomorphism.

Proposition 1.2. Let G be a topological group and H ≤ G an open subgroup. Then H is also closed.

Proof. Let G be a subset of G consisting of representatives of the quotient G/H and let e ∈ G be the representative of H . Then for all g ∈ G we have a cover G = S

g∈GgH

which are disjoint open sets. Since G − H =S

g∈G\{e}gH and

S

g∈G\{e}gH is also open,

we have that H is indeed closed in G.

Example 1.3. Any group G can be made into a topological group by assigning to G the discrete topology. Other examples of topological groups are the p-adic numbers Qpunder

addition and the algebraic group GLn(Qp) for p a prime number under multiplication.

1.2 l-groups and l-spaces

In this section we will define locally profinite groups also known as l-groups. We will first discuss profinite groups.

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Definition 1.4. A profinite group is a topological group which is compact, Hausdorff and totally disconnected.

Remark. Every profinite group is the inverse limit of a collection of finite discrete groups together with induced product topology. The opposite direction is also true. See Theo-rem 3.7 of [8] for a proof.

Definition 1.5. A topological space is called an l-space if it is Hausdorff, locally com-pact and totally disconnected. In particular, each point has a basis of open comcom-pact neighborhoods. A topological group which is an l-space is called an l-group.

Lemma 1.6. Let G be an l-group which acts continuously on an l-space X. Then for each compact open subset U in X there exists a neighborhood V of the identity element e ∈ G such that V · U = U .

Proof. The proof presented here is based on the proof of Lemma 4 of [10]. Since U is open and the action map G × X → X is continuous, we have for every point x ∈ U a neighborhood Vx ⊂ G of e and a neighborhood Ux ⊂ X of x such that Vx· Ux ⊂ U .

Also since U is compact, we can cover U by a finite number of Ux1, ..., Uxm and take the

intersection V = T

iVxi. Note that V · U ⊆ U and since e ∈ V , we have the equality

V · U = U .

Proposition 1.7. Let G be an l-group. Then e ∈ G has a basis of neighborhoods which are open compact subgroups.

Proof. The proof presented here is based on the proof of Proposition 5 of [10]. We need to prove that every open compact neighborhood of e contains an open compact subgroup. Let U ⊂ G be an arbitrary open compact neighborhood of e. As G acts on itself, we have by Lemma 1.6 that there exists an open neighborhood V ⊂ G of e such that V · U = U . Without loss of generality, we may assume that V ⊂ U and V−1 = V . Take H :=S∞

i=1Vi which is an open subgroup of G. By Lemma 1.6 it is also closed.

We will show that H ⊂ U . It will imply that H is compact, since closed subsets of compact sets are compact.

We see that if Vi−1 ⊂ U then V · Vi−1 _{⊂ V · U = U , so by mathematical induction}

we have indeed that H ⊂ G.

We have the following convenient properties as a direct result of Proposition 1.7: Corollary 1.8. Let G be an l-group and X an l-space.

• Every open neighborhood of e ∈ G contains a compact open subgroup; • The set B_X := {U ⊂ X | U open and compact} is a basis of X. More is true about l-spaces:

Lemma 1.9. Every cover of an l-space X has a countable refinement consisting of pairwise disjoint open compact sets.

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Proof. See Lemma 10 of [10].

Proposition 1.10. Let G be an l-group,

1. If G is compact, then G is a profinite group; 2. A closed subgroup of G is also locally profinite; Proof. We will prove the results separately.

1. We know that G is also Hausdorff and totally disconnected since G is an l-space. Hence if G is also assumed to be compact, G is profinite.

2. Let H < G be a closed subgroup. Note that H is also Hausdorff and totally disconnected. Locally compactness follows from the fact that the intersection of a compact subset and a closed subset is also compact.

1.3 The General Linear Group

In the Introduction we mentioned that we will be working with linear algebraic groups. In this section we will briefly discuss the general linear group and its relation to l-groups. Definition 1.11. The general linear group GLn over a field F is a scheme of the

form

GLn= Spec(F [{y, xij}1≤i,j≤n]/(y det(xij) − 1))

where det(xij) is the determinant polynomial of the n × n matrix (xij)1≤i,j≤n.

Definition 1.12. Let X be a scheme and F a field. The F -points of X denoted by X(F ) is the set

HomSch(Spec(F ), X)

which consists of morphisms between schemes.

We are interested in the algebraic group GLn(F ). The following result gives us a very

convenient description of GLn(F )

Proposition 1.13. Let F be a field. Then GLn(F ) is naturally bijective to the invertible

n × n-matrices with values in F .

To prove Proposition 1.13 we need the following well-known result from algebraic geometry:

Proposition 1.14. Let R and T be rings. Then there is a natural bijection between HomSch(Spec(T ), Spec(R)) and HomRing(R, T ).

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Proof of Proposition 1.13. Applying Proposition 1.14 we see that GLn(F ) := HomSch(Spec(F ), GLn)

is naturally bijective to

HomRing(F [{y, xij}1≤i,j≤n]/(y det(xij) − 1), F ).

Observe that any ring homomorphism

φ : F [{y, xij}1≤i,j≤n]/(y det(xij) − 1) → F

is uniquely determined by the values of {φ(xij)}1≤i,j≤n and φ(y). Hence we can assign

to φ the matrix Mφ:= (φ(xij))1≤i,j≤n which has determinant equal to φ(y)−1 ∈ F since

we are studying modulo (y det(xij) − 1). This is a well-defined invertible matrix since

φ(y) 6= 0 and .

This map is a bijection. An inverse map is to assign to each invertible n × n-matrix M the map φM such that φM(xij) = Mij and φM(y) = det(M )−1.

Thus we can view GLn(F ) as a subset of the product space Fn

2

which in terms of algebraic geometry is denoted by An2(F ). Assume now that F is a finite extension of Qp which is an l-group. Hence Fn

2

endowed with the product topology is also an l-space. The following propositions shows why we are interested in studying l-groups and l-spaces:

Proposition 1.15. Let F be a finite extension of Qp. Assign to GLn(F ) the subspace

topology of Fn2_{. Then GL}

n(F ) is an l-group. Moreover, GLn(F ) has an open profinite

subgroup of countable index.

Proof. See Sections 1.4 and 7.3 of [4].

Remark. The hypothesis G has an open profinite subgroup of countable index is thus satisfied for all the l-groups we care about. Moreover the assumption simplifies the theory. For example without the assumption, Schur’s lemma need not hold (see section 2.6 of [4]).

1.4 Smooth Representations

Let us briefly recall the general definition of a representation of a group G.

Definition 1.16. Let V be a complex vector space (hence a vector space over C) and let G be a group. A representation (π, V ) of G is a group homomorphism

π : G → GL(V ).

The vector space V is called the representation space of G. The representation (π, V ) induces the following action on G which we denote by ˜π:

˜

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A morphism between two representations (π, V ) and (π0, V0) of G is a linear map φ : V → V0 such that the following diagram commutes for all g ∈ G:

V V0

φ

π(g) π0_(g)

φ

The set of those morphisms is denoted by HomG(π, π0).

Two representations are called isomorphic if the linear morphism is also an isomor-phism as vector spaces.

A subrepresentation of (π, V ) is a representation (π |W, W ) where W is a stable

G-subspace of V and π |W is simply the restriction of π to W .

When assuming that G is a l-group and taking into account its topological properties, we come across the notion of a smooth representation of G:

Definition 1.17. Let G be a l-group and let (π, V ) be a representation of G. The representation (π, V ) is called smooth if, for every v ∈ V there is a compact open subgroup Kv of G depending on v such that π(x)v = v for all x ∈ Kv.

Note. For the remainder of this chapter, G will be a l-group.

Remark. Some remarks on smooth representation theory are worth mentioning: • Definition 1.17 is equivalent to the following:

Let K0 be a subgroup of G and let VK0 denote the space of π(K0)-fixed vectors in V . Then smoothness of the representation (π, V ) is equivalent to

V =[

K

VK

where K ranges over the compact open subgroups of G;

• Observe that a G-representation (π, V ) is smooth if and only if the action ˜π : G × V → V in Definition 1.16 is continuous where V is endowed with the discrete topology;

• Let G be a finite group with discrete topology. Note that G is also a l-group. Since G is discrete, the trivial subgroup {e} is an open compact subgroup of G. This implies that all representations (ρ, V ) of G are smooth.

The morphisms between smooth representations of G are the G-morphisms as in Defi-nition 1.16. All smooth representations together with morphisms form a category called the Category of Smooth Representations. We are particularly interested in the structure of this category.

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Lemma 1.19. Let G be a compact l-group (hence a profinite group) and let (π, V ) be an irreducible representation of G. Then V is finite dimensional.

Proof. Let v ∈ V be non-zero. Let K < G be a compact open subgroup of G which is contained in the stabilizer of v (which exists since the representation is smooth). Observe that π(G/K)v spans V . Since G is compact, we have that [G : K] is finite, hence V is finite dimensional.

1.5 Induced Representation

The material presented in this section is mainly based on Section 2.4 of [4]. Let H be a closed subgroup of G, thus H is also a l-group. Let (σ, W ) be a smooth representation of H. Like in the case of finite groups, we want to construct a smooth representation of G such that its restriction to H equals σ. This will be done as follows:

Let X be the C-vector space consisting of functions f : G → W satisfying the following conditions:

1. f (hg) = σ(h)f (g);

2. There is a compact open subgroup K of G such that f (gk) = f (g) for all k ∈ K and g ∈ G.

There is a natural homomorphism Σ : G → AutC(X) defined by

Σ(g)f : x 7→ f (xg), g, x ∈ G.

Observe that the pair (Σ, X) provides a smooth representation of G where smoothness is due to the second condition of X. Hence the following definition is well-defined. Definition 1.20. The representation of (Σ, X) constructed above is called the repre-sentation of G smoothly induced by σ, and is denoted by

(Σ, X) = IndG_Hσ. Note that the map σ 7→ IndG_Hσ gives a functor

IndG_H : Rep(H) → Rep(G). Also, there is a canonical H-homomorphism

ασ : IndGHσ −→ W, f 7−→ f (1)

The functor IndG_H and the map ασ have the following important property called the

Frobenius reciprocity.

Theorem 1.21 (Frobenius Reciprocity). Let H be a closed subgroup of G. For a smooth representation (σ, W ) of H and a smooth representation (π, V ) of G, the canonical map

HomG(π, IndGHσ) −→ HomH(π |H, σ), φ 7−→ ασ◦ φ

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Proof. Let f ∈ HomH(π |H, σ), hence f : V → W is an H-homomorphism. Consider the

G-homomorphism f? : V → IndGHσ by letting f? be the function g 7→ f (π(g)v). Note

that the map f 7→ f? is the inverse of σ 7→ ασ◦ σ. Hence the theorem follows.

1.6 Compactly Induced Representation

There is an important variation of induced representations. Again, let H be a closed subgroup of G and recall (σ, W ) and X. Now consider the space Xc⊆ X consisting of

functions which are compactly supported modulo H. These are functions f ∈ X such that the image of its support in H \ G is compact. Equivalently, this means that there exists a compact subset C ⊆ G such that suppf ⊆ HC.

Note that the space Xc is stable under the action of G and thus provides another

smooth representation of G.

Definition 1.22. The representation (Σ, Xc) constructed above is called compact

in-duction and is denoted by

(Σ, Xc) = c-IndGH.

Again, the map σ 7→ c-IndG_Hσ gives a functor

c-IndG_H : Rep(H) → Rep(G). Proposition 1.23. The functor c-IndG_H is additive and exact. Proof. See section 2.5 of [4].

Studying compactly induced representations becomes especially interesting when H is also an open subgroup of G. In this case, there is a canonical H-homomorphism

α_σc : W −→ c-IndG_HW, w 7−→ fw

where fw ∈ Xc is supported in H and fw(h) = σ(h)w for h ∈ H. We have the following

result regarding the structure of the vector space Xc which can be found in section 2.5

of [4].

Proposition 1.24. Let H be an open subgroup of G which is also closed by Proposition 1.2 and let (σ, W ) be a smooth representation of H.

1. The map αc_σ : w 7→ fw is an H-isomorphism of W with the space of functions

f ∈ c-IndG_Hσ such that suppf ⊆ H.

2. Let W be a C-basis of W and G a set of representatives for G/H. The set {gfw:

w ∈ W, g ∈ G} is a C-basis of c-IndG_Hσ.

Proof. The proof is based on the proof of the Lemma in section 2.5 of [4]. For (1), observe that αc_σ is a H-homomorphism to the space of functions supported in H since suppfw⊆ H for all w ∈ W . The inverse map is given by f 7→ f (1).

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By definition, the support of a function f ∈ c-IndG_Hσ is a finite union of cosets Hg−1, for various g ∈ G, and the restriction of f to any one of these also lies in c-IndG_Hσ. If suppf = Hg−1, then g−1f has support in H, and so is a finite linear combination of functions fw, w ∈ W. The set of functions gfw where w ∈ W and g ∈ G is linearly

independent, hence we have proven (2).

Proposition 1.25. Let H be an open subgroup of G. Then the functor c-IndG_H is faithful. Proof. We know by Proposition 1.23 that c-IndG_H is additive. Hence we only need to show that for any two smooth H-representations (ρV, V ) and (ρW, W ) and for any

H-morphism φ : V → W such that ρ 6= 0, we have that c-IndG_H(φ) 6= 0. So assume that ρ 6= 0. Then there exists a v ∈ V such that φ(v) 6= 0. Recall the function fv ∈ c-IndGHV .

Then c-IndG_Hρ maps fv to the map c-IndGHφ(fv) = fφ(v) ∈ c-IndGHW which is not zero

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2 Sheaves on Topological Spaces

Sheaves play a very important role in this thesis. More specifically, we are particularly interested in the category of sheaves on topological spaces or stacks. In this chapter, we will discuss some important results of sheaves on topological spaces.

2.1 Presheaves

Definition 2.1. Let X be a topological space. A presheaf F on X consists of the following data:

1. For all open U ⊆ X, we assign a complex vector space F (U ); 2. For all pairs of open sets U1 ⊂ U2, there is a restriction map

ρU2,U1 : F (U2) → F (U1)

which is a linear map satisfying the following conditions:

• ρU,U = idF (U );

• if U1 ⊆ U2 ⊆ U3, then ρU2,U1 ◦ ρU3,U2 = ρU3,U1

An element s ∈ F (U ) over some open subset U ⊂ X is called a section on U .

Note. In general sheaf theory, F (U ) can be a set. We however are only interested in sheaves assigning F (U ) to a complex vector space for all open subsets U ⊂ X! This will be true for the remainder of this thesis.

Remark. Although the restriction map ρ depends on the sheaf, we will use the same notation for the restriction map ρ when working with other sheaves. Moreover, when we have a section s ∈ F (U ) and V ⊂ U , then by the terminology ”s restricted to V ” refers to the element ρU,V(s). A conventional alternative notation we often use is s |V.

Definition 2.2. Let F and G be two presheaves on X. A morphism φ : F → G between the presheaves consists of a family of maps φU : F (U ) → G(U ) for every open

set U ⊆ X such that if U ⊂ V , the following diagram commutes: F (V ) G(V )

F (U ) G(U )

φV

ρV,U ρV,U

φU

If for all U ⊂ X the map φU : F (U ) → G(U ) is an isomorphism of vector spaces, then φ

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Having defined morphisms between presheaves, we have a category consisting of presheaves on X.

Definition 2.3. The category of presheaves on X is denoted by P Sh(X).

Definition 2.4. Let X be a topological space and F ∈ P Sh(X). The stalk of F at a point x ∈ X is defined to be the set

Fx := lim_←− x∈U,U open

F (U )

For s ∈ F (U ), the corresponding element in the stalk Fx at x will be denoted by sx or

s |x

Remark. From Definition 2.4 we see that if a ∈ Fx, then there exists an open

neighbor-hood U of x and a section sU ∈ F (U ) such that sU

x = a.

Proposition 2.5. Let φ : F → G be a morphism between the sheaves F and G over some topological space X. Then for every x ∈ X, φ induces a linear map

φx: Fx → Gx.

Proof. Let a ∈ Fx. By definition there exists an open neighborhood U of x and a section

aU ∈ F (U ) such that aU

x = a. Now consider φU(s) ∈ G(U ) and assign

φx(a) = φU(aU)x.

This map is well-defined by applying the direct limit to get the germ of φU(aU) at x.

Stalks play a very important role in sheaf theory, mainly due to its convenient nature that it is preserved after sheafification which we will address later.

Definition 2.6. Let X and Y be topological spaces and let f : X → Y be a continuous map. Let F ∈ P Sh(Y ). The pullback presheaf f−1F ∈ P Sh(X) is defined as follows:

For every open U ⊂ X we assign

f−1F (U ) := lim_←−

f (U )⊆V,V open

F (V ) where the inverse limit is taken over the restriction maps.

Observe that f−1F is indeed a presheaf on X.

Lemma 2.7. Let f : X → Y be a continuous map and let x ∈ X and y = f (x). Then f−1Fx = Fy.

Proof. The proof consists of the following computation: f−1F_x := lim_←− x∈U,U open f−1F (U ) = lim_←− x∈U,U open lim ←− f (U )⊂V,V open F (V ) ! = lim_←− f (x)∈V,V open F (V ) := F_y

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Lemma 2.8. Let f : X → Y be an open map. Then f−1F (U ) = F (f (U )). In particular, if we consider the identity map id : X → X, then

id−1F (U ) = F (U ).

Proof. The proof is a direct consequence of the definition of the inverse limit over all open sets containing the open image f (U ).

Lemma 2.9. Consider the composition

X−→ Yφ −→ Z.λ Then we have

φ−1(λ−1F ) = (λ ◦ φ)−1F .

Proof. Observe that since X, Y and Z are Hausdorff, we have the identities \

V :φ(U )⊆V

V = φ(U ) and \

W :λ(V )⊆W

W = λ(V )

where U ⊂ X, V ⊂ Y and W ⊂ Z are open subsets. Hence this validates the following computation: φ−1(λ−1F )(U ) = lim_←− φ(U )⊆V λ−1F (V ) = lim_←− φ(U )⊆V lim ←− λ(V )⊆W F (W ) ! = lim_←− λ◦φ(U )⊆W F (W ) := (λ ◦ φ)−1F (U ).

2.2 Sheaves

We will now define sheaves which are our main point of interest.

Definition 2.10. Let X be a topological space and F a presheaf on X. Then F is called a sheaf if it satisfies the following conditions:

• If (Ui)i∈I is an open covering of an open set U , and if s, t ∈ F (U ) are sections such

that s |Ui= t |Ui for all Ui, then s = t;

• If (Ui)i∈I is an open covering of an open set U , and if we have for each Ui a section

si∈ F (Ui) such that si |Ui∩Uj= sj |Ui∩Uj for all pairs (i, j) ∈ I

2_{, then there exists}

a unique section s ∈ F (U ) such that s |Ui= si for all i ∈ I.

Morphisms between sheaves are the morphisms viewed as presheaves. Hence there is also a subcategory of P Sh(X) consisting only of sheaves.

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Definition 2.11. The category consisting of sheaves is denoted by Sh(X).

Sheaves are convenient to work with since it allows us to retrieve sections given a covering with compatible sections. They also behave nicely when working with stalks. Lemma 2.12. Let X be a topological space and F a sheaf on X. Then for every open U ⊂ X, the map

F (U ) → Y

u∈U

Fu, s 7→ (su)u∈U

is injective.

Proof. Let s1, s2 ∈ F (U ) be two elements with the same image in Q_u∈UFu. Then by

definition of a stalk there exists an open covering (Ui)i∈I of U and sections ti ∈ F (Ui)

for every Ui such that

• For every i ∈ I and x ∈ U_i, ti |x= s1 |x= s2 |x;

• s1|Ui= s2|Ui= ti.

Since the elements ti ∈ F (Ui) are the restriction of some element in F (U ) (in this case

s1 and s2) we have for every pair (i, j) ∈ I2 that ti |Ui∩Uj= tj |Ui∩Uj. Hence since F is

a sheaf, there exists a unique element s ∈ F (U ) such that s |Ui= ti for all i ∈ I which

implies that s = s1 = s2 and thus we showed injectivity.

Proposition 2.13. Let X be a topological space and let F and G be two sheaves on X. Moreover, let φ : F → G be a morphism between the sheaves. Then φ is an isomorphism of sheaves if and only if φx: Fx→ Gx is an isomorphism for all x ∈ X.

Proof. See Proposition 5.1 of [6]

Definition 2.14. Let X be topological space and let B be a basis on X. We say that F is a restricted sheaf on B if it behaves in the same way as a sheaf on X. To be more precise, for every U ∈ B we assign a complex vector space F (U ), and for all U, V ∈ B with V ⊂ U we have a restriction map ρU,V : F (U ) → F (V ) such that the following

properties are satisfied:

• For every U ∈ B, the map ρ_U,U is the identity on F (U );

• For all U, V, W ∈ B with W ⊂ V ⊂ U , we have ρU,W = ρV,W ◦ ρU.V;

• If U ∈ B is the union of open sets (Vi)i∈I with Vi ∈ B, and s ∈ F (U ) is such that

ρU,Vi(s) = 0 ∈ F (Vi) for all i ∈ I, then s = 0;

• If U and (Vi)i∈I are as above, and si ∈ F (Vi) are given such that ρVi,W(si) =

ρVj,W(sj) for all i, j ∈ I and all W ∈ B with W ⊂ Vi ∩ Vj, then there exists

s ∈ F (U ) such that ρU,Vi(s) = si, for all i ∈ I.

Proposition 2.15. Let X be topological space and let B be a basis on X. Let F |B be a

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Proof. See the notes Construction of a sheaf from the data on a basis of open sets [5] of Bas Edixhoven for a detailed proof.

Remark. Let U ⊂ X open and let F be a sheaf on X. An alternative way of thinking about F is to think of elements f ∈ F (U ) as functions with domain U and such that every u ∈ U is mapped to fu ∈ Fu (hence f (u) = fu). For V ⊂ U ⊂ X, applying ρU,V

to f equals restricting the domain of f (seen as a function) to V . This point of view of sheaves will be really useful in Chapter 3. Hence it makes sense to talk about the support of f ∈ F (U ) which is used in Definition 3.7.

2.3 Sheafification

As seen in the previous section, not all presheaves are sheaves. However, there is a natural way to construct a sheaf from a given presheaf. This process is called sheafification or sheaving which will be described in this section. There are a few ways to describe this process. The one presented here is the same as the process described in Part 1 section 6.17 of The Stacks Project [1].

The basic construction is the following. Let F be a presheaf on sets on a topological space X. For every open U ⊂ X we define

F#(U ) = {(su) ∈

Y

u∈U Fu such that (∗)}

where (∗) is the property:

For every u ∈ U , there exists an open neighbourhood u ∈ V ⊂ U , and a section σ ∈ F (V ) such that for all v ∈ V we have sv = σ |v in Fv.

Note that (∗) is a condition for each u ∈ U , and that given u ∈ U the truth of this condition depends only on the values sv for v in any open neighbourhood of u. Thus if

V ⊂ U ⊂ X are open inclusions, the projection maps Y

u∈UFu −→

Y

v∈V Fv

maps elements of F#(U ) into F#(V ). In other words, we get the structure of a presheaf on sets on F#.

Furthermore, the image of the map F (U ) → Q

u∈UFu is contained in F#(U ). In

addition, if V ⊂ U ⊂ X are open then we have the following commutative diagram F (U ) // F#_{(U )} // Q u∈U Fu F (V ) //F#_{(V )} //Q v∈V Fv

where the vertical maps are induced from the restriction mappings. Thus there is a canonical morphism of presheaves F → F#.

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Proof. The first property of a sheaf is satisfied by evaluating coordinate wise. Let U =S

i∈IUi be an open covering of U and assume we have section si ∈ F#(Ui)

such that for all (i, j) ∈ I2 we have si |Ui∩Uj= sj |Ui∩Uj. Thus we have si |Ui∩Uj= (si |x

)x∈Ui∩Uj. Observe that there is a unique element s = (su)u∈U ∈

Q

u∈U Fu such that

restricting to Ui for every i ∈ I we get s |Ui= si. It remains to show that s ∈ F

#_{. For}

every u ∈ U there is some Ui such that u ∈ Ui. By hypothesis, there exists an open

V ⊂ Ui and a section σ ∈ F#(V ) such that σ |v= si |v= s |v. Since this is true for all

u ∈ U , we have that s satisfies property (∗) and thus s ∈ F#(U ).

Proposition 2.17. Let X be a topological space and let F be a presheaf on X. Then for all x ∈ X we have

F_x = F_x#

Proof. As we have seen above, we have that the map φ : F → F# defined by φU : F (U ) → F#(U ), s 7→ (su)u∈U

is a morphism of presheaves. Hence all elements in Fx# are represented by elements in

F_x. It remains to show that φx: Fx → Fx# is an isomorphism for all x ∈ X.

Injectivity is due to the construction of φ. To show surjectivity, let a ∈ Fx#. By

definition, there exists an open neighborhood U of x and a section s ∈ F#(U ) such that sx = a. By property (∗) there exists an open neighborhood V ⊂ U of x and a

section σ ∈ F (U ) such that σv = sv for all v ∈ V . Clearly, φ(σ) = σ ∈ F# and so

φx(σx) = σx= a which shows surjectivity.

Hence φx is indeed an isomorphism which concludes the proof.

Sheafification also has the following universal property:

Proposition 2.18 (Universal property of sheafification). Let F be a presheaf and G be a sheaf on X. Assume that we have a morphism φ : F → G as presheaves. Then there exists a unique morphism τ : F#→ G such that we have the following factorization

φ : F → F# τ−→ G

Proof. See Lemma 6.17.5 of Part 1 Section 6.17 of The Stacks Project [1].

2.4 Pullback of Sheaves

Let f : Y → X be a continuous map of topological spaces, and let F be a sheaf on X. We want to construct a sheaf on Y in a natural way such that it is compatible with f .

We can try to construct a sheaf on Y in the same way as done with presheaves: To every open subspace U of X, we assign the complex vector space

lim −→

f (U )⊆V

F (V ).

This does give us a presheaf f−1(F ) on Y . However, it is not necessarily a sheaf. Using the sheafification process described above on f−1F , we are able to assign a proper sheaf to Y .

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Definition 2.19. Let f : Y → X be a continuous map of topological spaces, and let F be a sheaf on X. Consider the presheaf f−1F on Y . We call the sheaf

f∗F := (f−1F )# on Y the pullback sheaf on Y .

Lemma 2.20. Let f : Y → X be a continuous map and let F be a sheaf on X. Also let y ∈ Y and x = f (y) ∈ X. Then we have a canonical isomorphism

(f∗F )_y−→F˜ _x

Proof. This follows directly from Lemma 2.7 and Proposition 2.17. Lemma 2.21. Consider the composition

X−→ Yg −→ Z.f Then we have a natural isomoprhism

g∗(f∗F ) ' (g ◦ f )∗F .

Proof. This is due to Lemma 2.9 and that sheafification preserves isomorphisms.

2.5 Sheaves on an l-space

Let X be an l-space. By Corollary 1.8 we have that the set consisting of open compact subsets of X forms a basis of X. Since X is Hausdorff, all such open compact subsets are also closed subsets (a result from basic topology states that all compact subsets of a Hausdorff space is closed).

Lemma 2.22. Let X be an l-space, F a sheaf on X and U ⊂ X an open compact set. Consider a section s ∈ F (U ). Then s can be glued together with 0 ∈ F (X \ U ) to get a global section. Hence we have an injective map

iU : F (U ) → F (X).

Proof. Since U is a compact subset of X, U is also closed since X is a Hausdorff space. Hence X \ U is open and so F (X \ U ) is well-defined. Moreover, Since X is covered by U and X \ U and since U ∩ (X \ U ) = ∅, we can indeed glue together s ∈ F (U ) with 0 ∈ F (X \ U ) to get a global section.

Definition 2.23. The map iU in Lemma 2.22 is called extension by zero.

Corollary 2.24. Let X be an l-space, F a sheaf on X and U ⊂ X an open compact set. Then ρX,U ◦ iU = idF (U ). Hence the map ρX,U : F (X) → F (U ) is surjective.

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Proof. This is immediate when viewing a section s ∈ F (U ) as a function with domain U .

Remark. Corollary 2.24 is in sharp contrast to the usual sheaf theory. For instance the sheaf of holomorphic functions on P1(C) does not have this property. For U ⊂ P1(C) an open subset, consider a constant function fc: U → C, u 7→ c where c 6= 0. Extending

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3 Sheaves and Representation Theory

The idea of using stacks in representation theory is to translate the classical representa-tion theory into a more geometric framework. In this secrepresenta-tion, We will provide two very important results which links smooth representation theory to sheaves which is a very strong indication of a proper geometric framework of representation theory.

Let G be a l-group and X be a topological space together with a continuous group action a : G × X → X. We will provide proof of the following two results:

Fact 1. Let X be the one-point space. Then the category ShG(X) of G-equivariant

sheaves is equivalent to the category Rep(G) of smooth representations of G.

Fact 2. Let H be an open subgroup of G and let X = H \ G with induced topology. Consider the action a : G × X → X, (g, x) 7→ x · g−1. Then we have natural equivalences of categories

ShG(X) ' ShH(x) ' Rep(H)

In the first part we will discuss the notion of a G-equivariant sheaf followed by the proof of the two facts.

3.1 G-equivariant Sheaves Part I

Let G be an l-group and X a l-space together with a continuous action a : G × X → X. The two maps of interest from G × X to X are the action map a and the projection map

π : G × X → X, (g, x) 7→ x. Moreover, we have three natural maps

p1, p2, p3 : G × G × X → G × X

defined as follows:

• p1(g1, g2, x) = (g1g2, x);

• p₂(g1, g2, x) = (g1, a(g2, x));

• p₃(g1, g2, x) = (g2, x).

Lemma 3.1. We have the following identities: 1. a ◦ p1 = a ◦ p2 : G × G × X → X;

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2. π ◦ p1 = π ◦ p3 : G × G × X → X;

3. π ◦ p2 = a ◦ p3: G × G × X → X.

Proof. We will prove Lemma 3.1 by computing the functions and show that the identities hold for all (g1, g2, x) ∈ G × G × X.

1. Computing a ◦ p1 and a ◦ p2 separately gives us

a ◦ p1(g1, g2, x) = a(g1g2, x)

a ◦ p2(g1, g2, x) = a(g1, a(g2, x)) = a(g1g2, x).

which implies the identity a ◦ p1 = a ◦ p2.

2. Computing π ◦ p1 and π ◦ p3 separately gives us

π ◦ p3(g1, g2, x) = π(g2, x) = x

π ◦ p1(g1, g2, x) = π(g1g2, x) = x.

which implies the identity π ◦ p1 = π ◦ p3.

3. Computing π ◦ p2 and a ◦ p3 separately gives us

π ◦ p2(g1, g2, x) = π(g1, a(g2, x)) = a(g2, x)

a ◦ p3(g1, g2, x) = a(g2, x).

which implies the identity π ◦ p2 = a ◦ p3.

Let R be a sheaf on X and assume moreover that we are given an isomorphism between sheaves

α : π∗R −→ a∗R.

Applying the pullback functor and lemma 3.1, we get the following diagram of sheaves on G × G × X: p∗₁π∗R p ∗ 1α // p∗₁a∗R p∗₃π∗R p ∗ 3α// p∗₃a∗R = p∗₂π∗Rp ∗ 2α // p∗₂a∗R (3.1)

Remark. The diagram is not commutative in general! Consider the following example: Example 3.2. Let X = {x} be a one-point space, G = Z/2Z be the group or order 2 with elements {1, −1} with discrete topology and let a : G × X → X be the trivial action. In this case, a coincides with π. Let R be the sheaf on X which assigns {x} to the one-dimensional complex vector space C. Observe that a−1X and π−1X are naturally homeomorphic to G as a topological space. Let α : π∗R → a∗R be an isomorphism of sheaves defined by

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• α_(1,x) : C → C, z 7→ z;

• α_(−1,x): C → C, z 7→ z · i;

at the level of stalks. Observe that

α_{(−1,a(−1,x))}◦ α_(−1,x)(z) = (z · i) · i = −z 6= z := α_(1,x)

while −1 · −1 = 1. Hence at the level of stalks, diagram 3.1 is not commutative in this case!

The composition p∗₂α ◦ p∗₃α can be viewed as a morphism from the sheaf p∗₁π∗R to p∗₁a∗R since p∗₁π∗R = p∗₃π∗R and p∗₁a∗R = p∗₂a∗R. Consider the morphisms

ν, µ : p∗₁a∗R −→ p∗₁π ∗ R ν = p∗₁α

µ = p∗₂α ◦ p∗₃α We now give the definition of a G-equivariant sheaf.

Definition 3.3. A G-equivariant sheaf over X consists of a pair (F , α) where F is a sheaf on X and α : π∗F → a∗_XF an isomorphism of sheaves on G × X such that the morphisms of sheaves ν, µ : p∗₁a∗R −→ p∗₁π∗R as defined above coincide. The latter condition is also called the cocycle condition.

Remark. A few remarks regarding G-equivariant sheaves:

1. An equivariant sheaf is thus already equiped with a specific given isomorphism α of sheaves. Just being isomorphic is not enough;

2. The cocycle condition is equivalent to diagram 3.1 being a commutative diagram. In terms of stalks, this is equivalent to the equality

α(g1·g2,x)= α(g1,a(g2,x))◦ α(g2,x)

for all g1, g2∈ G and x ∈ X.

Definition 3.4. Let (F1, α1) and (F2, α2) be two G-equivariant sheaves on X. A

mor-phism φ from (F1, α1) to (F2, α2) is a morphism

φ : F1 → F2

such that the following diagram commutes: π∗F₁ α1 // π∗_φ a∗F₁ a∗_φ π∗F2 α2 // a∗F2 (3.2)

Definition 3.5. G-equivariant sheaves on X together with the morphisms defined in definition 3.4 forms a category called the category of G-equivariant sheaves on X denoted by ShG(X).

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3.2 G-equivariant Sheaves Part II

In this section, we will discuss an alternative definition for G-equivariant sheaves. Again assume that G is an l-group and X an l-space together with a continuous action a : G × X → X.

Definition 3.6. Let F be a sheaf on X. An action of G on F consists of the following data: For every g ∈ G and for every open subset U ⊂ X we have a morphism

φg : F (U ) → F (gU )

such that

1. For all V ⊂ U open in X, the diagram F (U ) φg // ρU,V F (gU ) ρgU,gV F (V ) φg //F (gV ) commutes;

2. φg2·g1(s) = φg2 ◦ φg1(s) for every U ⊂ X and s ∈ F (U ).

We will also denote φg by g and φg2 ◦ φg1 by g2· g1.

Definition 3.7. Let F be a sheaf on X. Let S(X, F ) be the vector space consisting of the compactly supported sections s ∈ F (X). This means that for every s ∈ S(X, F ) there exists a compact (hence closed) subspace U ⊂ X such that ρX,Uc(s) is the zero

map. Hence S(X, F ) is a linear subspace of F (X).

Corollary 3.8. Let X be a one-point space and let F be a sheaf on X. Then S(X, F ) = F (X).

Proof. This is immediate since every one-points space is compact.

Proposition 3.9. Let F be a sheaf on X together with a G-action as in Definition 3.6. The action of G on the sheaf F induces a G-representation on the vector space S(X, F ). Proof. By Lemma 1.9, supp(f ) can be covered by a finite open compact coveringS

j∈JUj

which are pairwise disjoint. Hence f equals the finite sum of sections sj := iUj(f |Uj) ∈ S(X, F ).

By compatibility of the G-action on F , we are only required to show the proposition for sections s ∈ S(X, F ) with support in an open compact subset U of X. By Definition 3.7 and Lemma 2.22, such a section s is in the image of the map iU in F (X) from a unique

section sU ∈ F (U ) for some open compact subset U ⊂ X. By Property 1 of Definition

3.6, g · s ∈ F (X) is the image of igU from a unique section g · sU ∈ F (gU ). Since gU is

also compact, we have that g · f ∈ S(X, F ). Assosiativity of the action of G is due to Property 2 of Definition 3.6.

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Definition 3.10. Let F be a sheaf on X together with a G-action on F . Then F is called a G-equivariant sheaf over X if the induced G-representation on S(X, F ) defined in Proposition 3.9 is a smooth representation.

Definition 3.11. Let F1 and F2 be two G-equivariant sheaves on X. A morphism φ

from F1 to F2 is a morphism of sheaves

φ : F1 → F2

such that for every g ∈ G and for every open U ⊂ X, the following diagram commutes: F₁(U ) g // φU F₁(gU ) φgU F2(U ) g _// F2(gU )

Definition 3.12. G-Equivariant sheaves on X defined in Definition 3.11 together with the morphisms defined in Definition 3.11 forms a category which we will denote by ShG(X).

Proposition 3.13. Let φ : F1 → F2 be a morphism in ShG(X). The morphism φX on

global sections induces a morphism S(X, F1) → S(X, F2).

Proof. We need to show that for all f ∈ S(X, F1) we have that φX(f ) ∈ S(X, F2).

To achieve this, we only need to verify the above statement for sections f ∈ S(X, F1)

with support on some compact open subset U . By Lemma 2.22 and Definition 3.7 we have that f is contained in the image of (i1)U : F1(U ) → F1(X) (extending by zero).

Hence by the compatibility conditions of a morphism of sheaves, we have that φX(f ) is

contained in the image of (i2)U : F2(U ) → F2(X) which implies that φX(f ) ∈ S(X, F ).

Hence φX induces a map S(X, F1) → S(X, F2).

3.3 Equivalence of Sh

G

(X) and Sh

G

(X)

In the previous sections, we have seen two notions of G-equivariant sheaves. The follow-ing theorem shows that these notions are equivalent:

Theorem 3.14. We have a natural equivalence of categories ShG(X) ' ShG(X).

We will prove theorem 3.14 by constructing two functors F : ShG(X) → ShG(X)

and

H : ShG(X) → ShG(X)

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Constructing the functor F

Let (F , α) ∈ ShG(X). Our goal is to naturally construct a G-action on F from α.

Consider the homeomorphism

j : G × X → G × X, (g, x) 7→ (g, g−1x) and the morphism

ig : X → G × X, x 7→ (g, x).

We have the identity a ◦ j = prX.

The remainder of the proof is based on the following commutative diagram:

X oo _a G × X G × X j oo π uu _X ig oo _.

At first, we see that j∗a∗F ' π∗F since a ◦ j = π. Observe that for any open O ⊂ G × X we have

a∗F (O) = j∗a∗F (j−1(O)) = j∗a∗F ({(g, gx) | (g, x) ∈ O}) since j is a homeomorphism. We also have the natural isomorphism

j∗a∗F ({(g, gx) | (g, x) ∈ O}) ' π∗F ({(g, gx) | (g, x) ∈ O}) since a ◦ j = π.

Observe that α induces an isomorphism

αO: π∗F (O) → a∗F (O).

Thus composing with αO we get the isomorphism

ϕO : π∗F (O)

αO

−−→ a∗F (O)−∼→ π∗F ({(g, gx) | (g, x) ∈ O}). Consider the following operations: for g ∈ G an s ∈ F (U ), we define

g · s = i∗_g◦ φ_G×U(π∗(s)).

Lemma 3.15. The operation g · s defined above is an G-action on F .

Proof. We only need to show that this operation satisfies the two properties of Definition 3.6.

1. Property 1 of Definition 3.6 is satisfied since α and the natural isomorphism j∗a∗F −∼→ π∗F are compatible with restriction maps;

2. Property 2 of Definition 3.6 is also satisfied. If this wasn’t the case, there exists g1, g2 ∈ G and U ⊂ X open such that (g1· g2)s 6= g1· (g2· s). In particular there

exists a point x ∈ U such that

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which implies that (since on the level of stalks we have g · sx = α(g,x)(sx) for

s ∈ F (U ) and x ∈ U )

α_(g₁_,a(g₂_,x))◦ α_(g₂_,x)(sx) 6= α(g1g2,x)(sx)

which contradicts the cocycle condition. Hence property 2 must be true.

More is true: the sheaf F together with the action defined in Lemma 3.15 is an object in ShG(X). To show this, we need to show that the induced G-representation on S(X, F ) is smooth. Take an open compact subset U ⊂ X and identity e ∈ G. Consider a neighborhood V of e such that V U = U which exists from Lemma 1.6. Hence for all v ∈ V , we have F (vU ) = F (U ) which implies that v induces a map

v : F (U ) → F (U ).

Consider an arbitrary section s ∈ F (U ). The goal is to show that there is a smaller neighborhood V0 of e such that for all v0 ∈ V, v0· s = s. As every element of S(X, F ) has support on some open compact set by definition, it will prove the smoothness condition. Lemma 3.16. Let U ⊂ X open and compact and consider a section f ∈ π∗F (G × U ). Then the map

µ : G → F (U ), g 7→ i∗_g(f ) is a locally constant map.

Proof. By sheafification, there exists an open compact coveringS

i∈Iπ(Gi× Ui) of G × U

such that f |Gi×Ui is represented in F (Ui). For g ∈ G, let Ig := {i ∈ I | g ∈ Gi}. Note

that

U = [

i∈Ig

π(Gi× Ui).

Since U is compact and π is an open mapping, there exists I_g0 ⊂ I_g finite such that U =S

i∈I0

gπ(Gi× Ui). Taking Gg =

T

i∈I0

gGi we have that Gg is an open neighborhood

of g ∈ G such that µ is constant on Gg. Since this is true for all g ∈ G, µ is locally

constant.

Corollary 3.17. The G-representation on the complex vector space S(X, F ) constructed in Proposition 3.9 is smooth.

Proof. Using Lemma 3.16, the map

G → F (gU ), g 7→ i∗_g(φV ×U(prX∗(s))) := g · s

for g ∈ V is locally constant. Thus there exists an open neighborhood Ge⊂ G of e ∈ G

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Using Lemma 3.15 and Corollary 3.17 we can conclude that (F , α) ∈ ShG(X) naturally

induces a G-action on F such that F ∈ ShG(X).

Lemma 3.18. Let φ : (F1, α1) → (F2, α2) be a morphism in ShG(X). Then the

under-lying morphism φ : F1 → F2 is a morphism in ShG(X).

Proof. We will do this by looking at the stalks. Let U ⊂ X open and s ∈ F1(U ). At the

level of stalks, we have that

su 7→ g · su= α(g,u)(su)

for all u ∈ U . The morphism φ satisfies the commutative diagram (F1)u (α1)(g,u)_// φu (F1)a(g,u) φa(g,u) (F2)u (α2)(g,u)_// (F2)a(g,u)

but commutativity of the diagram is exactly the condition of φ being a morphism in ShG_.

Hence we define functor F as follows:

• F sends (F , α) to F with induced G-action as described above; • F sends φ to φ.

Constructing the functor H

Consider a sheaf F together with a fixed G-action. We want to naturally construct an isomorphism

α : π∗(F ) → a∗(F ).

Using the action of G on F we will construct the family of maps: ϕO: π∗F (O) → π∗F ({(g, gx) | (g, x) ∈ O})

for all O ⊂ G × X open and compact.

At first we construct this map only for sections π∗(f ) ∈ π∗F (V × U ) where V and U are compact open subsets of G and X respectively, f ∈ F (U ), V · U = vU for some v ∈ V and such that

v0· f = v · f for all v0 ∈ V . For such sections π∗(f ) we put

ϕV ×U(π∗(f )) = π∗(v · f ) ∈ π∗F (vU ).

This construction allows us to construct φO for all compact open subsets O ⊂ G × X and

for all sections in F (O). To see this, consider a section s ∈ F (O) and letS

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a finite open compact cover of O such that s |Vi×Ui= π

∗_(f

i) where fi ∈ F (Ui) and such

that Vi· Ui = vi· Ui for some vi ∈ Vi (such a finite cover exists by compactness of O).

Since the induced representation of G on S(X, F ) is smooth, we can find a refinement of the covering such that v_i0· f = v_i· f for all v0

i ∈ Vi.

Hence let ϕO(s) be the finite sum of sections ϕVi×Ui(s |Vi×Ui) for which we have already

defined ϕ. This construction is independent of the chosen cover of O by compatibility of the action of G on fi∈ F (Ui) (property 1 of definition 3.6).

Lemma 3.19. The above constructed map ϕO is an isomorphism. Moreover, the maps

ϕO induces the anticipated isomorphism α : π∗F → a∗F .

Proof. The construction of the inverse map of ϕOis analogous to the construction of ϕO

itself where we now consider V × vU such that V · vU = U instead.

Notice that (j−1)∗π∗F = a∗_{F since π = a ◦ j. It gives us an isomorphism between}

π∗F ({(g, gx) | (g, z) ∈ O}) and a∗F (O) for O ⊂ G × X. If we compose it with ϕO, we

receive an isomorphism of vector spaces αO : π∗F (O) → a∗F (O) for O ⊂ X open and

compact. Since all open compact subsets of X form a basis of X, this extends uniquely to an isomorphism of sheaves α : π∗F → a∗F .

This isomorphism α is the isomorphism we were looking for.

Lemma 3.20. The pair (F , α) as constructed above is a G-equivariant sheaf according to Definition 3.3.

Proof. It remains to show that it satisfies the cocycle condition in Definition 3.3. Looking at the level stalks, we get

α_(g,x)(s |x) = g · s |x

for all sections s ∈ F (U ) such that x ∈ U . Together with property 2 of Definition 3.6, this implies that

α(g1·g2,x)= α(g1,a(g2,x))◦ α(g2,x)

which is exactly the cocycle condition.

Lemma 3.21. Let φ : F1 → F2 be a morphism in ShG(X). Then φ induces a morphism

from (F1, α1) to (F2, α2).

Proof. We need to show that the diagram π∗F₁ π ∗_φ // α1 π∗F₂ α2 α∗F₁ α ∗_φ //_a∗_F

commutes. This is true when the induced diagrams at the stalks commute: (F1)x φx // ·g (F2)x ·g (F1)a(g,x) φa(g,x)_// (F2)a(g,x)

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But this is true by Definition 3.11 which finishes the proof. We define functor the H as follows:

• H sends F to (F , α) as described above; • H sends φ to φ.

Proof of Theorem 3.14. We will finish the proof of Theorem 3.14 by showing that F ◦H = Id_ShG_(X) and H ◦ F = Id_Sh

G(X).

Both functors F and H maps the underlying sheaf to the same sheaf. The G-action on F and the isomorphism α both induces the same maps on stalks which implies that applying G ◦ F to α we get the same α back. The same is true for applying F ◦ H on the action on F . This implies that both F ◦ H = Id_ShG_(X) and H ◦ F = Id_Sh_G_(X) are

true.

Remark. From the proof of Theorem 3.14 we conclude that the isomorphism α implies smoothness of the G-action on S(X, F ) and that the cocycle condition implies associa-tivity of the G-action on F . The other way around is also true

Note. Since ShG(X) and ShG(X) are naturally equivalent we can use any of the two

definitions of a G-equivariant sheaves. For this reason, we won’t make a distinction between these categories and use the notation ShG(X) when talking about the category

of G-equivariant sheaves.

3.4 Proof of Fact 1

We will prove Fact 1 by constructing two functors F : ShG(X) → Rep(G)

and

H : Rep(G) → ShG(X)

and show that these functors induce an equivalence between these categories.

Constructing the functor F

Consider the functor

F : ShG(X) −→ Rep(G)

defined by:

• For F a G-equivariant sheaf, let F (F ) be the vector space F (X) = S(X, F ) (Corol-lary 3.8) where G acts on F (X) as described in Theorem 3.14.

• For φ : F → G a morphism in Sh_G(X), let F (φ) : F (X) → G(X) be the induced morphism between the global sections by Theorem 3.14.

Theorem 3.14 also implies that this is a well-defined functor which sends G-equivariant sheaves to smooth G-representations.

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Constructing the functor H

Consider the functor

H : Rep(G) −→ ShG(X)

defined by

• For (ρ, V ) a smooth G-representation, let H(ρ, V ) be the sheaf G on X where we assigned the vector space V to the one-point space X together and with the G-action ρ on G(X) = V . Since X is a one-point space, ρ induces an G-action on the sheaf G(X) which is smooth on S(X, G) = G(X). Hence H(ρ, V ) is a G-equivariant sheaf;

• For θ : (ρ, V ) → (ρ0, V0) a morphism between smooth G-representations, let H(θ) : H(ρ, V ) → H(ρ0, V0) be the unique morphism such that the induced morphism between the global sections is the map

θ : H(ρ, V )(X) → H(ρ0, V0)(X).

Since X is a one-point space, any sheaf F on X is entirely determined by the global sections and so any morphism between sheaves on X is determined by the morphism between global sections. Moreover, a G-action on a sheaf F is a G-action on the global sections of F . This observation together with Theorem 3.14 implies that this is also a well-defined functor.

Proof of Fact 1. It remains to check whether G ◦ H and F ◦ H are isomorphic to IdShG

and IdRep(G) respectively.

Let F be a G-equivariant sheaf with a smooth G-action ρ : G → Aut(G(X)). By applying the functor F we get the representation (ρ, F (X)) and after applying the functor G, we get a G-equivariant functor with global sections F (X) together with the G-action ρ which is the functor F again. In a diagram, we get

F_ρ−F→ (ρ, F (X))−H→ F_ρ.

We apply a similar procedure for checking the case with morphisms which is described in the diagram below

F G φ −→F F (X) G(X) φ H −→ F G φ

Thus we see that H ◦ F equals the identity functor IdShG.

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3.5 Proof of Fact 2

Proving the second fact requires a bit more work. These are the data we will be working with:

1. Let G be a l-group and let H < G be an open (hence also closed) subgroup; 2. We consider the quotient space X = H \ G with the induced topology together

with the quotient map q : G → H \ G. Note that X is also an l-space;

3. We use the continuous action a : G × X → X, (g, x) → x · g−1 _{where the overline}

indicates the corresponding class.

We have the following trivial result which is a direct consequence of our assumptions above:

Corollary 3.22. The stabilizer of e ∈ X with respect to the action a is the group H. We have a functor

F : ShG(X) → Rep(G)

consisting of the following data:

• For F a G-equivariant sheaf, let F (F ) be the vector space S(X, F ) where G acts on S(X, F ) as described in theorem 3.14;

• For φ : F → G a morphism in Sh_G(X), let F (φ) : S(X, F ) → S(X, G) be the induced morphism between the global sections by Theorem 3.14. This is well-defined by Proposition 3.13.

We will use the following result from category theory:

Corollary 3.23. Let T : A → B be faithful such that for two non-isomorphic objects A, B ∈ A we have that T (A) 6= T (B) (we call the latter property injective on objects). Then T (A) is a category and A is equivalent to T (A).

Proof. See Remark 1.5.8. of [9].

Proposition 3.24. The functor F defined above is faithful and injective on objects. Hence ShG(X) is equivalent to the subcategory F (ShG(X)) of Rep(G).

Proof. Since (ρ1) : S(X, F1) → F1(U ) is surjective, a G-map φX : S(X, F1) → S(X, F2)

uniquely determines a map F1(U ) → F2(U ). Since this is true for all open compact

subsets U ⊂ X and since these sets form a basis of X (1.8), φX uniquely determines a

morphism φ : F1→ F2 between the displayed sheaves which implies faithfulness of F .

Observe that if we have two G-equivariant sheaves F1 and F2 such that F (F1) =

F (F2), we would have that F1 equals F2 since the compactly supported sections

deter-mine the sheaf structure (this is because the open compact subsets of X form a basis of X). Hence F is indeed injective on objects and so by Corollary 3.23 we have that ShG(X) is equivalent to the subcategory F (ShG(X)) of Rep(G).

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Next we consider the functor c-IndG_H : Rep(H) → Rep(G) from Definition 1.22. Corollary 3.25. Rep(H) is equivalent to the subcategory c-Ind(Rep((H)) of Rep(G). Proof. The functor c-IndG_H is faithful by Proposition 1.25. Also Proposition 1.24 implies injectivity of objects. Thus Corollary 3.23 implies that Rep(H) is equivalent to the subcategory c-Ind(Rep((H)) of Rep(G).

We will prove Fact 2 by showing that the two subcategories F (ShG(X)) and

c-Ind(Rep((H)) are equivalent in Rep(G). This is done by showing that every element in F (ShG(X)) is isomorphic to an element in c-Ind(Rep((H)) in the category Rep(G).

The following result will be useful.

Lemma 3.26. Let F ∈ ShG(X). Consider the vector space V := Fe. Define an action

of H on V as follows:

σ : H × V → V, (h, v) 7→ α_(h,e)(v)

which is indeed an action by the cocycle condition. Denote the corresponding represen-tation by σ. Then (σ, V ) is a smooth H-represenrepresen-tation.

Proof. Let v ∈ V := Fe. By assumption, there exists an open compact set U ⊂ X and a

section vU _{∈ F (U ) such that v}U

e = v. Moreover, we can view vU as a section contained

in S(X, F ). By theorem 3.14, G acts smoothly on S(X, F ) so there exists a compact open subgroup K < G such that k · vU = vU for all k ∈ K. Observe that K0 := K ∩ H is also compact open in H since H is closed. By observing the proof of theorem 3.14, we have that for all k0 ∈ K0 that

(k0· vU)e= αa(k0_,e)(v).

Since k0 ∈ K, we have that k0· vU _{= v}U _{which implies that (k}0_{· v}U₎

e= veU for all k0 ∈ K0.

Thus αa(k0_,e)(v) = v which proves that (σ, V ) is indeed smooth.

We will view a section f ∈ S(X, F ) as a function with domain X and f (x) = fx ∈ Fx

for all x ∈ X which has compact support in H \ G (we will also denote this function by f ). Observe that if we let g ∈ G act on f , we get the function

g · f : x 7→ α_(g,a(g−1_,x))(f (a(g−1, x)) ∈ F_x

which has now compact support in a({g} × U ) = U · g−1.

Note. Let V := Fe. The goal is to extend the function f to a function ˜f : G → V . To

do this, we need to make a choice for the representatives of X. Later we will see that the choice of representatives is independent modulo isomorphism which is enough when working in categories.

Let Q be a choice of representatives of X in G. The function f can naturally be composed to a function fQ: X → V as follows: For u ∈ Q we get

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Moreover, f can be naturally extended to a function ˜fQ : G → V in the following way:

For every element z ∈ G, there exists an unique element u ∈ Q and an unique element h ∈ H such that

z = h · u. Thus let

˜

fQ: G → V, z 7→ σ(h) · α(u,˜z)f (˜z) = σ(h) · α(u,˜u)f (˜u).

By the cocycle condition, we know that

σ(h) · α_(x,x)= α_(h,e)◦ α_(x,x)= α_(x·h,x) which implies that

˜

f : z 7→ α(x·h,x)f (x)

We can also define an action on ˜f . For g ∈ G, let g · ˜f : z 7→ ˜f (z · g).

Lemma 3.27. For every y ∈ Q viewed as a subset of G, we have the equality g · ˜fQ(y) = (g · f )Q(y)

which implies

g · ˜fQ(z) = ^(g · f )Q(z)

for all z ∈ G.

Proof. To start off, note that for every z ∈ G there exists a unique element hz ∈ H and

a unique element uz∈ Q such that z · g = hz· uz. This implies that

uz = z · g = a(g−1, z)

On the right hand side, we have

g · ˜fQ(z) = ˜fQ(z · g) = ˜fQ(hz· uz) = α(hz,e)f˜Q(uz)).

Also we have that

α(z,z)α(g,uz) = α(z·g,uz)= α(h,e)α(uz,uz).

which implies that for z ∈ Q we get

(g · f )Q(z) = g · ˜fQ(z)

due to the following computation:

(g · f )Q(z) = α(z,z)α(g,uz)f (uz) = α(h,e)α(uz,uz)f (uz) := α(h,e)f˜Q(uz) := g · ˜fQ(z).

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Proposition 3.28. Let (σ, V ) as in Lemma 3.26 and let Xc = c-IndGH(V ), hence Xc

is the set of compactly supported functions on G modulo H over the H-representation (σ, V ). Then the map

CQ : S(X, F ) −→ Xc, f 7−→ ˜fQ

is a well-defined isomorphism in Rep(G).

If Q∗ is another set of representatives of X in G, then we have an isomorphism CQ(S(X, F )) ' CQ∗(S(X, F ))

in Rep(G).

Proof. By the discussion above, we see that ˜f ∈ Xc (in fact, the construction of ˜f was

made in such a way that it would be an element in Xc). The same argument is true for

showing that C is a G-map.

An inverse of the morphism C is the following natural map: C−1 : Xc−→ S(X, F ), f 7−→ [f : x ∈ X 7→ ˜˜ f (x)].

Hence C is an isomorphism in Rep(G). Now consider the morphism

T : CQ(S(X, F )) → CQ∗(S(X, F )), f˜_Q7→ ˜f_Q∗

Lemma 3.27 implies that this is indeed a morphism in Rep(G) (compatible with the G-action). An inverse map is the map

T−1 : CQ∗(S(X, F )) → C_Q(S(X, F )), f˜_Q∗ 7→ ˜f_Q

which shows that T is an isomorphism in Rep(G).

The final step of proving Fact 2 is to show that given a representation (ρ, V ) ∈ Rep(H), there exists a sheaf F ∈ ShG(X) such that

C(F (F )) = c-IndG_H(V ). We start by constructing a natural sheaf on G.

Lemma 3.29. Consider the presheaf S on G such that for every open O ⊆ G we assign the set S(O) consisting of functions ˜f : G → V satisfying the following properties:

1. ∀(h, g) ∈ H × G : ˜f (hg) = ρ(h) ˜f (g); 2. ˜f is locally constant;

3. supp( ˜f ) ⊆ O.

Then S together with the natural maps is a sheaf. Proof. See [10, Propostion 11].

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Moreover, we define an action of G on the sheaf S. For g ∈ G and ˜f ∈ S(O) we have g · ˜f : G → V, z 7→ ˜f (z · g).

This implies that g · ˜f ∈ S(O · g−1) hence we have that S ∈ ShG(G) by Theorem 3.14.

Observe that a function ˜f ∈ S(O) is uniquely determined by the values of ˜f over a set of representatives of X in G (call this set Q) hence we only need to consider ˜f |Q

instead. For U ⊆ X open, let F (U ) consist of functions ˜f |Q where ˜f ∈ S(q−1(U )),

hence we have an isomorphism

S(q−1(U )) −→ F (U ), f 7−→ f |Q.

where the G-action on S naturally induces a G-action on F . Thus by Theorem 3.14 we have that F ∈ ShG(X). Observe that

S(X, F ) = c-IndG_H(V )

and that the action of G constructed on S(X, F ) coincides with the G-action c-IndG_H(ρ) on c-IndG_H(V ). Thus we have proven the following result:

Proposition 3.30. Given a representation (ρ, V ) ∈ Rep(H), there exists a sheaf F ∈ ShG(X) such that

Bernstein's New Approach to Representation Theory

MSc Mathematics

Master Thesis