Geometry of canonical bases and mirror symmetry

# Geometry of canonical bases and mirror symmetry

Alexander Goncharov, Linhui Shen
###### Abstract

A decorated surface is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let be a split reductive group over .

A pair gives rise to a moduli space , closely related to the moduli space of -local systems on . It is equipped with a positive structure [FG1]. So a set of its integral tropical points is defined. We introduce a rational positive function on the space , called the potential. Its tropicalisation is a function . The condition defines a subset of positive integral tropical points . For , we recover the set of positive integral -laminations on from [FG1].

We prove that when is a disc with special points on the boundary, the set parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence [L4], [G], [MV], [BD] they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group :

 (Vλ1⊗…⊗Vλn)GL. (1)

When , , there is a special coordinate system on [FG1]. We show that it identifies the set with Knutson-Tao’s hives [KT]. Our result generalises a theorem of Kamnitzer [K1], who used hives to parametrise top components of convolution varieties for , . For , , we prove Kamnitzer’s conjecture [K1]. Our parametrisation is naturally cyclic invariant. We show that for any and it agrees with Berenstein-Zelevinsky’s parametrisation [BZ], whose cyclic invariance is obscure.

We define more general positive spaces with potentials , parametrising mixed configurations of flags. Using them, we define a generalization of Mirković-Vilonen cycles [MV], and a canonical basis in , generalizing the Mirković-Vilonen basis in . Our construction comes naturally with a parametrisation of the generalised MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer [K].

We prove that the set parametrises top dimensional components of a new moduli space, surface affine Grasmannian, generalising the fibers of the convolution maps. These components are usually infinite dimensional. We define their dimension being an element of a -torsor, rather then an integer. We define a new moduli space , which reduces to the moduli spaces of -local systems on if has no special points. The set parametrises a basis in the linear space of regular functions on .

We suggest that the potential itself, not only its tropicalization, is important – it should be viewed as the potential for a Landau-Ginzburg model on . We conjecture that the pair is the mirror dual to . In a special case, we recover Givental’s description of the quantum cohomology connection for flag varieties and its generalisation [GLO2]. [R2]. We formulate equivariant homological mirror symmetry conjectures parallel to our parametrisations of canonical bases.

## 1 Introduction

### 1.1 Geometry of canonical bases in representation theory

#### 1.1.1 Configurations of flags and parametrization of canonical bases

Let be a split semisimple simply-connected algebraic group over . There are several basic vector spaces studied in representation theory of the Langlands dual group :

1. The weight component in the universal enveloping algebra of the maximal nilpotent Lie subalgebra in the Lie algebra of .

2. The weight subspace in the highest weight representation of .

3. The tensor product invariants

4. The weight subspaces in the tensor products .

Calculation of the dimensions of these spaces, in the cases 1)-3), is a fascinating classical problem, which led to Weyl’s character formula and Kostant’s partition function.

The first examples of special bases in finite dimensional representations are Gelfand-Tsetlin’s bases [GT1], [GT2]. Other examples of special bases were given by De Concini-Kazhdan [DCK].

The canonical bases in the spaces above were constructed by Lusztig [L1], [L3]. Independently, canonical bases were defined by Kashiwara [Ka]. Canonical bases in representations of were defined by Gelfand-Zelevinsky-Retakh [GZ], [RZ].

Closely related, but in general different bases were considered by Nakajima [N1], [N2], Malkin [Ma], Mirković-Vilonen [MV], and extensively studied afterwords. Abusing terminology, we also call them canonical bases.

It was discovered by Lusztig [L] that, in the cases 1)-2), the sets parametrising canonical bases in representations of the group are intimately related to the Langlands dual group .

Kashiwara discovered in the cases 1)-2) an additional crystal structure on these sets, and Joseph proved a rigidity theorem [J] asserting that, equipped with the crystal structure, the sets of parameters are uniquely determined.

One of the results of this paper is a uniform geometric construction of the sets parametrizing all of these canonical bases, which leads to a natural uniform construction of canonical bases parametrized by these sets in the cases 2)-4). In particular, we get a new canonical bases in the case 4), generalizing the Mirković-Vilonen (MV) basis in . To explain our set-up let us recall some basic notions.

A positive space is a space, which could be a stack whose generic part is a variety, equipped with a positive atlas. The latter is a collection of rational coordinate systems with subtraction free transition functions between any pair of the coordinate systems. Therefore the set of the integral tropical points of is well defined. We review all this in Section 2.1.1.

Let be a positive pair given by a positive space equipped with a positive rational function . Then one can tropicalize the function , getting a -valued function

 Wt:Y(Zt)⟶Z.

Therefore a positive pair determines a set of positive integral tropical points:

 Y+W(Zt):={l∈Y(Zt) | Wt(l)≥0}. (2)

We usually omit in the notation and denote the set by .

To introduce the positive pairs which play the basic role in this paper, we need to review some basic facts about flags and decorated flags in .

##### Decorated flags and associated characters.

Below is a split reductive group over . Recall that the flag variety parametrizes Borel subgroups in G. Given a Borel subgroup B, one has an isomorphism .

Let be the adjoint group of . The group acts by conjugation on pairs , where is an additive character of a maximal unipotent subgroup in . The subgroup stabilizes each pair . A character is non-degenerate if is the stabilizer of . The principal affine space111Inspite of the name, it is not an affine variety. parametrizes pairs where is a non-degenerate additive character of a maximal unipotent group . Therefore there is an isomorphism

 iχ:AG′\lx@stackrel∼⟶G′/U.

This isomorphism is not canonical: the coset does not determine a point of . To specify a point one needs to choose a non-degenerate character . One can determine uniquely the character by using a pinning, see Sections 2.1.2-2.1.3. So writing we abuse notation, keeping in mind a choice of the character , or a pinning.

Having said this, one defines the principal affine space for the group by We often write instead of . The points of are called decorated flags in G. The group acts on from the left. For each , let be its stabilizer. It is a maximal unipotent subgroup of . There is a canonical projection

 π:A⟶B,    \rmπ(A):= the % normalizer of UA. (3)

The projection gives rise to a map whose fibers are torsors over the center of . Let . Here is a maximal unipotent subgroup of . It is identified with a similar subgroup of , also denoted by . So a decorated flag in provides a non-degenerate character of the maximal unipotent subgroup in :

 χA:UA⟶A1. (4)

Clearly, if , then , and

 χA(u)=χg⋅A(gug−1). (5)
##### Example.

A flag for is a nested collection of subspaces in an -dimensional vector space equipped with a volume form :

 F∙=F0⊂F1⊂…⊂Fm−1⊂Fm,    dimFi=i.

A decorated flag for is a flag with a choice of non-zero vectors for each , called decorations. For example, parametrises non-zero vectors in a symplectic space . The subgroup preserving a vector is given by transformations . Its character is given by .

Our basic geometric objects are the following three types of configuration spaces:

 Confn(A)=G∖An,    Conf(An,B):=G∖(An×B),    Conf(B,An,B):=G∖(B×An×B). (6)
##### The potential W.

A key observation is that there is a natural rational function

 χo:Conf(B,A,B)=G∖(B×A×B)⟶A1.

Let us explain its definition. A pair of Borel subgroups is generic if is a Cartan subgroup in . A pair is generic if the pair is generic. Generic pairs form a principal homogeneous -space. Thus, given a triple such that and are generic, there is a unique such that

 {A2,B3}=u⋅{A2,B1}. (7)

So we define Using it as a building block, we define a positive rational function on each of the spaces (6).

For example, to define the on the space we start with a generic collection , set , and define as a sum, with the indices modulo :

 W:Confn(A)⟶A1,    W(A1,...,An):=n∑i=1χo(Bi−1,Ai,Bi+1). (8)

Note that the potential is well-defined when each adjacent pair is generic, meaning that is generic. Assigning the (decorated) flags to the vertices of a polygon, we picture the potential as a sum of the contributions at the -vertices (shown boldface) of the polygon, see Fig 1.

By construction, the potential on the space is the pull back of the potential for the adjoint group via the natural projection :

 WG=p∗G→G′WG′. (9)

Potentials for the other two spaces in (6) are defined similarly, as the sums of the characters assigned to the decorated flags of a configuration. A formula similar to (9) evidently holds.

##### Parametrisations of canonical bases.

It was shown in [FG1] that all of the spaces (6) have natural positive structures. We show that the potential is a positive rational function.

We prove that the sets parametrizing canonical bases admit a uniform description as the sets of positive integral tropical points assigned to the following positive pairs . To write the potential we use an abbreviation , with indices mod :

1. The canonical basis in :

 Y=Conf(B,A,B),   W(B1,A2,B3):=χA2.
2. The canonical basis in :

 Y=Conf(A,A,B),   W(A1,A2,B3):=χA1+χA2.
3. The canonical basis in invariants of tensor product of irreducible -modules:

 Y=Confn(A),   W(A1,…,An):=n∑i=1χAi. (10)
4. The canonical basis in tensor products of irreducible -modules:

 Y=Conf(An+1,B),   W(A1,…,An+1,B):=n+1∑i=1χAi. (11)

Natural decompositions of these sets, like decompositions into weight subspaces in 1) and 2), are easily described in terms of the corresponding configuration space, see Section 2.3.2.

Let us emphasize that the canonical bases in tensor products are not the tensor products of canonical bases in irreducible representations. Similarly, in spite of the natural decomposition

 Vλ1⊗...⊗Vλn=⊕λVλ⊗(V∗λ⊗Vλ1⊗...⊗Vλn)GL,

the canonical basis on the left is not a product of the canonical bases on the right.

Descriptions of the sets parametrizing the canonical bases were known in different but equivalent formulations in the following cases:

In the cases 1)-2) there is the original parametrization of Lusztig [L].

In the case 3) for , there is Berenstein-Zelevinsky’s parametrization [BZ], referred to as the BZ data. We produce in Appendix an isomorphism between our parametrization and the BZ data. The cyclic symmetry, evident in our approach, is obscure for the BZ data.

The description in the case in 3) seems to be new.

The cases 1), 2) and 4) were investigated by Berenstein and Kazhdan [BK1],[BK2], who introduced and studied geometric crystals as algebraic-geometric avatars of Kashiwara’s crystals. In particular, they describe the sets parametrizing canonical bases in the form (2), without using, however, configuration spaces. We show in Appendix A that the space of generic configurations with the potential is a positive decorated geometric crystal in the sense of [BK3]. Interpretation of geometric crystals relevant to representation theory as moduli spaces of mixed configurations of flags makes, to our opinion, the story more transparent.

To define canonical bases in representations, one needs to choose a maximal torus in and a positive Weyl chamber. Usual descriptions of the sets parametrizing canonical bases require the same choice. Unlike this, working with configurations we do not require such choices.222 We would like to stress that the positive structures and potentials on configuration spaces which we employ for parametrization of canonical bases do not depend on any extra choices, like pinning etc., in the group. See Section 6.3.

Most importantly, our parametrization of the canonical basis in tensor products invariants leads immediately to a similar set which parametrizes a linear basis in the space of functions on the moduli space of -local systems on a decorated surface . Here the approach via configurations of decorated flags, and in particular its transparent cyclic invariance, are essential. See the example when in Section 1.3.1.

Summarizing, we understood the sets parametrizing canonical bases as the sets of positive integral tropical points of various configuration spaces. Let us show now how this, combined with the geometric Satake correspondence [L4], [G], [MV], [BD], leads to a natural uniform construction of canonical bases in the cases 2)-4).

We explain in Section 1.1.2 the construction in the case of tensor products invariants. A canonical basis in this case was defined by Lusztig [L3]. However Lusztig’s construction does not provide a description of the set parametrizing the basis. Our basis in tensor products is new – it generalizes the MV basis in . We explain this in Section 2.4.

#### 1.1.2 Constructing canonical bases in tensor products invariants

We start with a simple general construction. Let be a positive space, understood just as a collection of split tori glued by positive birational maps [FG1]. Since it is a birational notion, there is no set of -points of , where is a field. Let . In Section 2.2.1 we introduce a set We call it the set of transcendental -points of . It is a set making sense of “generic -points of ”. In particular, if is given by a variety with a positive rational atlas, then . The set comes with a natural valuation map:

 val:Y∘(K)⟶Y(Zt).

For any , we define the transcendental cell assigned to :

 C∘l:=val−1(l)⊂Y∘(K),    Y∘(K)=∐l∈Y(Zt)C∘l.

Let us now go to canonical bases in invariants of tensor products of -modules (1). The relevant configuration space is The tropicalized potential determines the subset of positive integral tropical points:

 Conf+n(A)(Zt):={l∈Confn(A)(Zt) | Wt(l)≥0}. (12)

We construct a canonical basis in (1) parametrized by the set (12).

Let . In Section 2.2.2 we introduce a moduli subspace

 ConfOn(A)⊂Confn(A)(K). (13)

We call it the space of -integral configurations of decorated flags. Here are its crucial properties:

1. A transcendental cell of is contained in if and only if it corresponds to a positive tropical point. Moreover, given a point , one has

 l∈Conf+n(A)(Zt) ⟺ C∘l⊂ConfOn(A) ⟺ C∘l∩ConfOn(A)≠∅. (14)
2. Let be the affine Grassmannian. It follows immediately from the very definition of the subspace (13) that there is a canonical map

 κ:ConfOn(A)⟶Confn(Gr):=G(K)∖(Gr)n.

These two properties of allow to transport points into the top components of the stack . Namely, given a point , we define a cycle

 Ml:=closure of M∘l⊂Confn(Gr),    where M∘l:=κ(C∘l).

The cycle is defined for any . However, as it clear from (14), the map can be applied to it if and only if is positive: otherwise is not in the domain of the map .

We prove that the map provides a bijection

 Conf+n(A)(Zt)\lx@stackrel∼⟶{\rm closures of the top dimensional components of the stack Confn(Gr)}. (15)

Here the very notion of a “top dimensional” component of a stack requires clarification. For now, we will bypass this question in a moment by passing to more traditional varieties.

We use a very general argument to show the injectivity of the map . Namely, given a positive rational function on , we define a -valued function on . It generalizes the function on the affine Grassmannian for and its products defined by Kamnitzer [K], [K1]. We prove that the restriction of to is equal to the value of the tropicalization of at the point . Thus the map (15) is injective.

Let us reformulate our result in a more traditional language. The orbits of acting on are labelled by dominant weights of . We write if is in the orbit labelled by . Let be the identity coset in . A set of dominant weights of determines a cyclic convolution variety, better known as a fiber of the convolution map:

 Grc(λ––):={(L1,…,Ln) | L1\lx@stackrelλ1⟶L2\lx@stackrelλ2⟶…\lx@stackrelλn⟶Ln+1, L1=Ln+1=[1]}⊂[1]×Grn−1. (16)

These varieties provide a -equivariant decomposition

 [1]×Grn−1=∐λ––=(λ1,...,λn)Grc(λ––). (17)

Since is connected, it preserves each component of . Thus the components of live naturally on the stack

 Confn(Gr)=G(O)∖([1]×Grn−1).

We prove that the cycles assigned to the points are closures of the top dimensional components of the cyclic convolution varieties. The latter, due to the geometric Satake correspondence, give rise to a canonical basis in (1). We already know that the map (15) is injective. We show that the -components of the sets related by the map (15) are finite sets of the same cardinality, respected by the map. Therefore the map (15) is an isomorphism.

Our result generalizes a theorem of Kamnitzer [K1], who used hives [KT] to parametrize top components of convolution varieties for , .

Our construction generalizes Kamnitzer’s construction of parametrizations of Mirković-Vilonen cycles [K]. At the same time, it gives a coordinate free description of Kamnitzer’s construction.

When , there is a special coordinate system on the space , introduced in Section 9 of [FG1]. We show in Section 3 that it provides an isomorphism of sets

 Conf+3(A)(Zt) \lx@stackrel∼⟶ {Knutson-Tao's hives \@@cite[cite]{[% \@@bibref{}{KT}{}{}]}}.

Using this, we get a one line proof of Knutson-Tao-Woodward’s theorem [KTW] in Section 2.1.6.

For , , we prove Kamnitzer conjecture [K1], describing the top components of convolution varieties via a generalization of hives – we identify the latter with the set via the special positive coordinate systems on from [FG1].

### 1.2 Positive tropical points and top components

#### 1.2.1 Our main example

Denote by the subvariety of parametrizing configurations of decorated flags such that the flags are in generic position for each modulo . The potential was defined in (8). It is evidently a regular function on .

Let be the cone of dominant coweights. There are canonical isomorphisms

 α:Conf×2(A)\lx@stackrel∼⟶H,    Conf2(Gr)=P+. (18)

Configurations sit at the vertices of a polygon, as on Fig 2. Let be the projection corresponding to a side of the polygon. Denote by its restriction to . The collection of the maps , followed by the first isomorphism in (18) provides a map

 π:Conf×n(A)⟶Conf×2(A)n\lx@stackrelα=Hn.

Using similarly the second isomorphism in (18), we get a map

 πGr:Confn(Gr)⟶Conf2(Gr)n=(P+)n.

Let be a basis of the cone of positive dominant weights of . The functions are equations of the irreducible components of the divisor :

 D:=Confn(A)−Conf×n(A)=∪E,iDEi.

Equivalently, the component is determined by the condition that the pair of flags at the endpoints of the edge belongs to the codimension one -orbit corresponding to the simple reflection . 333Indeed, if and only if the corresponding pair of flags belongs to the codimension one -orbit corresponding to a simple reflection .

The space has a cluster -variety structure, described for in [FG1], Section 10. An important fact [FG5] is that any cluster -variety has a canonical cluster volume form , which in any cluster -coordinate system is given by

 ΩA=±dlogA1∧…∧dlogAn.

The functions are the frozen -cluster coordinates in the sense of Definition 12.5. This is equivalent to the claim that the canonical volume form on has non-zero residues precisely at the irreducible components of the divisor .444Indeed, it follows from Lemma 12.3 and an explcit description of cluster structure on that the form can not have non-zero residues anywhere else the divisors . One can show that the residues at these divisors are non-zero.

All this data is defined for any split semi-simple group over . Indeed, the form on for the simply-connected group is invariant under the action of the center the group, and thus its integral multiple descends to a form on . The potential is defined by pulling back the potential for the adjoint group . We continue discussion of this example in Section 1.4, where it is casted as an example of the mirror symmetry.

##### The simplest example.

Let be a two dimensional vector space with a symplectic form. Then , and . Next, is the space of configuration of non-zero vectors in . Set . Then the potential is given by the following formula, where the indices are mod :

 W:=n∑i=1Δi,i+2Δi,i+1Δi+1,i+2. (19)

The boundary divisors are given by equations . To write the volume form, pick a triangulation of the polygon whose vertices a labeled by the vectors. Then, up to a sign,

 Ω:=⋀EdlogΔE.

where are the diagonals and sides of the -gon, and if . The function (19) is invariant under , and thus descends to .

#### 1.2.2 The general framework

Let us explain main features of the geometric picture underlying our construction in most general terms, which we later on elaborate in details in every particular situation. First, there are three main ingredients:

1. A positive space with a positive rational function called the potential, and a volume form with logarithmic singularities. This determines the set of positive integral tropical points – the set parametrizing a canonical basis.555The set , the tropicalization , and thus the subset can also be determined by the volume form , without using the positive structure on .

2. A subset of -integral points . Its key feature is that, given an ,

 l∈Y+W(Zt) ⟺ C∘l⊂YO ⟺ C∘l∩YO≠∅. (20)
3. A moduli space , together with a canonical map

 κ:YO⟶GrY,W. (21)

These ingredients are related as follows:

• Any positive rational function on gives rise to a -valued function on , such that for any , the restriction of to equals .

So we arrive at a collection of irreducible cycles

 M∘l:=κ(C∘l)⊂GrY,W,    Ml:=closure of M∘l,   l∈Y+W(Zt).

Thanks to the , the assignment is injective.

Consider the set of all irreducible divisors in such that the residue of the form at is non-zero. We call them the boundary divisors of . We define

 Y×:=Y−∪Dc. (22)

By definition, the form is regular on . In all examples the potential is regular on .

There is a split torus , and a positive regular surjective projection

 π:Y×⟶H.

The map is determined by the form . For example, assume that each boundary divisor is defined by a global equation . Then the regular functions define the map , i.e. .

Next, there is a semigroup containing , defining a cone

 P:=HO/H(O)⊂H(Zt):=H(K)/H(O)=X∗(H),

such that the tropicalization of the map provides a map , and there is a surjective map . Denote by restricting of to . These maps fit into a commutative diagram

 Y+W(Zt)\lx@stackrelval⟵YO\lx@stackrelκ⟶GrY,Wπt↓πO↓↓πGrP\lx@stackrelval⟵HO\lx@stackrelval⟶P (23)

We define and as the fibers of the maps and over a . So we have

 GrY,W=∐λ∈PGr(λ)Y,W,    Y+W(Zt)=∐λ∈PY+W(Zt)λ. (24)

The following is a key property of our picture:

• The map provides a bijection

 Y+W(Zt)λ  ⟷  {\rm Closures of top dimensional components of Gr(λ)Y,W}.

Although the space is usually infinite dimensional, it is nice. The map slices it into highly singular and reducible pieces. However the slicing makes the perverse sheaves geometry clean and beautiful. It allows to relate the positive integral tropical points to the top components of the slices.

##### Example.

In our main example, discussed in Section 1.1 we have

 Y=Confn(A),    Y×=Conf×n(A),    YO=ConfOn(A),    GrY,W=Confn(Gr),    H=Hn,    P=(P+)n.

The potential is defined in (8), and decomposition (24) is described by cyclic convolution varieties (17).

#### 1.2.3 Mixed configurations and a generalization of Mirković-Vilonen cycles

Let us briefly discuss other examples relevant to representation theory. All of them follow the set-up of Section 1.2. The obtained cycles can be viewed as generalisations of Mirković-Vilonen cycles. Let us list first the spaces and . The notation indicates that the pair of the first and the last flags in configuration is in generic position.

i) Generalized Mirković-Vilonen cycles:

 Y:=Confw0(A,An,B),    GrY,W:=Confw0(A,Grn,B)=Grn.

If , we recover the Mirković-Vilonen cycles in the affine Grassmannian [MV].

ii) Generalized stable Mirković-Vilonen cycles:

 Y:=Conf