Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types
Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel, Tubbenhauer, Xiaoting Zhang
TL;DR
This paper proves that for finite Coxeter types, Soergel bimodules have finitely many simple transitive 2-representations and completes their classification for most types.
Contribution
It provides a classification of simple transitive 2-representations of Soergel bimodules for finite Coxeter types, except H3 and H4.
Findings
Finite Coxeter types have finitely many simple transitive 2-representations.
Complete classification achieved for all types but H3 and H4.
Abstract
In this paper we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive -representations and we complete their classification in all types but and .
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Simple transitive -representations of Soergel bimodules for finite Coxeter types
Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz,
Daniel Tubbenhauer and Xiaoting Zhang
Abstract.
In this paper we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive -representations and we complete their classification in all types but and .
{NoHyper}††Mathematics Subject Classification 2020. Primary: 18N10, 18D25; Secondary: 16D20, 18A50, 18N25, 20C08.{NoHyper}††Keywords. -categories, -representations, Soergel bimodules, Lusztig’s asymptotic bicategories.
Contents
- 1 Introduction
- 2 Recollections
- 3 Soergel bimodules and the asymptotic bicategory
- 4 The role of the Duflo involution
- 5 Lifted simple transitive 2-representations
- 6 Separability and semisimplicity
- 7 The main theorem
- 8 Classification results
1. Introduction
The series of papers [MM1], [MM2], [MM3], [MM4], [MM5], [MM6] laid the foundations for a systematic theory of finitary -representations of finitary -categories, which is a categorical analog of the theory of finite dimensional representations of finite dimensional algebras. This theory is a part of modern -representation theory which originated in seminal papers like [Os3], [ENO], [CR] and [KhLa]. The main difference to the “alternative” theory of module categories over tensor categories, see [EGNO] and references therein, is not that -categories can have more than one object: every finitary -category gives rise to a multifinitary monoidal category with the same finitary -representation theory and vice versa, see [EGNO, Remark 4.3.7] and [MMMTZ, Subsection 2.4]. The main difference is that finitary -representations and -categories are not assumed to be abelian, but additive and idempotent complete. This is important for the main character of this paper, the one-object -linear -category of Soergel bimodules for a Coxeter system of finite type over the coinvariant algebra of , which is finitary but not abelian. (Our choice of ground field in this paper is crucial, as S behaves very differently over a field of positive characteristic.) One can, of course, consider an abelianization of a finitary -category C , but such abelianizations do not admit adjunctions in general, which is a serious obstruction. Recall that a finitary -category is (quasi) fiat if it has a duality structure satisfying conditions akin to those for (rigid or) pivotal monoidal categories (this is the categorical analog of an algebra having an involution). Tensor categories are rigid by assumption and many of the structural results on their module categories do not hold without that assumption, see [EGNO], but is not equivalent to a multitensor category for a general quasi fiat -category C . For this reason, there are many differences between the representation theories of quasi fiat -categories and multitensor categories, e.g. cell structures have played a key role in the first one but no role at all in the second. Even the formulation of analogous results in the two theories, and the respective proofs, can have important differences, e.g. the role of coalgebras in quasi fiat -categories is analogous to that of algebras in multitensor categories, but technically more involved, compare e.g. [MMMT, Section 4] and [EGNO, Subsections 7.8 to 7.10]. However, the two representation theories coincide when C is fusion or pivotal fusion, i.e. locally semisimple (meaning that morphism categories are semisimple) and quasi fiat or fiat, which is very helpful for the purpose of this paper, as we will explain below.
A key result in finitary -representation theory is the (weak) Jordan–Hölder theorem [MM5, Theorem 8] for finitary -representations, in which the role of the simples is played by the so-called simple transitive -representations. Classifying the simple transitive -representations of a given finitary -category C is therefore a fundamental problem in -representation theory, which we will refer to as the Classification Problem. When C is fusion, there are only finitely many equivalence classes of simple transitive -representations by (a consequence of) Ocneanu rigidity, see e.g. [EGNO, Proposition 3.4.6 and Corollary 9.1.6]. For certain fusion -categories those equivalence classes have even been classified explicitly, see Section 8. However, when C is not locally semisimple, that finiteness result need not be true, see e.g. [EO, Subsection 4.3], and in most cases the Classification Problem is unsolved.
In this paper we address the Classification Problem for the fiat -category S of Soergel bimodules. This -category, which is naturally graded and categorifies the corresponding Hecke algebra , was introduced by Soergel [So1], [So3] to give an alternative proof of the famous Kazhdan–Lusztig conjectures of positive integrality in the case of Weyl groups and provides a framework in which to generalize and approach these conjectures for arbitrary Coxeter groups, see e.g. [So2]. A general and completely algebraic proof using Soergel bimodules was eventually found by Elias and Williamson [EW3]. Since their introduction, Soergel bimodules and their (graded) -representations have played a fundamental role in representation theory, both in characteristic zero, see e.g. [KiMa], [KMM] (and the above mentioned papers), and, more recently, in positive characteristic, see e.g. [LW], [RW]. It is therefore interesting to study its (graded) -representation theory, initially in characteristic zero. However, only very partial results on the classification of its simple transitive -representations were known. To be precise, the Classification Problem was completely solved for finite Coxeter type A in [MM5] both in the graded and the ungraded setting, for finite dihedral type in [KMMZ] and [MT] in the graded setting and almost completely in the ungraded setting, and for a small number of other special cases in other Coxeter types, see e.g. [KMMZ] and [MMMZ], both in the graded and the ungraded setting. In all these cases, the classification turned out to be finite. However, in general it was not even known if the number of equivalence classes of simple transitive -representations of S is finite, left aside any classification. In this paper, we show that there are indeed only finitely many equivalence classes of simple transitive -representations of S for any finite Coxeter type, and classify them completely for all finite Coxeter types except and . All our results are proved in the graded setting, but they hold in the ungraded setting as well, see Subsection 7.4.
Let us explain our results in some more detail. First of all, the Classification Problem for S can be reduced to that of certain subquotients. To do that, one uses the cell structure of S , which is the categorical analog of the Kazhdan–Lusztig cell structure of . By [ChMa, Subsection 3.2], for every (graded or ungraded) simple transitive -representation of S there is a unique two-sided cell, called apex, that is not annihilated by and is maximal in the two-sided order with respect to that property. The Classification Problem for S can therefore be addressed apex by apex, and for a fixed two-sided cell we denote the -category of (graded) simple transitive -representations of S with apex by . Next, we can reduce the Classification Problem even further by (strong) -reduction, see [MMMZ, Theorem 15] or [MMMTZ, Theorem 4.32]. For any diagonal -cell , which, by definition, the intersection of a left cell in and the dual right cell, there is a subquotient of S whose only left, right and two-sided cells are the trivial cell and . By [MMMTZ, Theorem 4.32], there is a biequivalence
[TABLE]
This is a major reduction, because is much smaller than S and one can pick any diagonal -cell in (which is helpful in practice, because not all diagonal -cells of necessarily have the same structure, see Section 8).
By the results in this paper, the benefits turn out to be even bigger. Based on Elias and Williamson’s results in [EW3], Lusztig defined in [Lu2, Subsection 18.15] a locally semisimple bicategory , for any two-sided cell of any Coxeter group. This bicategory categorifies the summand of the asymptotic Hecke algebra corresponding to the cell . The bicategory is pivotal fusion by [EW1] and contains a one-object pivotal fusion full subbicategory for any diagonal -cell , which we call the asymptotic bicategory associated to . Being fusion, the bicategory has only one cell, corresponding to , which is left, right, two-sided and diagonal simultaneously, so all its simple transitive -representations have apex . The main insight of this paper is that these asymptotic bicategories completely determine the (graded) simple transitive -representations of S . To be precise, our main result (Theorem 7.1) is the existence of a biequivalence of graded -categories
[TABLE]
for every diagonal -cell of S for any finite Coxeter type. Here is the graded -category with translation obtained from by a well-known and straightforward construction, which we will recall in Subsection 2.6. As a matter of fact, and are biequivalent as ungraded -categories, but not as graded -categories. In the ungraded case, therefore, the above biequivalence becomes
[TABLE]
see Subsection 7.4. By [EGNO, Corollary 9.1.6 and Proposition 3.4.6] and strong -reduction, this implies that the number of equivalence classes of (graded) simple transitive -representations of S is finite for any finite Coxeter type, c.f. Corollary 7.2 and the end of Subsection 7.4. Moreover, for all finite Coxeter types but and , all two-sided cells contain a diagonal -cell for which both and are known explicitly, as summarized in Section 8. For some two-sided cells in Coxeter types and this is unfortunately not true, see Section 8. Therefore, we get a complete solution of the Classification Problem for S for all finite Coxeter types but and , and even for those two Coxeter types we get a complete solution for more than half the number of apexes, see Section 8. Finally, Theorem 7.1 also implies (see Theorem 7.8) that is locally (graded) semisimple, meaning that its morphism categories are all (graded) semisimple, for any two-sided cell and any finite Coxeter type.
Let us briefly sketch the key ingredients of the proof of Theorem 7.1. Let be the cell -representation of with apex , which categorifies the Kazhdan–Lusztig cell module of the Hecke algebra associated to . The main ingredient in the proof of Theorem 7.1 is the existence of a graded biequivalence
[TABLE]
c.f. Corollary 6.10. (For the experts, we remark that the proof of this proposition uses that the Duflo involution is a separable Frobenius algebra in , as we show in Subsection 4.4 and Proposition 6.2.) In the ungraded setting, the above biequivalence becomes
[TABLE]
Another important ingredient in the proof of our main result is the double centralizer theorem, see [MMMTZ, Theorem 5.2], which implies that there is a biequivalence
[TABLE]
as we show in Proposition 7.7. (Note that the bicategory in that proposition, which will be defined in (4.6), is graded biequivalent to \big{(}\sc\mbox{E}\hskip 1.0pt\mathrm{nd}_{\scc\mbox{S}_{\mathcal{H}}\hskip 1.0pt}(\mathbf{C}_{\mathcal{H}})\big{)}^{\mathrm{op}} by Corollary 4.7.)
Along the way, we also show several results that are interesting in their own right, e.g. the aforementioned fact that is a separable Frobenius algebra in . In order to achieve this, we prove in Proposition 4.3 that the underlying algebra of is a finite dimensional positively graded weakly symmetric Frobenius algebra of graded length , where is the value of Lusztig’s -function on . In certain cases is known to be symmetric, see Remark 4.1. However, we do not know if is symmetric in general.
Acknowledgments. All computer assisted calculations in this paper were done using SageMath.
We thank Ben Elias, Meinolf Geck, Matt Hogancamp, Hankyung Ko, Raphael Rouquier and Geordie Williamson for stimulating discussions and helpful exchanges of emails. D.T. would also like to thank Ben Elias for mellifluous influence which is hardly superfluous. We thank the referee for a very careful reading of the paper and many helpful comments.
Significant parts of this research were done when all the authors met at the University of Zürich in September 2018, at Uppsala University in April 2019 and at the University of East Anglia in July 2019. Hospitality and support of these universities are gratefully acknowledged.
M.M. was supported in part by Fundação para a Ciência e a Tecnologia (Portugal), projects UID/MAT/04459/2013 (Center for Mathematical Analysis, Geometry and Dynamical Systems - CAMGSD) and PTDC/MAT-PUR/31089/2017 (Higher Structures and Applications). Vo.Ma. is partially supported by the Swedish Research Council and Göran Gustafsson Stiftelse. Va.Mi. is partially supported by EPSRC grant EP/S017216/1, which also funded the workshop at UEA in July 2019. D.T. is supported by SageMath. X.Z. is supported by Göran Gustafsson Stiftelse and National Natural Science Foundation of China (Grant No.12101422).
2. Recollections
2.1. Categorical conventions
Categories and -categories, -semicategories and bicategories C in this paper are assumed to be essentially small. We also view a monoidal category as a -category with one (possibly unspecified) object; a perspective which we will use throughout, e.g. for Soergel bimodules. We will also use the following notation:
- •
objects in categories (which are not morphism categories in -categories) are denoted by letters such as , and morphisms by ; 2. •
objects in -categories are denoted by , -morphisms by and -morphisms by Greek letters such as ; 3. •
for any , we denote by the corresponding morphism category; 4. •
identity -morphisms are denoted by and identity -morphisms by , where the subscripts are sometimes omitted; 5. •
we write for composition of -morphisms, and and denote vertical and horizontal compositions of -morphisms, respectively.
We will also use bicategories, silently adapting definitions and results to the weaker setting if necessary, using [MMMTZ]. We will stress when we do not work with genuine -categories, -functors etc. The reader is referred to e.g. [ML], [Le] or [Ben] for these and related notions.
2.2. Finitary and fiat -categories, and their -representations
Let be an algebraically closed field.
A category is called finitary (over ) if it is equivalent to the category of finitely generated, injective (or projective) modules over some associative, finite dimensional -algebra. These categories assemble into a -category having additive, -linear functors and natural transformations as - and -morphisms, respectively. Similarly, a -category C is finitary (over ) if it has finitely many objects, all identity -morphisms are indecomposable and each morphism category is finitary over with all compositions being (bi)additive and -(bi)linear. We further say that C is fiat if it has a weak antiinvolution ⋆, reversing the direction of both - and -morphisms, and adjunction -morphisms associated to ⋆. If ⋆ is just a weak antiequivalence of finite order, then C is called quasi fiat. Finally, a finitary -category C is called locally semisimple if its morphism categories are all semisimple. Let us note that our use of the term “locally semisimple” is similar to such standard categorical terminology as “locally small”, with “locally” referring to -morphism categories.
Remark 2.0*.*
For completeness, dropping the assumption of the identity -morphisms being indecomposable gives what we call multifinitary, multifiat or quasi multifiat -categories. Most of the theory goes through for these as well, see [MMMTZ], but we will not need that level of generality in this paper.
Example 2.0*.*
For a finite group , the (strictified) one-object -category of finite dimensional representations of over is fiat if and only if the algebra has finite representation type. This is true, for example, if , in which case is locally semisimple.
Let . Another example of a fiat -category is , the one-object -category of Soergel bimodules over the coinvariant algebra of a finite Coxeter group, see Section 3.
Example 2.0*.*
For any finite dimensional algebra we have an associated one-object -category , called the -category of projective functors, whose -morphisms are direct sums of functors with summands isomorphic to the identity functor or to tensoring with projective --bimodules. (Despite the name of , the identity functor is not a projective functor, but is needed to make a genuine -category instead of a -semicategory.) Assume that is basic and connected and that is a splitting of the identity into orthogonal primitive idempotents, then is a complete and irredundant set of indecomposable projective --bimodules. By e.g. [MM1, Lemma 45 and its proof], for every ,
[TABLE]
The left injective -modules , for , are projective if and only if is Frobenius (since is assumed to be basic, it is self-injective if and only if it is Frobenius). Therefore, is quasi fiat if and only if is Frobenius.
Assume that is quasi fiat. Then , for , where is the Nakayama permutation. Applying ⋆ twice yields
[TABLE]
for all . We thus see that is fiat if and only if weakly symmetric. For more details, see [MM1, Subsection 7.3].
A locally semisimple quasi fiat -category is called a fusion -category and a locally semisimple fiat -category can be equipped with the structure of a pivotal fusion -category. Note that one-object (pivotal) fusion -categories, a.k.a. (pivotal) fusion categories, form an important class of tensor categories which has been intensively studied, see e.g. [EGNO].
Example 2.0*.*
The -category is pivotal fusion unless . By contrast, the -category S is not (pivotal) fusion, since it is not locally semisimple.
Remark 2.0*.*
Throughout this paper, we will consistently use the above -categorical terminology. Let us, for convenience, list the correspondence between our terminology and the one used in [EGNO] in the abelian setting for one-object -categories/monoidal categories:
- •
quasi fiat finitary and rigid; 2. •
fiat finitary and pivotal; 3. •
locally semisimple semisimple; 4. •
(pivotal) fusion -category (pivotal) fusion category.
Note that the notion of fiat only requires the existence of a natural isomorphism between the identity and ⋆⋆, whereas the notion of pivotal requires a choice of such a natural isomorphism. In practice, this subtle difference is not important for us because the abstract results hold for fiat -categories, whereas in the examples the pivotal structure is known and fixed.
For a finitary -category C , a finitary -representation is an additive, -linear -functor from C to . Finitary -representations of C form a -category, as we will explain in Subsection 2.5; in particular, there is an appropriate notion of equivalence. The underlying category of is defined as
[TABLE]
The rank of is by definition the number of isomorphism classes of indecomposable objects in . Moreover, we will often use the action notation for -representations.
Example 2.0*.*
If C is finitary, then the so-called principal or Yoneda -representation is finitary, for all .
A finitary -representation is called transitive if, for any and any non-zero object , the additive closure (in the sense of being closed under direct sums, direct summands and isomorphisms)
[TABLE]
coincides with . Recall that an ideal of is by definition a C -stable ideal of . A transitive -representation is said to be simple transitive provided that it has no non-trivial ideals. Moreover, every transitive -representation has a unique simple transitive quotient.
The importance of simple transitive -representations is explained, in particular, by the existence of a weak version of the Jordan–Hölder theorem. Namely, for any finitary -representation of C , there is a finite filtration
[TABLE]
where every -representation generates an ideal in such that is transitive, and thus, has a unique associated simple transitive quotient . Up to equivalence and ordering, the set is an invariant of .
The above motivates the Classification Problem, i.e. the classification of simple transitive -representations for a fixed finitary -category.
Example 2.0*.*
For the -category the Classification Problem has a well-known solution in terms of subgroups of and their group cohomology, see Section 8 for details.
By contrast, solving the Classification Problem for S is the main objective of this paper.
See [MM1, Subsections 2.2, 2.3 and 2.4], [MM6, Subsection 2.5], [MM3, Subsection 2.3] and [MM5, Subsection 3.5] for details.
2.3. Cells and cell -representations
For every finitary -category C , one has the notion of cells: For any pair of indecomposable -morphisms and , we define if is isomorphic to a direct summand of , for some -morphism . This produces the left preorder , for which the equivalence classes are called left cells. Similarly one obtains right cells and two-sided cells. By [ChMa, Subsection 3.2], for any transitive -representation there is a unique two-sided cell , an invariant of called the apex, which is not annihilated by and is maximal, in the two-sided order, with respect to this property.
Example 2.0*.*
Any one-object (pivotal) fusion -category C , a.k.a. (pivotal) fusion category, has only one cell, which is left, right and two-sided. This follows from the fact that and both contain the identity -morphism as a direct summand, for all indecomposable .
Example 2.0*.*
When , the -category is pivotal fusion and has, therefore, only one cell, which is left, right and two-sided. When , the -category has more than one cell, for example, the projective modules form a two-sided cell.
As we will recall in Subsection 3.1, the cells of S are given by the Kazhdan–Lusztig cells in the sense of [KaLu].
Each left cell of C can be used to define a cell -representation as follows. First we note that all -morphisms in have the same domain, say . Define a -subrepresentation of using the induced action of C on
[TABLE]
The -representation has a unique maximal ideal and we define
[TABLE]
which is a simple transitive -representation by construction. If C is quasi fiat, then the apex of is the two-sided cell containing .
Example 2.0*.*
The (unique) cell -representation of coincides with its unique principal -representation.
The cell -representations of S categorify the Kazhdan–Lusztig cell representations, see Subsection 3.1.
Finally, recall that a -category is called -simple, where is a two-sided cell, if any non-zero -ideal contains the identity -morphisms of all -morphisms in . If C is fiat, then C has an associated -simple subquotient -category , whose only two-sided cells are and the two-sided cells containing , such that is the source of some -morphism in . Note that these cells may all coincide, e.g. if C is fusion.
For further details we refer to [MM1, Subsection 4.5], [ChMa, Subsection 3.2], [MM5, Section 3], [MMMZ, Subsection 4.2] and [MMMTZ, Subsections 2.5 and 2.6].
2.4. Coalgebra and algebra -morphisms
Finitary -categories can be injectively or projectively abelianized. The injective abelianization is denoted and the projective abelianization is denoted . Moreover, C embeds into or into , and the isomorphism closure of the image of this embedding is the -full -subcategory of injective or projective -morphisms, respectively. In particular, in these abelianizations, each indecomposable -morphism has an associated simple socle or head, respectively.
These abelianizations are rather technical and not all properties of C carry over to the abelianizations. In particular, the abelianizations of fiat -categories are usually not even finitary and the involution ⋆ only gives rise to an antiequivalence between and .
The same abelianizations, mutatis mutandis, exist for finitary -representations, where we use the same notation.
See [MMMT, Section 3] for details.
Example 2.0*.*
When , we have , because is locally semisimple. In contrast, S is not abelian and neither nor are equivalent to it.
Abelianizations play a key role in the construction and study of -representations: Recall that a coalgebra in is a -morphism in some equipped with -morphisms , called comultiplication, and , called counit, satisfying the usual conditions. Dually, one can define algebras as well. A right -comodule is a -morphism in some together with a right coaction , again satisfying the usual coherence conditions. For each object , these assemble into a category
[TABLE]
whose morphisms are called right -comodule morphisms and satisfy the usual intertwining condition. All of these notions can be defined, mutatis mutandis, for left coactions as well, and we put the coalgebras on the left-hand side to stress that we have a left coaction. Let
[TABLE]
denote the full subcategory of consisting of all injective objects. We set
[TABLE]
Note that left composition defines a natural -action on , which restricts to a natural left C -action on . We use a bold font whenever we want to consider these as -representations, e.g. we write instead of .
Example 2.0*.*
The identity -morphism , for any , has a natural structure of a coalgebra, with both the comultiplication and the counit being given by the identity -morphism. The -representation of C is equivalent to and the -representation of C is equivalent to .
Another natural source of coalgebras in is provided by the internal cohom construction: Let be a finitary -representation of C and consider objects , . Their internal cohom is uniquely determined, up to isomorphism, by the isomorphism
[TABLE]
for all . If it is not immediately obvious in which -category or bicategory the internal cohom is taken, we will use a subscript to clarify. The -morphism has a natural structure of a coalgebra in . Moreover, if C is fiat and generates in the sense that , then . In this way, we can realize any finitary -representation of a fiat -category C , up to equivalence, as for some coalgebra in (or in the injective abelianization of the additive closure of C , if no generator in a single exists), and simple transitive -representations of C (which always have a generator) correspond to cosimple coalgebras in . If is a simple transitive -representation of C with apex and C is -simple, then is in C for every .
Finally, we recall that coalgebras in C have an associated Morita–Takeuchi theory. In particular, and are Morita–Takeuchi equivalent if and only if . See also e.g. [EGNO, Chapter 7], [MMMT, Sections 4 and 5] and [MMMZ, Subsection 3.6].
For later use, we recall one technical result and record three others. The first result, Theorem 2.1, is the content of Theorem [KMMZ, Theorem 2], which we will need repeatedly throughout the paper.
Theorem 2.1**.**
Let be a simple transitive -representation of a fiat -category C with apex . For every -morphism , the endofunctor is projective (in the category of right exact linear endofunctors).
In particular, suppose that C has only one object and that is the basic connected underlying algebra of , i.e., and . Theorem 2.1 then implies that, for every -morphism , the action of on is given by tensoring over with a projective --bimodule . Note that for any .
Corollary 2.2**.**
The algebra is Frobenius.
Proof.
Let be a splitting of the unit into orthogonal primitive idempotents in . Then is a complete and irredundant set of indecomposable projective --bimodules.
Choose . By Theorem 2.1 and transitivity of , for every with , there is an such that is a direct summand of , so contains a direct summand of the form
[TABLE]
for some . As in Example 2.2, the decomposition of thus contains a direct summand of the form
[TABLE]
for some . This shows that is a projective left -module, by Theorem 2.1.
Since this holds for all , every left injective -module is also projective, i.e., is self-injective. Since every self-injective basic algebra is Frobenius, the result follows. ∎
Corollary 2.2 shows that Theorem 2.1 can also be formulated in terms of the injective abelianization , so and , and the action of on is also given by , which is injective-projective as an --bimodule. This consequence of Theorem 2.1 and Corollary 2.2 is crucial and will be used repeatedly throughout the paper.
Remark 2.2*.*
The -category in Example 2.2 is fiat if and only if is weakly symmetric, whereas simple transitivity of only implies that is Frobenius. The reason is that, for any , the --bimodule is a direct sum of indecomposable --bimodules. When C is fiat, then , but ⋆⋆ can still permute the indecomposable direct summands, so the Nakayama permutation of need not be trivial. In Proposition 5.7, however, we will show that is weakly symmetric if is a (graded) simple transitive -representation of Soergel bimodules of finite Coxeter type.
Proposition 2.3**.**
Suppose that C is a one-object pivotal fusion -category and a finitary -representation of C . If is simple transitive, then is semisimple.
Proof.
Suppose that is simple transitive. As explained in Example 2.3, C has only one cell, which is thus the apex of . By Theorem 2.1, this implies that is a projective endofunctor of , where is the unique identity -morphism of C . This implies that every simple is projective, since . Therefore, is semisimple. ∎
Corollary 2.4**.**
Suppose that C is a one-object pivotal fusion -category and a cosimple coalgebra in C . Then is semisimple.
Proof.
If is a cosimple coalgebra in C , then is simple transitive by [MMMZ, Corollary 12]. Therefore, its underlying category is semisimple by Proposition 2.3. ∎
2.5. -Categories of -representations
The Classification Problem suggests studying the following -categories. Given two -categories C and D , let denote the -category of -functors together with -natural transformations and modifications.
If C is finitary, then let denote the additive, -linear -category of its finitary -representations. We let denote the -full -subcategory of consisting of all simple transitive -representations. Further, we indicate by the subscript , where is a given two-sided cell of C , the -full -subcategories consisting of the finitary -representations of C whose weak Jordan–Hölder constituents all have apex . Finally, we use the superscript to indicate the -full -subcategories having the same objects, but -morphisms being exact, in the sense of their component functors extending to exact functors in the (injective) abelianization.
Example 2.4*.*
Suppose that C is fiat and a left cell inside a two-sided cell of C . Then .
As discussed in detail in [MMMTZ, Subsection 2.6], there are various relations between the above -categories. We recall two of the most crucial ones here in the form in which we need them. The first relates to all morphisms between simple transitive -representations with the same apex being exact:
Proposition 2.5**.**
Suppose that C is fiat. For any two-sided cell of C , the -categories and are equal.
The second one is the so-called strong -reduction [MMMTZ, Theorem 4.32]: Suppose that C is fiat and let be a two-sided cell of C . Then ⋆ preserves and the -simple subquotient , see the end of Subsection 2.3, is also fiat and, by [MMMTZ, Theorem 4.28], there is a biequivalence
[TABLE]
The point of this biequivalence is that the structure of is simpler than that of the whole C in general.
However, there are even simpler -categories, which still contain enough information to solve the Classification Problem. We define -cells as the intersection of left and right cells. Suppose that C is fiat. Then, for every left cell , we define the associated diagonal -cell
[TABLE]
By construction, lies in the same two-sided cell as . Further, recall that each contains a unique distinguished -morphism called Duflo involution and that, in fact, .
Given a left cell in some two-sided cell , we define to be the -full -subcategory of generated by all -morphisms in together with the identity -morphism , where is the unique domain and codomain of all -morphisms in . The -category is fiat, has as its maximal two-sided cell and is -simple. The importance of is expressed by the following result, called strong -reduction.
Theorem 2.6**.**
Let C be a fiat -category and fix a two-sided cell of C . For any diagonal -cell , there is a biequivalence
[TABLE]
Example 2.6*.*
The main purpose of this paper is to study . Theorem 2.6 tells us that, by varying over all two-sided cells and choosing an arbitrary but fixed diagonal -cell in each , this problem can be reduced to studying . The main result of this paper, which is Theorem 7.1, shows that the latter problem can be reduced even further.
The main reference for the above is [MMMTZ, Subsections 2.6, 4.7 and 4.8].
2.6. Grading conventions
Let denote the category of finite dimensional (-)graded -vector spaces. An object in has the form , where denotes the linear subspace of elements of which are homogeneous of degree . Morphisms in are -linear maps (not necessarily homogeneous, but each morphism is a linear combination of homogeneous morphisms). The group acts on by grading shift via the rule , for all . From now on, if and is a graded vector space, then we use the notation
[TABLE]
With this notation, we have e.g. . Further, if is a finite dimensional graded vector space, then its graded dimension is uniquely defined by the property that the graded vector spaces and are isomorphic by a homogeneous isomorphism of degree [math], where is concentrated in degree zero. A finite dimensional algebra is called positively graded if it is non-negatively graded, i.e. , and its degree [math] component is semisimple. A finite dimensional positively graded algebra is called a graded Frobenius algebra of length if as graded left -modules, where again the isomorphism is homogeneous of degree zero.
A graded -linear category is a category enriched over (in this paper, most categories will have finite dimensional morphism spaces, but there are some exceptions). This means that
[TABLE]
and the composite of two homogeneous morphisms of degrees and is homogeneous of degree . We let
[TABLE]
By definition, is the (non-full) subcategory of given by taking the degree zero morphisms between all objects of , i.e.
[TABLE]
The grading on is, of course, trivial.
We call a functor between graded -linear categories a graded functor, provided the assignment
[TABLE]
on morphisms is homogeneous of degree zero.
Following [CiMa], a -linear category is a -category if it is equipped with a group homomorphism from to the group of autoequivalences of . It is a free -category if the action of thus obtained is free. For , we again write for the action of on .
We say that a graded -linear category admits translation if it is a free -category and for every there is a homogeneous natural isomorphism of degree , such that and , for . In particular, we have
[TABLE]
for all objects and . If comes with a fixed choice of isomorphisms , we say that is a category with translation.
A graded functor between two graded categories with translation automatically preserves translations (up to natural isomorphism), as follows from the existence of the homogeneous degree-zero natural isomorphism
[TABLE]
for any and .
Suppose that admits translation. Then in the objects and are isomorphic if and only if , so does not admit translation, although it is a free -category. Note also that is never finitary, because has infinitely many isomorphism classes of indecomposable objects. We say that a free -category is -finitary if there is a finite set of indecomposable objects , , such that every indecomposable object in is isomorphic to , for some and , and every object in has an essentially unique decomposition into indecomposables objects. We will call graded finitary provided that is -finitary.
Given a graded -linear category , we can define a new graded -linear category with objects , where and , and
[TABLE]
with the evident composition. Then has a translation, with given by the identity on shifted by degrees, and the natural embedding of into sending to is an equivalence of ungraded categories.
Remark 2.6*.*
Note that the above embedding is degree-preserving but it need not be an equivalence of graded categories in general, e.g. if , then the functorial inverse of the embedding would have to be given by , for all and . However, that functor is not degree-preserving, as it maps any homogeneous morphism of any degree in to a homogeneous morphism of degree zero in . Nonetheless, if admits translation, then it is equivalent to as a graded category, because defines a functorial inverse of the above embedding.
Observe that, given a -linear category admitting translation, is equal to the skew-category in the sense of [CiMa, Definition 2.3]. Recall that for any -linear free -category , the objects of coincide with those of and the morphism spaces are defined by
[TABLE]
with the composition being induced by the one in after shifting. The skew-category is naturally graded and comes with the natural translation induced by the identity on . By construction, the isomorphism classes of its objects correspond bijectively to the -orbits of objects in .
For future use, we record the following easy lemma.
Lemma 2.7**.**
Let be a trivially graded -linear category, and let be defined as above. Then is equivalent, as a -linear free -category, to , where is the full subcategory of with objects for .
Moreover, let . Then and are equivalent as graded -linear categories.
Proof.
The first statement is clear and the second is a special case of [CiMa, Proposition 2.6]. ∎
Another notion that we will need is that of graded semisimplicity. A graded finitary category is called graded semisimple if is semisimple. Note that in the latter case there is a finite number of simple object , , and every simple object in is isomorphic to , for some and .
Graded -linear categories admitting translation, together with graded functors and all natural transformations form a -category, which we will denote by . We denote by the -full -subcategory whose objects are graded finitary categories admitting translation.
As our notation emphasizes, all non-trivially graded -linear categories in this paper will be assumed to admit translation.
References for the above are for example [He, Section 1] and [CiMa].
2.7. Graded -categories of -representations
Many -categories in this paper will be graded, meaning their morphism categories are in and horizontal composition is a graded bifunctor.
By a graded finitary -category, we mean a graded -linear -category which has finitely many objects, on which identity -morphisms are indecomposable, and morphism categories in with horizontal composition being a graded -bilinear bifunctor.
If C is a graded -linear -category, not necessarily admitting translation, we define the -category with translation , with the same objects as C and morphism spaces defined by and the evident horizontal composition.
For a graded -category C , we denote by the -subcategory whose morphism categories are given by . As in the case of categories, even if C is graded finitary, the -subcategory will never be finitary, since and are not isomorphic in unless . However, is a -finitary -category, meaning that its morphism categories are -finitary categories and horizontal composition is compatible with the free -actions. Moreover, the skew -category , whose morphism categories are given by , is biequivalent to C (since is a free -category).
A graded finitary -category C is called locally graded semisimple if all its morphism categories are graded semisimple. Equivalently, this means that is locally semisimple.
A graded finitary -representation of a graded finitary -category C is a graded -linear -functor , where by a graded -functor we mean that all its component functors are graded.
Similarly, a morphism of graded finitary -representations is a collection of graded functors together with homogeneous natural isomorphisms of degree zero providing the compatibility with the -actions.
Graded finitary -representations of a graded finitary -category C , together with morphisms of graded finitary -representations and modifications, form a graded -category, since spaces of modifications are naturally enriched over . This -category is denoted by .
The concepts of transitive, simple transitive and cell -representations are defined as in the ungraded case. In particular, we can define the -full -subcategory , whose objects are graded simple transitive -representations of C . As in the ungraded case, a subscript indicates that we are restricting our attention to simple transitive -representations with apex .
For a graded finitary -representation of a graded finitary -category C , we define a graded finitary -representation of by , for every , with the obvious extension of the action.
Similarly, for a graded finitary -representation of a graded finitary -category C , we define a -representation of by for every . Again, is a -finitary -representation, meaning that is a -finitary category, for every , and all other data of the -representation are compatible with the free -actions. Conversely, given a -finitary -representation of , we define a graded finitary -representation of C by setting and extending the other data of the -representation in the obvious way. Finally, we say that a graded finitary -representation is graded semisimple if is semisimple.
Following [McPh, Subsection 6.4], given a graded finitary -representation , we can associate a coalgebra in to , and . In other words, we can associate to a coalgebra whose comultiplication and counit are homogeneous of degree zero. We, therefore, call it a graded coalgebra. Now, suppose that is simple transitive, then is cosimple in , see Subsection 2.4. This implies that is also cosimple in , of course, whence is also simple transitive (again, see Subsection 2.4). In all this, however, the definition of the graded injective abelianization requires some care. It is defined as
[TABLE]
This means that the -morphisms in are complexes of the form
[TABLE]
where , are -morphisms in C and , and -morphisms in are defined in the usual way without the restriction of homogeneity. With this definition, is indeed a graded abelian -category and, for any graded coalgebra in , the category is indeed a graded subcategory of .
3. Soergel bimodules and the asymptotic bicategory
3.1. Soergel bimodules
From now on we set (or any other algebraically closed field of characteristic zero), and all categories and -categories etc. are over unless indicated otherwise.
Let be a Coxeter system of finite type. We fix a reflection faithful representation of and let be the corresponding coinvariant algebra with the usual (-)grading. We denote by the -category of Soergel bimodules over , by which we mean the -category with one object , which we identify with the category of finite dimensional graded -modules, whose -morphisms are endofunctors of the latter category isomorphic to tensoring with Soergel bimodules, and whose -morphisms are natural transformations, which we can identify with bimodule homomorphisms. The -category S is graded fiat (this follows from e.g. [EW3], see also [MM1, Subsection 7.1] for the case of Weyl groups).
By [So3] (see also [EW2]), there is a -linear isomorphism between the split Grothendieck ring of and the Hecke algebra of such that, for each , there is a unique (up to homogeneous isomorphism) indecomposable Soergel bimodule in whose Grothendieck class is sent to the Kazhdan–Lusztig basis element corresponding to , and for any -morphism in . We call this fact the Soergel–Elias–Williamson categorification theorem and we call the isomorphism the character isomorphism. As a consequence, the cell structure of S is given by the Kazhdan–Lusztig combinatorics. In particular, Lusztig’s conjectures [Lu2, Conjecture 14.2], which we will use several times, hold in our case, see e.g. [Lu2, Subsection 15.1] or [duCl, Corollary 1.4].
We set
[TABLE]
where we recall that, to each , Lusztig [Lu1] assigns a number , called its -value, such that the function is constant on two-sided cells. In the projective -morphism has a simple head, and dually in the injective -morphism has a simple socle.
Example 3.0*.*
For we have in , but . The heads of and , seen as projective -morphisms in , are concentrated in degrees and [math], respectively.
For , we let and be given by
[TABLE]
where is Lusztig’s -function. Since the are bar invariant, i.e. invariant with respect to the symmetry , we also have
[TABLE]
By the Soergel–Elias–Williamson categorification theorem, the are also the structure constants of with respect to the basis and the are the structure constants of Lusztig’s asymptotic Hecke algebra, denoted , with respect to the basis . (Lusztig [Lu3] called the asymptotic Hecke algebra the -ring, denoting its basis elements by in [Lu3].)
Recall from Subsection 2.5 that, for every two-sided cell , there is a -simple subquotient of S and, for every left cell in , there is a diagonal -cell and a -full -subcategory of which is -simple. Recall also that every left cell contains a unique Duflo involution and that . Let denote the set of Duflo involutions in . Both and have asymptotic counterparts too, which we denote by and , respectively, and will recall in Subsection 3.2. For every two-sided cell of , the split Grothendieck group of is a subalgebra of , denoted . The unit of is equal to and there is a decomposition of algebras
[TABLE]
where the direct product runs over all two-sided cells of . Note that is an idempotent subalgebra of . Similarly, for every diagonal -cell , the split Grothendieck group of is an idempotent subalgebra of , denoted by . The unit of is equal to , but the direct product of these is strictly smaller than in general. In fact, decomposes as
[TABLE]
as a vector space, where the direct sum runs over all pairs of left cells in and is the direct summand of generated by , which is an --bimodule. Alternatively, we can see as a finitary category, whose objects are indexed by the left cells and whose morphism categories are defined by
[TABLE]
The identity morphism on is given by .
From Soergel’s hom formula, see [EW3, Theorem 3.6], we obtain
[TABLE]
By slight abuse of terminology, we will also refer to (3.3) as Soergel’s hom formula. Consequently, for any subset , the -endomorphism algebra of in S is positively graded. This property is inherited by .
Let for any . Finally, the following lemmas are evident and we state them for later use.
Lemma 3.1**.**
In , for all , we have that
[TABLE]
In particular, if is isomorphic to a direct summand of , then .
Lemma 3.1 has the following “negative” counterpart.
Lemma 3.2**.**
For , set . Then, in , for all , we have that
[TABLE]
In particular, if is isomorphic to a direct summand of , then .
3.2. The asymptotic bicategory
We define two -semicategories inside :
[TABLE]
The -semicategory is, in fact, a lax monoidal category with lax identity -morphism and strict associators. Let us explain this in detail. From [MM3, Subsection 7.6] and positivity of the grading on Soergel bimodules, it follows that there is a unique, up to a non-zero scalar, -morphism in . The lax structure of the identity -morphism of on is now defined, for , by the two -morphisms in
[TABLE]
with the unitality condition expressed, for , by the diagram
[TABLE]
which commutes by associativity and the interchange law.
The bicategory is defined as the quotient , where denotes the -ideal of generated by . We denote by the natural projection.
Note that is only a bicategory as it does not contain any strict identity -morphism, but, at the same time, the composition in is strictly associative. Up to isomorphism, the identity -morphism in is the image of the lax identity -morphism of . Finally, note that there is no non-trivial -action on and the grading is trivial, i.e. all -morphisms are homogeneous of degree zero.
By the Soergel–Elias–Williamson categorification theorem and (3.1), categorifies the asymptotic Hecke algebra associated to in the sense that categorifies the basis element for . The algebra is a fusion algebra and the following categorifies this fact.
Proposition 3.3**.**
* is a (one-object) pivotal fusion bicategory. Moreover, if is a finite Weyl group, then is biequivalent to from [BFO, Section 5] and [Os4].*
Note that this implies that , and are canonically isomorphic.
Proof.
The bicategory is fiat by [EW1, Theorem 5.2] (see also [Lu2, Subsection 18.19], [Lu4, Subsection 9.3] and [BFO, Subsection 4.3]). Moreover, due to the definition of and Soergel’s hom-formula (see (3.3)), is locally semisimple. Finally, the connection to [BFO, Section 5] and [Os4] follows from Soergel’s identification of S and the category of semisimple perverse sheaves on the associated flag variety, see [So1, Erweiterungssatz 5]. ∎
Remark 3.3*.*
For completeness, we note that there are analogs of the above for two-sided cells instead of diagonal -cells. In fact, for finite and affine Weyl groups, Lusztig and Bezrukavnikov–Finkelberg–Ostrik (see [BFO] and the references therein) showed that the pivotal fusion bicategories corresponding to can be described geometrically as bicategories of vector bundles on a square of a finite set equivariant with respect to an algebraic group.
A small remark about terminology: When is finite, e.g. for finite Coxeter type, one can also see as a pivotal multifusion category with decomposable identity object isomorphic to , where the finite direct sum is over all Duflo involutions in . In fact, this is Lusztig’s point of view, see [Lu2, Subsection 18.15]. When has infinitely many Duflo involutions (which, of course, can only happen when is infinite), there is no identity object, as Lusztig indeed remarks. However, for all , finite or infinite, one can consider to be a pivotal fusion bicategory, whose (possibly infinitely many) objects are indexed by the left cells of , which correspond bijectively to the Duflo involutions in .
Finally, if is finite, by strong -reduction (which we recalled in Theorem 2.6), Proposition 3.3 and the fact that one-object fusion bicategories have only one two-sided cell, there is a biequivalence for any diagonal -cell . As we will summarize in Section 8, in each two-sided cell of a (finite) Weyl group one can pick a nice diagonal -cell , i.e. one such that is a well-known pivotal fusion bicategory for which the Classification Problem has been solved. For (finite) non-Weyl groups this is no longer true in general. However, even for those groups, roughly half the number of the two-sided cells contains a nice diagonal -cell. This is why we restrict our attention to in this paper.
3.3. Going up
This subsection investigates a certain oplax pseudofunctor from to which we loosely call “going up”.
By the definition of , the projection is the identity on and zero on , so it is a genuine (in contrast to lax) -semifunctor. It sends the lax identity of to the identity of (which is a weak identity, in the sense that there are non-trivial left and right unitors) and, for any , we have , by definition.
Indeed, for any , we can define the -morphism
[TABLE]
Then is a natural transformation from to the identity -functor on and we have . Similarly, one can define a natural transformation from to the identity -functor on via . Here and are the left and right unitors for the bicategory . In details, by applying to (3.8), we obtain the commutative diagram
[TABLE]
for all .
Observing that and , we obtain the commutative diagrams
[TABLE]
which, together with the fact that preserves strict associativity, imply that is a lax pseudofunctor.
Positivity of the grading on Soergel bimodules shows that the functor underlying has a left adjoint
[TABLE]
which is unique up to natural -isomorphisms. Up to isomorphism, is determined by , in particular, is an embedding.
Since is lax, the doctrinal adjunction [Ke] (see also [SS, Formula (3.5)]) implies that is an oplax pseudofunctor. Note that, for each , the object is isomorphic to , where is the subobject of generated in degree [math]. As , for some , the adjunction morphisms guarantee that .
We can further use to extend to an oplax pseudofunctor from to via the embeddings .
We use the notation , where is the Duflo involution in . Note that is only a weak identity -morphism. Then .
Lemma 3.4**.**
For any coalgebra in , the oplax -functor induces the structure of a coalgebra in on . Moreover, a (left or right) -comodule in is sent to a (left or right) -comodule in , which can, of course, be viewed as a comodule in .
Proof.
Mutatis mutandis as in [JS, Proposition 5.5]. ∎
Proposition 3.5**.**
If as in Lemma 3.4 is cosimple in , then is cosimple in .
Note that Proposition 3.5 rules out the existence of ungraded subcoalgebras of as well.
Proof.
By construction, belongs to . Let be a cosimple subcoalgebra of in . By [MMMZ, Corollary 12], the corresponding -representation is simple transitive. By [MMMTZ, Example 4.20], we have , which, by [MMMTZ, Theorem 4.19], implies that is a direct summand of in . Hence, is of the form , for some -morphism in . It is easy to check that is a subcoalgebra of in and is thus isomorphic to . The claim follows. ∎
Example 3.5*.*
Being an identity -morphism, is a cosimple coalgebra in . By Lemma 3.4 and Proposition 3.5, this implies that has the structure of a cosimple coalgebra in whose structure -morphisms are homogeneous of degree zero. By duality, this implies that has the structure of a simple algebra in , again with structure -morphisms homogeneous of degree zero.
In Subsection 4.4, we will additionally see that has the structure of a Frobenius algebra in .
We denote by the cotensor product, see [Ta, Section 0].
Lemma 3.6**.**
Let be a coalgebra in , a right -comodule and a left -comodule. Then there is a -isomorphism in
[TABLE]
that is natural in and .
Proof.
Consider the commutative diagram in
[TABLE]
where the top row is given by applying to the definition of , the bottom row is the definition of and the two solid inclusions are given by the oplax structure of . The dashed inclusion is induced by commutativity of the solid square and the universality property of the cotensor product. As all solid arrows are natural in and , the same holds for the dashed inclusion.
Note that the vertical arrows in the solid square restrict to -isomorphisms in degree [math]. Consequently, the dashed arrow is also -isomorphism in degree [math]. To prove the lemma, we need to show that is injective when restricted to summands of generated in positive degrees (as -morphisms in ). We will verify this by passing to the cell -representation .
Let be the underlying algebra of . Then is naturally a positively graded algebra. The bimodules representing , and are all projective by Theorem 2.1 and generated in degree [math], by construction. Let be an indecomposable summand of \mathbf{C}_{\mathcal{H}}\big{(}\Theta(\mathrm{X})\Theta(\mathrm{Y})\big{)} not generated in degree [math]. Being a summand in the tensor product over of two projective --bimodules generated in degree [math], we can view as a summand of . Writing any non-zero element of in a fixed homogeneous basis of , the element will have a non-zero coefficient at some , where , and are elements from a fixed homogeneous basis of and has positive degree. Under , such is sent to
[TABLE]
where and are non-zero elements of degree [math] in . The element is clearly non-zero in \mathbf{C}_{\mathcal{H}}\big{(}\Theta(\mathrm{X})\Theta(\mathrm{A})\Theta(\mathrm{Y})\big{)}. All other basis elements appearing with non-zero coefficient in will be mapped to elements with zero coefficients at and . The claim follows. ∎
Proposition 3.7**.**
Let and be Morita–Takeuchi equivalent coalgebras in . Then and are Morita–Takeuchi equivalent in .
Proof.
By [MMMT, Theorem 5.1], there exist a --bicomodule and an --bicomodule such that
[TABLE]
Applying Lemmas 3.4 and 3.6, we obtain
[TABLE]
The claim of the proposition now follows from [MMMT, Theorem 5.1]. ∎
Proposition 3.8**.**
If as in Lemma 3.4 is cosimple and , then is in \mathrm{inj}_{\underline{\scc\mbox{S}_{\mathcal{H}}\hskip 1.0pt}}\big{(}\Theta(\mathrm{A})\big{)}.
Note that Lemma 3.4 shows that the right coaction of on is, in fact, homogeneous of degree zero.
Proof.
By additivity, it suffices to prove that is in \mathrm{inj}_{\underline{\scc\mbox{S}_{\mathcal{H}}\hskip 1.0pt}}\big{(}\Theta(\mathrm{A})\big{)}, for any .
Set . By [MMMTZ, Corollary 4.17], the Morita–Takeuchi equivalence between and is given by the -bicomodule and the -bicomodule . By Lemma 3.6 and Proposition 3.7, the -bicomodule and the -bicomodule provide a Morita–Takeuchi equivalence between and in . In particular, is in \mathrm{inj}_{\underline{\scc\mbox{S}_{\mathcal{H}}\hskip 1.0pt}}\big{(}\Theta(\mathrm{A})\big{)}. ∎
Corollary 3.9**.**
If as in Lemma 3.4 is cosimple, then the ranks of and \mathrm{inj}_{\underline{\scc\mbox{S}\hskip 1.0pt_{\mathcal{H}}}}\big{(}\Theta(\mathrm{A})\big{)} are equal.
Proof.
The fact that and Proposition 3.8 imply that application of induces an injection from the set of isomorphism classes of indecomposable objects in to the set of isomorphism classes of indecomposable objects in \mathrm{inj}_{\underline{\scc\mbox{S}\hskip 1.0pt_{\mathcal{H}}}}\big{(}\Theta(\mathrm{A})\big{)}. Let us now argue that this is also surjective.
Let M\in\mathrm{inj}_{\underline{\scc\mbox{S}\hskip 1.0pt_{\mathcal{H}}}}\big{(}\Theta(\mathrm{A})\big{)} be indecomposable. Up to an overall grading shift, we may assume that there is a decomposition in of the form
[TABLE]
where the belong to and are such that is non-zero for at least one . Here means the evaluation of at . Let
[TABLE]
By positivity of the grading on and Lemma 3.1, is a -subcomodule of in .
Note that {\Pi}\big{(}\Theta(\mathrm{A})\big{)}\cong\mathrm{A} by construction. Similarly, {\Pi}\big{(}M^{0}\Theta(\mathrm{A})\big{)}\cong{\Pi}(M^{0}){\Pi}\big{(}\Theta(\mathrm{A})\big{)}. Hence, there is an induced -comodule structure on . By Corollary 2.4, is injective as an -comodule since is cosimple and is fusion. Now, \Theta\big{(}{\Pi}(M^{0})\big{)}\cong M^{0} is an injective -comodule in by Proposition 3.8. Consequently, is isomorphic to in . ∎
3.4. Going down
In this subsection we describe, see Lemma 3.10, and investigate what we loosely call “going down”. As the name suggests, this is, in some sense, the opposite of going up.
Let be a graded simple transitive -representation of with apex . Let be a graded coalgebra in such that is equivalent to . Let be a complete and irredundant list of representatives of isomorphism classes (up to grading shift) of indecomposable objects in , normalized such that as -morphisms in they are concentrated in non-negative degrees with non-zero degree zero part. Set .
Lemma 3.10**.**
The quotient
[TABLE]
carries an induced action of . Here \big{(}\mathrm{add}\big{(}\{X^{\oplus\mathsf{v}^{k}}\mid k>0\}\big{)}^{(0)}\big{)} is the ideal generated by \mathrm{add}\big{(}\{X^{\oplus\mathsf{v}^{k}}\mid k>0\}\big{)}^{(0)}.
Proof.
Lemma 3.1 implies that, if is isomorphic to a direct summand of in , where , then . Therefore \mathrm{add}\big{(}\{X^{\oplus\mathsf{v}^{k}}\mid k\geq 0\}\big{)}^{(0)} is stable under the action of \mathrm{add}\big{(}\{\mathrm{C}_{w}^{\oplus\mathsf{v}^{l}}\mid w\in W,l\geq 0\}\big{)}^{(0)}. Moreover, \big{(}\mathrm{add}\big{(}\{\mathrm{C}_{w}^{\oplus\mathsf{v}^{l}}\mid w\in W,l>0\}\big{)}^{(0)}\big{)} maps \mathrm{add}\big{(}\{X^{\oplus\mathsf{v}^{k}}\mid k\geq 0\}\big{)}^{(0)} to \big{(}\mathrm{add}\big{(}\{X^{\oplus\mathsf{v}^{k}}\mid k>0\}\big{)}^{(0)}\big{)}. The claim follows. ∎
We state and prove the following lemma for or , although we only need it in the latter case, since it is a useful result in its own right.
Lemma 3.11**.**
Let or . Given any graded simple transitive -representation of C with apex , there exists a coalgebra in C such that is equivalent to and is the image of a multiplicity free direct sum of representatives of isomorphism classes of indecomposable objects in under the forgetful functor to C .
Proof.
Let be any coalgebra in C and a multiplicity free direct sum of representatives of isomorphism classes of indecomposable objects in . Then the coalgebra in C is Morita–Takeuchi equivalent to , and the equivalence between and identifies with , so is the image of a multiplicity free direct sum of representatives of isomorphism classes of indecomposable objects in under the forgetful functor to C , as required. ∎
Proposition 3.12**.**
There is an injection from the set of equivalence classes of simple transitive -representations of to the set of equivalence classes of graded simple transitive -representations of with apex .
Proof.
Let be a simple transitive -representation of . Let be a coalgebra in such that is equivalent to . Then \mathbf{inj}_{\underline{\scc\mbox{S}\hskip 1.0pt_{\mathcal{H}}}}\big{(}\Theta(\mathrm{A})\big{)} is a graded simple transitive -representation of (with apex ) due to Proposition 3.5 and [MMMZ, Corollary 12]. By Proposition 3.7, this yields a well-defined map .
Now assume is chosen such that is the image of a multiplicity free direct sum of representatives of isomorphism classes of indecomposable objects in under the forgetful functor to , see Lemma 3.11. Set . Then, by Proposition 3.8 and Corollary 3.9, the object defined in the going down procedure is isomorphic to . The -representation of obtained by going down is now, clearly, equivalent to . Therefore, the map defined in the previous paragraph is injective, as claimed. ∎
As we will see later, Theorem 7.1 actually implies that is a bijection.
4. The role of the Duflo involution
Throughout this section we set for our fixed diagonal -cell, and let be the Duflo involution in .
4.1. Cell -representations and Duflo involutions
Let and be the corresponding simple object in , concentrated in degree zero. By [MM3, Section 7] (see also [MM1, Subsection 4.5]), is an indecomposable injective object in with simple socle concentrated in degree [math].
Dually, let be the simple object in , corresponding to , concentrated in degree zero. Then is an indecomposable projective object in with simple head concentrated in degree [math].
Lemma 4.1**.**
For any ,
- (i)
the injective objects and in are concentrated between the degrees and [math] and between the degrees [math] and , respectively; 2. (ii)
the projective objects and in are concentrated between the degrees and [math] and between the degrees [math] and , respectively.
Proof.
Let us prove the first statement in (i). As the -category of Soergel bimodules is positively graded, the fact that is concentrated in non-positive degrees follows from Lemma 3.2. This in turn implies that is concentrated in non-positive degrees as well.
By adjunction, we have
[TABLE]
As the right-hand side of the above isomorphism is zero for , we deduce that is also zero if .
The second statement in (i) follows from the first one and the fact that .
The dual statements in (ii) are proved in exactly the same way, using Lemma 3.1 instead of Lemma 3.2. ∎
Let be the principal -representation of and its injective abelianization, see Example 2.2. Denote by and the corresponding injective object in with respect to and , respectively, see [MMMT, Subsection 3.1] for details. Note that has simple socle concentrated in degree 0 and has simple socle concentrated in degree .
Lemma 4.2**.**
For any , the following hold.
- (i)
The injective object in has simple head concentrated in degree . 2. (ii)
The projective object in has simple socle concentrated in degree .
Proof.
We only prove statement (i) since the dual statement (ii) follows by similar arguments. As is fiat, the underlying algebra of is self-injective, by Corollary 2.2, which implies that the indecomposable injective object is also projective and thus, has a simple head. Therefore, it suffices to prove that is non-zero. By adjunction, this is equivalent to proving that
[TABLE]
which holds if and only if appears as a direct summand of in .
Note that the latter holds if and only if the same is true in . (Recall the two equivalent constructions of cell -representations, see [MM1, Subsection 4.5].) In the principal -representation, we can use the following fact. Since , we have
[TABLE]
whose right-hand side is zero unless and . In other words, the only such that has a composition factor isomorphic to , up to a shift, is . Therefore, to prove (4.1), it is enough to show that is non-zero. By adjunction, we have
[TABLE]
where the right-hand side is non-zero. Hence, appears in the head of in . ∎
Assume that is the underlying basic algebra of . Then the indecomposable objects in are identified with indecomposable injective -modules.
Proposition 4.3**.**
The algebra is a finite dimensional positively graded weakly symmetric Frobenius algebra of graded length .
Proof.
By construction, the algebra is non-negatively graded and its degree [math] part, which is isomorphic to , is semisimple, so is positively graded.
Since is fiat, the algebra is Frobenius, see Corollary 2.2.
The fact that is weakly symmetric follows from Lemma 4.2(i). Together with Lemma 4.1(ii), this implies is of graded length . ∎
Remark 4.3*.*
In some cases we know that is symmetric:
- (i)
Let be the left cell for . From [MS, Theorem 4.6] we know that is symmetric if is a finite Weyl group and
[TABLE] 2. (ii)
For any finite Coxeter group , is symmetric if is a diagonal -cell in the subregular two-sided cell of S , see [KMMZ, Corollary 5, Proposition 14 and the comment below it, and Remarks 29, 32 and 38] and [MT, Theorem I].
However, we do not know if this holds in general.
Since is fiat, the action of on the category of -modules is exact. Further, by Theorem 2.1, we know that the action of via is given by tensoring with a projective-injective bimodule. It follows from Lemma 4.1(i) that the bimodule representing is isomorphic to a direct sum of bimodules of the form , possibly with multiplicities but without grading shifts, where are some primitive idempotents of . By Proposition 4.3 and the fact that is a faithful -functor which is degree-preserving on -morphisms, this implies that the -morphism in has graded length at most .
Recall from Example 3.3 that is a cosimple coalgebra -morphism in . By [MMMZ, Corollary 12], is a graded simple transitive -representation of with apex . We denote by the basic algebra underlying .
Proposition 4.4**.**
The algebra is a positively graded Frobenius algebra of graded length .
Proof.
The algebra is graded, by definition, and Frobenius by Corollary 2.2.
By Lemma 3.11, we can choose a coalgebra in such that is equivalent to and is the image of a multiplicity free direct sum of representatives of isomorphism classes of indecomposable objects in under the forgetful functor to . This implies that as -morphisms in , because is the identity -morphism of .
Define in . Since , for all , there is an isomorphism in and, by Proposition 3.7, there is an equivalence . From the definition of , for all , we have
[TABLE]
If , then Soergel’s hom formula [EW3, Theorem 3.6] (here we need the full formula, not just the restricted version which we recalled in (3.3)) implies that the left-hand side of (4.2) is zero. If , then Soergel’s hom formula implies that the left-hand side of (4.2) has dimension one, because
[TABLE]
This implies that is positively graded.
Since in , the left-hand side of (4.2) is zero if and hence, the graded length of is at most . As the algebra underlying the cell -representation has graded length , see Proposition 4.3, we know that there exists a such that , which implies that the left-hand side of (4.2) is non-zero for . Thus, the graded length of is exactly . This completes the proof. ∎
Lemma 4.5**.**
For all , the -morphism in is of graded length .
Proof.
Recall from [MMMTZ, Example 4.20] that in , where the internal cohom-construction is with respect to . We first claim that
[TABLE]
is non-zero, where is as in the proof of Proposition 4.4. By Lemma 3.1, we have
[TABLE]
in . Noting that (3.1), (3.2) and [Lu2, Conjecture 14.2.P2, P5 and P7] imply that
[TABLE]
we deduce that
[TABLE]
Therefore, the object is isomorphic to a direct summand of in . The head of the indecomposable injective object in is isomorphic to a direct summand of the socle of in , see Proposition 4.4 and its proof. Hence, the right-hand side of (4.3) is non-zero, which implies that the left-hand side is non-zero. This shows that has a direct summand in isomorphic to with graded length at least . Therefore must have graded length exactly , as we already know that it is at most . By adjunction, we have
[TABLE]
yielding that the graded length of is at least . Thus, as above, it must be equal to . Note that for any we have . Therefore each contains a direct summand in . By (4.4), we have
[TABLE]
By Lemma 3.1, the direct summands of in have non-negative shift. Again as before, this shows that the graded length of is at least , which implies that it must be equal to . ∎
Recall from Example 3.3 that is a simple algebra -morphism in .
Proposition 4.6**.**
- (i)
The -representation is equivalent to the cell -representation . 2. (ii)
The -representation is also equivalent to the cell -representation .
Proof.
Again, we will prove the statement (i) and the dual statement (ii) follows verbatim. Consider as an object of and set (note that here the internal cohom-construction is with respect to and not as in the proof of Lemma 4.5). As a -morphism in we have
[TABLE]
with . Furthermore,
[TABLE]
By positivity of the grading on and Lemma 3.1, we see that .
If , then the right-hand side of (4.5) is zero, because is an indecomposable injective object of graded length by Lemma 4.1, and the action of increases the graded length by at most by Lemma 3.1. On the left-hand side, the indecomposable injective object has graded length , see Lemma 4.5, which implies that lives in non-negative degrees, that is, .
For , the right-hand side of (4.5) is one dimensional if , and zero otherwise. This implies in . Since the degree [math] maps and are unique up to scalar, it follows that as coalgebra -morphisms in . ∎
Let
[TABLE]
denote the graded one-object bicategory of biinjective --bicomodules in , where horizontal composition is given by the cotensor product and the identity -morphism is . By definition, a biinjective --bicomodule is injective as a left and as a right -comodule, but not necessarily as a --bicomodule, e.g. is biinjective but not injective as a --bicomodule when . The bicategory will play an important role in this article, c.f. Proposition 6.9 and Theorem 7.1. For now, it is important to record the following corollary.
Corollary 4.7**.**
There is a graded biequivalence
[TABLE]
Proof.
By Proposition 4.6(i), there is a graded biequivalence
[TABLE]
Using this identification, the assignment
[TABLE]
defines a biequivalence, as follows from Proposition 2.5 and [MMMTZ, Theorems 4.26 and 4.31]. ∎
4.2. The categorified bar involution
Proposition 4.8**.**
There exists a functorial involution , which is covariant on -morphisms and contravariant and degree-preserving on -morphisms, such that we have
[TABLE]
in for all and .
Proof.
Let D be the diagrammatic (one-object) Soergel -category as in [EW2], whose -morphisms are of the form , where is a word in the simple reflections of and indicates a formal degree shift, and whose -morphisms are Soergel diagrams, defined by generators and relations. By [EW2, Theorem 6.30], we can identify S with . (Strictly speaking, we have to quotient by the -ideal generated by the totally invariant polynomials with no constant term in the base ring in that paper, because S was defined over the coinvariant algebra. This is a technical detail, which we will suppress from now on.) Now, is defined by , for all words and , and by flipping the Soergel diagrams upside-down. By definition, is covariant on -morphisms, contravariant and degree-preserving on -morphisms. For any word in the simple reflections of , let be the corresponding Bott–Samelson bimodule in S . Then \big{(}\mathrm{BS}(\underline{w})^{\oplus\mathsf{v}^{k}}\big{)}^{\vee}\cong\mathrm{BS}(\underline{w})^{\oplus\mathsf{v}^{{\,\text{-}k}}} under the identification .
Extend to . To show (4.7), we use induction on the length of , the case being immediate. Assume that and that (4.7) holds for all with and all . By [So2, Satz 6.14] (see also [EW2, Theorem 3.16] and the text around it) and [EW2, Corollary 6.27], we have
[TABLE]
for every , where is an arbitrary reduced expression for , the are invariant under the bar involution (which follows from [Lu2, Chapter 4] and the Soergel–Elias–Williamson categorification theorem) and is the Bruhat order. As remarked above, we have in . By induction, we also have in for all , using the bar invariance of . Since is Krull–Schmidt, we deduce that in . This implies that (4.7) holds for all . ∎
Recall that the bar involution on the Hecke algebra is uniquely determined by the fact that it is -linear, sends for all and fixes the Kazhdan–Lusztig basis elements, see e.g. [Lu2, Chapters 4 and 5]. By Proposition 4.8 and [EW2, Corollary 6.27 and Theorem 6.30], the duality thus categorifies the bar involution on the Hecke algebra. We will therefore refer to it as the categorified bar involution.
Remark 4.8*.*
The origin of the categorified bar involution is [So2]. In the diagrammatic setting appears in [EW2, Definition 6.22], where it is denoted and gives an antiinvolution on double light leaves.
Proposition 4.9**.**
The categorified bar involution on S induces a functorial involution on , also denoted , which is covariant on -morphisms and contravariant and degree-preserving on -morphisms, such that in , for all and . This functorial involution extends to an equivalence between and which sends injective -morphisms in the first -category to projective -morphisms in the second.
Proof.
Since is covariant on -morphisms and in , for and , it preserves left, right and two-sided cells.
As sends identity -morphisms to identity -morphisms, it also preserves the maximal -ideal in S that does not contain the identity -morphisms on the -morphisms in , whence it descends to the -simple quotient . Since also preserves the two-sided cell and its diagonal -cells, it restricts to the -full -subcategories and , which proves the first claim.
Finally, since is contravariant on -morphisms, it extends to an equivalence between and . ∎
Corollary 4.10**.**
The functorial involution induces a functorial involution on , also denoted , which is contravariant and degree-preserving on morphisms and satisfies in , for all and . This functorial involution extends to an equivalence between and which sends injective objects in the category underlying the first to projective objects in the category underlying the second.
Proof.
As already remarked, the functorial involution also preserves left cells. The rest now follows as in the proof of Proposition 4.9. ∎
Remark 4.10*.*
The existence of implies that any statement in Subsection 4.1 has a dual counterpart. In particular, the equivalence induces a degree-preserving isomorphism of algebras , see Proposition 4.3.
4.3. Explicit bimodules for the cell -representation
In the following, we will use projective abelianizations instead of injective ones. As we are in the fiat setup, the difference does not play an essential role on an abstract level, but with this choice we can conveniently describe the action of by projective bimodules and their composition by tensoring over the underlying algebra, instead of injective bicomodules and cotensoring over the underlying coalgebra.
Denote by
[TABLE]
the algebra underlying the cell -representation. Fix a complete set of orthogonal primitive idempotents , for , corresponding to the indecomposable projective objects . Set and let (recall the notation in Subsection 4.1) be the simple head of in . Note that is one-dimensional and isomorphic to as a vector space. Note that for , see Subsection 2.3 and [MM3, Section 7]. Lemma 4.2(ii) implies that the socle of is isomorphic to .
For every pair , we have
[TABLE]
By [Lu2, Subsection 13.6], we know that
[TABLE]
Recall that, by definition of the Kazhdan–Lusztig basis, the are invariant under the bar involution.
Remark 4.10*.*
By [Lu2, 13.1(e)], we have , which corresponds to the fact that
[TABLE]
for all .
For the formulation of the next proposition, we recall that the are defined in Subsection 3.1.
Proposition 4.11**.**
For any , the action of on the category of finite dimensional graded -modules is isomorphic to tensoring (over ) with the graded projective --bimodule
[TABLE]
Proof.
By Theorem 2.1, we know that the action of is given by tensoring with a --bimodule of the form
[TABLE]
for certain . We also know that, for any , we must have
[TABLE]
On the other hand, using (4.8), we obtain
[TABLE]
and hence, deduce that the have to satisfy
[TABLE]
for all .
For every fixed pair , this is a system of linear equations, indexed by , in variables, indexed by . We claim that , for , is the unique solution of (4.9).
Let us first show that is a solution of (4.9), i.e. that we have
[TABLE]
This equation is similar to one in [Lu2, Subsection 18.8] and can be proved in the same way, using:
- •
the equation at the beginning of the proof of Theorem 18.9(b) in [Lu2], i.e.
[TABLE] 2. •
the symmetries in [Lu2, 13.1(e)]; 3. •
[Lu2, Proposition 13.9(b) and Conjecture 14.2.P7], i.e.
[TABLE] 4. •
[Lu2, Conjectures 14.2.P2 and 14.2.P5], i.e.
[TABLE]
By (4.10), (4.11) and (4.12), we have
[TABLE]
Finally, note
[TABLE]
so the determinant of the matrix
[TABLE]
belongs to , and the matrix is hence invertible over . Our system of linear equations in (4.9) therefore has a unique solution and the statement of the proposition follows. ∎
4.4. The Frobenius structure on the Duflo involution
In this subsection, we describe the structure of a Frobenius algebra on the Duflo involution in explicitly. As is a graded coalgebra, the comultiplication and counit -morphisms are homogeneous of degree [math]. By duality, is a graded algebra. This implies that the multiplication and unit -morphisms of are homogeneous of degree and , respectively. It remains to show that the multiplication and the comultiplication of satisfy the Frobenius conditions.
Recall the algebra constructed in Subsection 4.3. Proposition 4.11, combined with (4.11) and (4.12), implies that acts via the --bimodule
[TABLE]
The comultiplication on this bimodule is given by
[TABLE]
and the counit by
[TABLE]
for certain . In general we do not have a presentation of in terms of generating -morphisms and relations, so we can not compute the explicitly. In the specific case of dihedral Soergel bimodules such a presentation does exist and the were computed explicitly in [MT, Subsection 4.2]. In general, all we can say is that if the bimodule map representing involves , then the bimodule map representing involves by counitality.
To describe the algebra structure, consider the Frobenius trace and note that for a homogeneous element unless (because is weakly symmetric) and the degree of is . Furthermore, for any , let and be dual bases of and , respectively, with respect to the Frobenius trace , i.e.
[TABLE]
Here , with defined as in (3.1). Following the same convention, the dual of is denoted by , for . Let be the Nakayama automorphism of defined by , for any . Note that, since is weakly symmetric, we have and for all . Without loss of generality, we therefore assume that and . The algebra structure is then given by the multiplication
[TABLE]
and the unit
[TABLE]
for certain . As for the comultiplication and counit -morphisms, we cannot determine the explicitly in general. For the specific case of dihedral Soergel bimodules, those scalars were computed explicitly in [MT, Subsection 4.2]. In the definition of , note that unless .
We already know that is a coalgebra and an algebra. One can immediately check that and satisfy the Frobenius conditions. Since the cell -representation is faithful on -morphisms, this proves that is a Frobenius algebra in .
The result in this subsection is weaker than the Klein–Elias–Hogancamp conjecture [Kl, Subsection 5.2], [EH, Conjecture 4.40], which conjectures the existence of a Frobenius structure on in S itself. Indeed, we do not know how to “lift” the Frobenius structure on from the -simple quotient to the whole of S . The problem is that , as noted in [EH, Conjecture 4.40], may contain indecomposable direct summands in isomorphic to , with and either or . Note that, in terms of , Soergel’s hom formula in (3.3) becomes:
[TABLE]
In particular, this implies that the dimension of
[TABLE]
need not be one, in general. Let us give one simple example.
Example 4.11*.*
For rank or lower never contains such direct summands, but for higher ranks it frequently does. For an explicit and minimal example, let be of type with simple reflections , , , where we write for short, and Coxeter diagram
[TABLE]
Set . Then . Consider also the longest element of (whose -value is ), which is strictly greater than in the two-sided order. We have
[TABLE]
where the minimal shift of is strictly smaller than and the maximal shift is strictly bigger than .
5. Lifted simple transitive 2-representations
5.1. The underlying algebra
Suppose is a cosimple coalgebra in . By [MMMZ, Corollary 12] and Corollary 2.4, is a simple transitive birepresentation of and is semisimple. By Lemma 3.4 and Proposition 3.5, is also a cosimple coalgebra, which implies that \mathbf{M}:=\mathbf{inj}_{\underline{\scc\mbox{S}\hskip 1.0pt_{\mathcal{H}}}}\big{(}\Theta(\mathrm{A})\big{)} is a graded simple transitive -representation of with apex , using [MMMZ, Corollary 12] again.
Since is locally semisimple, see Proposition 3.3, must contain a direct summand isomorphic to , which is the identity -morphism in , and is the projection, which is a morphism of coalgebras. Hence, we obtain a faithful morphism of birepresentations of
[TABLE]
which is the identity on morphisms and sends in to . Here denotes the cell birepresentation of .
Since is -linear, the above implies that contains a direct summand in isomorphic to and the counit is the composite of the projection and . In particular, we obtain a faithful, degree-preserving morphism of -representations of
[TABLE]
which is the identity on morphisms and sends the comodule in to the comodule \big{(}M,(\mathrm{id}_{M}\circ_{\mathrm{h}}\pi_{d})\circ_{\mathrm{v}}\delta_{M,\Theta(\mathrm{A})}\big{)}.
Altogether, this yields a commuting square
[TABLE]
Let be a complete set of pairwise non-isomorphic simple (indecomposable) objects in . For every , we have
[TABLE]
for certain .
Let , for . Then is a complete and irredundant set of indecomposable objects of up to isomorphism and grading shift, and
[TABLE]
for every .
Lemma 5.1**.**
For and , define by
[TABLE]
Then
[TABLE]
Proof.
The proof follows from comparing two decompositions in . On one hand, by (5.1) and (5.2), we have
[TABLE]
On the other hand, by (5.1) and the fact that is a morphism of -representations, we have
[TABLE]
Comparing (5.3) and (5.4) for a fixed , we obtain the equation
[TABLE]
The facts that and now imply the result, as for every there exists at least one such that . ∎
Define
[TABLE]
Then is equivalent to the category of finite dimensional graded injective -modules.
Proposition 5.2**.**
The algebra is a finite dimensional positively graded Frobenius algebra of graded length .
Proof.
The case of being the cell -representation is discussed in Proposition 4.3 (or in Proposition 4.4 because of Proposition 4.6). In the case when is not necessarily the cell -representation, the proposition follows from similar arguments as in the proof of Proposition 4.4. ∎
Remark 5.2*.*
In contrast to the situation in Subsection 4.1, we do not know a priori that is weakly symmetric. Therefore, we have to include a possible Nakayama permutation in Subsection 5.3 below. Only at the end of that section, we will be able to show that it is trivial.
5.2. A characterization of -representations in the image of
We note the following characterization of the graded simple transitive -representations in the image of the map from Proposition 3.12. As we will see in Theorem 7.1, these exhaust the graded simple transitive -representation of with apex , up to equivalence.
Theorem 5.3**.**
Let be a graded simple transitive -representation of with apex . Then is in the image of if and only if the following conditions are satisfied:
- (i)
there is a choice of representatives of isomorphism classes of indecomposable objects in such that the endomorphism algebra of in is positively graded, and, additionally; 2. (ii)
for every and , the object decomposes into a direct sum whose summands, up to degree-preserving isomorphism, are of the form , where and ; 3. (iii)
the graded length of is not greater than .
Proof.
For the “only if” part observe that, by Proposition 3.8, we can pick a choice of representatives of isomorphism classes of indecomposable objects in , which are in the image of . The condition in (ii) then follows from Lemma 5.1. Conditions (i) and (iii) hold by Proposition 5.2.
The “if” direction follows the proof of Proposition 4.6 closely. Set . Then
[TABLE]
Conditions (i) and (ii) imply that the right-hand side is zero if . Hence, writing
[TABLE]
we obtain .
Next we want to establish an analog of Lemma 4.1. Namely, we claim that, for any simple object in concentrated in degree [math], and for any , the injective object is concentrated between the degrees [math] and . Similarly to the proof of Lemma 4.1, the fact that is concentrated in positive degrees follows from Conditions (i) and (ii). The fact that is concentrated in degrees below follows from conditions (ii) and (iii).
Now, if , then the right-hand side of (5.5) is zero since, by condition (iii), is a projective-injective object of graded length at most and the action of is given by projective functors which increase the graded length by at most (see the previous paragraph). Given the graded length of in , see Lemma 4.5, this again shows that for all , so is in the image of , as claimed. ∎
5.3. Explicit bimodules for the -action
The degree-zero part of is isomorphic to , where is a complete and irredundant set of primitive, orthogonal idempotents corresponding to respectively. Due to (5.1), every is concentrated between degrees [math] and , whence
[TABLE]
where is the Nakayama permutation of . By Theorem 2.1, the action of on the category of finite dimensional graded injective -modules, for , is given by tensoring over with a --bimodule of the form
[TABLE]
for certain .
Proposition 5.4**.**
We have
[TABLE]
for all and .
Proof.
For and , we obtain two different expressions for in , the category of finite dimensional graded projective -modules. On one hand, by (5.2) and the fact that , we have
[TABLE]
On the other hand, we have
[TABLE]
Comparing the terms in (5.6) and (5.7) for a fixed , shows that
[TABLE]
Suppose that has a non-zero term belonging to for some . By (5.8) and the fact that has highest term , see Proposition 5.2, this implies that has a non-zero term belonging to . However, this contradicts Lemma 5.1.
Since , the equation in (5.8) implies that cannot have non-zero terms belonging to either, whence
[TABLE]
for all and . ∎
In particular, note that the fact that the constant term in \mathrm{grdim}\big{(}e_{\sigma(i)}\mathsf{B}^{\mathbf{M}}e_{\sigma(k)}\big{)} is if , and [math] otherwise, implies that
[TABLE]
where all summands of have coefficients in . Since the first summand descends to the action of , which is the identity -morphism in , on , by Lemma 3.10, we see that . By Proposition 5.4 and equation (5.8), this shows that the action of is given by tensoring with the bimodule
[TABLE]
and that
[TABLE]
In [Lu2, Theorem 18.9], Lusztig defined a homomorphism of -algebras. Its restriction to is given by
[TABLE]
where in the split Grothendieck group (which should not be confused with Lusztig’s ) and in . Let denote the pullback of .
Proposition 5.5**.**
We have
[TABLE]
Proof.
By (5.9), there is a degree-preserving isomorphism
[TABLE]
Using this, we obtain two expressions for in . On one hand,
[TABLE]
On the other hand,
[TABLE]
Comparing (5.11) and (5.12) for a fixed yields
[TABLE]
which is precisely what we had to prove. ∎
Corollary 5.6**.**
* is bar invariant.*
Proof.
Equation (5.13) implies that is bar invariant, since the and are bar invariant. This completes the proof. ∎
Proposition 5.7**.**
The algebra is weakly symmetric.
Proof.
Recalling (3.1), (3.2) and (4.12), we know that
[TABLE]
By (5.13) and the equality , (5.10) then shows that
[TABLE]
By Proposition 5.2, the highest non-zero term of \mathrm{grdim}\big{(}e_{i}\mathsf{B}^{\mathbf{M}}e_{j}\big{)} is if and only if . Therefore, we obtain for all and the claim follows. ∎
6. Separability and semisimplicity
Recall the definition of from (4.6). The main point of this section is to prove that the coalgebra in is separable, see Propositions 6.2 and 6.3, which in turn enables us to show that is locally graded semisimple, see Proposition 6.9.
6.1. Some (diagrammatic) preliminaries
Let be the basic underlying algebra of the cell -representation of , see Subsection 4.1. This in particular means that there is an equivalence of -representations of on
[TABLE]
such that the -action on , the category of finite dimensional graded projective -modules, is given by a graded pseudofunctor from to (projective --bimodules, see Example 2.2) which is -faithful, i.e. faithful on -morphisms. Recall that is a finite dimensional positively graded weakly symmetric Frobenius algebra of graded length , where is the value of Lusztig’s -function on , see Proposition 4.3. Note that therefore the objects in are projective-injective and thus, , the latter denoting the category of finite dimensional graded injective -modules.
By Subsection 4.4, is a graded Frobenius algebra in with homogeneous comultiplication and counit of degree [math], and homogeneous multiplication and unit of degree and , respectively. Recall that , so Proposition 4.11 shows that acts on by tensoring over with the projective --bimodule
[TABLE]
and, consequently, the action of is given by tensoring with
[TABLE]
The --bimodule maps corresponding to , , and , which we defined in Subsection 4.4, are graphically illustrated as follows (for convenience, we recall the formulas defining them):
[TABLE]
Here the source (bottom) and target (top) of the diagrams in (6.1), (6.2), (6.3) and (6.4) are composites of unshifted copies of .
Remark 6.0*.*
Unlike the degrees in e.g. [EH, Conjecture 4.40]), the ones above are “unbalanced”, i.e. and , because we are working with instead of .
To avoid overloading the diagrams with shifts, we will also sometimes use a diagram of a -morphism of degree to depict the -morphism . Note that the dual of is isomorphic to in , so we can see e.g. also as a -morphism of degree zero from or , which are isomorphic in , to . The coevaluation and evaluation of in are given by
[TABLE]
which we draw as
[TABLE]
Note that in and we could also define and in the evident way. Since , the diagrams for the evaluation and the coevaluations of and are the same, which justifies the lack of arrows on the diagrams.
Recall that being a Frobenius algebra in means that the (co)multiplication and the (co)unit satisfy the diagrammatic equations
[TABLE]
Recall further that (6.6), (6.7) and (6.8) imply
[TABLE]
By (6.12) we only need to consider isotopy classes of diagrams, and we simplify our diagrams by drawing them in a more topological fashion. However, some diagrams are not the same, as the calculus is not necessarily cyclic for general --bimodule maps. In particular, for any
[TABLE]
where the dots indicate multiple strands, we define the right mate and the left mate of via
[TABLE]
which have the same degree as the diagram in (6.13) itself. To be more concrete, let us compute the mates of in the cell -representation. Suppose that the --bimodule map is given by
[TABLE]
for and some homogeneous and such that . By (6.5), the --bimodule maps corresponding to and are given by
[TABLE]
for . Then and are given by
[TABLE]
where is the Nakayama automorphism. In other words, the --bimodule map associated to can be obtained from the one associated to by applying . To indicate this diagrammatically, we write
[TABLE]
Note that the equations in (6.12) imply that the left and right mates of the structural -morphisms of agree, i.e. and .
6.2. Dot diagrams
Next, recall that . For short, we denote the morphism spaces in the latter category by (for the enriched version) and (for morphism spaces of degree zero). A specific object in this category is , since the coalgebra structure of gives it the structure of an injective right comodule in over itself.
Lemma 6.1**.**
Let be a homogeneous endomorphism of in of degree , for some . Then we have if and only if the corresponding --bimodule map is of the form
[TABLE]
where is some homogeneous element in of degree .
Proof.
Let and suppose that the --bimodule map associated to is given by (6.14). Now compare the --bimodule maps associated to and , respectively
[TABLE]
By -faithfulness of the -functor defining the cell -representation (see the beginning of this section), we see that the equality holds if and only if
[TABLE]
which proves the lemma. ∎
Before the next statement, which is a version of separability, recall the following special cases of Soergel’s hom formula (3.3):
[TABLE]
Proposition 6.2**.**
There is a unique -morphism in , which we call the dot diagram and which we depict as
[TABLE]
such that
[TABLE]
Proof.
Choose any non-zero -morphism , which we depict as in (6.13). By (6.16), we have
[TABLE]
for some scalar . We claim that .
To prove this claim, we only need to compute the --bimodule map corresponding to the morphism on the left-hand side in (6.18). By Lemma 6.1, we have
[TABLE]
where for some scalar depending on . Hence, (6.1) and (6.2) imply that
[TABLE]
We see that for all . In particular, the scalar does not depend on the choice of . Since is not zero, there exists an such that , which implies that because for all . Note that this also means that for all .
Now define
[TABLE]
Clearly, this is the unique -morphism in satisfying (6.17). ∎
Remark 6.2*.*
One might be tempted to use the diagrammatic calculus to define the dot diagram as a multiple of
[TABLE]
but we do not know whether this -morphism in is non-zero in general.
For later use, note that
[TABLE]
for all , where . Note also that the fact that the dot diagram is a homomorphism of right -comodules in translates to
[TABLE]
Since fixes and , the equality in (6.15) and the --bimodule map in (6.19) show that the right and left mates of the dot diagram agree, i.e.
[TABLE]
Either of these diagrams we therefore call the dual dot diagram, and the following is an almost immediate consequence of Proposition 6.2, which can be proved diagrammatically using (6.12) and (6.20).
Proposition 6.3**.**
The dual dot diagram is a homomorphism of left -comodules in and satisfies
[TABLE]
Diagrammatically, this means that the dual dot diagram satisfies a sliding relation analogous to (6.20). Furthermore, as in (6.19), the image of the dual dot diagram in is given by
[TABLE]
for all .
A straightforward calculation in proves the following diagrammatic equations, which will be very useful in the following.
Lemma 6.4**.**
We have
[TABLE]
6.3. Graded semisimple bicomodules
We are now ready to determine the structure of , in particular, that it is locally graded semisimple.
Lemma 6.5**.**
The -morphism has a unique --bicomodule structure in , for every and
Proof.
For every , we have , see (3.9) and the text below it. Since is the identity -morphism in , the -morphism is naturally an --bicomodule. This, together with the fact that is an oplax pseudofunctor, implies that is a --bicomodule in .
Of course, this implies that is a --bicomodule in for all , with the same (degree zero) structure -morphisms. We denote the left and right -coactions on by and , respectively. Note that the bicomodule structure on each in is unique because Soergel’s hom formula (3.3) gives
[TABLE]
and the choice of and is fixed by counitality. ∎
Lemma 6.6**.**
For any and , we have
[TABLE]
Proof.
By arguments analogous to those in the proof of Lemma 6.1, but this time applied to the left and the right -coactions, we see that
[TABLE]
The result for follows from (3.3). ∎
Since is a graded Frobenius algebra in (c.f. Subsection 4.4), any right, left or bicomodule of has a compatible right, left or bimodule structure over up to a degree shift of . This is well-known, but for convenience and later use, we briefly repeat the main arguments and constructions. The shortest way to do this is diagrammatically. Let be a right -comodule in with right coaction . We depict as
[TABLE]
The coassociativity and counitality of are expressed by the diagrammatic equations
[TABLE]
We can then define the right -action of degree as
[TABLE]
Note also that (6.12) implies that
[TABLE]
Similarly, it is easy to see that
[TABLE]
i.e. is associative and unital and there is a Frobenius-type compatibility between and . Thus, (6.23), (6.24), (6.25) and (6.28) imply that the diagrammatic calculus is again topological in nature, which we will use to simplify our diagrammatic arguments.
By symmetry, similar results hold for left coactions and left actions of and one can check that any --bicomodule in has a compatible --bimodule structure and vice versa (which diagrammatically translates into a height exchange, see (6.29) below). To simplify our diagrams for bimodules and bicomodules, we also allow four valent vertices, e.g.
[TABLE]
The following lemma is the analog of Lemma 6.4.
Lemma 6.7**.**
We have
[TABLE]
Proof.
The trick is to reduce the statement to the one from Lemma 6.4. We show how to do this for the first diagrammatic equation.
[TABLE]
The proof of the second diagrammatic equation is analogous. ∎
The following lemma is the analog of (6.17) and (6.21).
Lemma 6.8**.**
We have
[TABLE]
Proof.
Let us prove the first equality.
[TABLE]
The second equation can be proved by symmetry. ∎
For any , let be the category defined as
[TABLE]
By definition, we will write
[TABLE]
for any . Note that horizontal composition in descends to , so the action of on gives rise to a semisimple simple transitive -representation of . In particular, is a -finitary locally semisimple bicategory, as defined in Subsection 2.7.
Recall also from Subsection 2.7 the definition of the graded finitary -category with translation for a given graded -linear -category C .
Proposition 6.9**.**
We have an equivalence of -finitary locally semisimple bicategories
[TABLE]
Moreover, this implies an equivalence of locally graded semisimple bicategories
[TABLE]
Proof.
Let be a -morphism in . Suppose that is a direct summand of in for some and . We first claim that is also a direct summand of in .
To this end, let and be the embedding and the projection respectively, i.e. . Depict them using our usual conventions, with dotted strands of a different color for (which we again denote by in the diagrams) and :
[TABLE]
Define and as
[TABLE]
Note that and are indeed -morphisms in , e.g.
[TABLE]
which shows that commutes with the right -coaction. Next, by (6.30), (6.31), the Frobenius properties and , we have . Indeed,
[TABLE]
which proves our claim. Hence, the decomposition of into indecomposables is the same in as in .
Together with Lemma 6.6, this implies that
[TABLE]
as -finitary categories.
To show that this is an equivalence of bicategories, recall the oplax pseudofunctor from (3.9). Since each has a canonical --bicomodule structure, has an induced -bicomodule structure in by the oplax condition of . By Lemma 3.6, there are homogeneous -isomorphisms of degree zero
[TABLE]
that are natural in and . This shows that gives rise to a -linear pseudofunctor
[TABLE]
for which we use the same notation.
Observe that extends uniquely to a -linear pseudofunctor
[TABLE]
which is compatible with the -actions, and again -full by Lemma 6.6 and -faithful by semisimplicity. By the claim at the beginning of the proof and Lemma 6.5, each has therefore a unique --bicomodule structure and is isomorphic to for some . This shows that is essentially surjective on -morphisms. Hence
[TABLE]
as free -bicategories.
Since by Lemma 2.7, this implies that there are biequivalences of free -bicategories
[TABLE]
Moreover, by the paragraph above Lemma 2.7, and, therefore, we obtain biequivalences of locally graded bicategories
[TABLE]
where the third biequivalence follows from Lemma 2.7 and the second from the previous chain of biequivalences. ∎
The following corollary follows immediately from Corollary 4.7 and Proposition 6.9.
Corollary 6.10**.**
There is a biequivalence of graded bicategories
[TABLE]
Remark 6.10*.*
If we forget the gradings, then the biequivalence (6.32) becomes a biequivalence of (ungraded) locally semisimple bicategories
[TABLE]
This induces the structure of a pivotal fusion bicategory on , by Proposition 3.3. Abusing the terminology slightly, we therefore say that is pivotal -fusion, in particular, it is a semisimple pivotal monoidal category.
Notice that the dual of in is isomorphic to , for any and . This in contrast to the dual of in , which is isomorphic to .
Remark 6.10*.*
The proof of Proposition 6.9 also shows that
[TABLE]
By the graded version of Theorem 2.6, there is a pair of graded biequivalences
[TABLE]
where is the two-sided cell containing . Among other things, the first biequivalence uses the fact that the additive closure of in is equal to the additive closure of in . The composite of both biequivalences sends to , where is the left cell such that . It also provides a graded equivalence between and , both being equivalent to .
We can now explain a generalization of Proposition 6.9. Let be the -category whose
- •
objects are left cells in , identified with the cell -representation of S ; 2. •
morphism categories with horizontal composition given by composition of morphisms of -representations.
Recalling that every left cell contains a unique Duflo involution and the equivalence , is biequivalent to the bicategory , where has the same objects but with morphism categories
[TABLE]
where horizontal composition is given by taking cotensor products over the relevant Duflo involution and the identity -morphism on is .
As already recalled in Remark 3.2, we can define the bicategory in analogy to using the perverse filtration of .
Theorem 6.11**.**
The -category is locally graded semisimple.
Proof.
As in Subsection 4.4 and Proposition 6.2, the proof follows from the fact that the Duflo involution is a separable Frobenius algebra in for every left cell in . Recall that is a -full one-object subbicategory (i.e. a full monoidal subcategory) of the bicategory . As we already mentioned in Remark 3.2, the latter can be regarded as a pivotal multifusion bicategory with one object (whose identity -morphism is decomposable in general) or, as we will do in this paper, as a pivotal fusion bicategory whose objects are the left cells and whose morphism categories consist of -morphisms in , for the -cell .
Analogously to Proposition 6.9, one can show that is locally graded semisimple and that there is a biequivalence of -finitary bicategories
[TABLE]
which implies that there is a graded biequivalence
[TABLE]
Since there is also a graded biequivalence , the result follows. ∎
Remark 6.11*.*
The previous theorem shows that is graded biequivalent to and hence locally graded semisimple. However the graded cell -representations of S with apex , which form the objects of , are not semisimple themselves in general, as follows from Proposition 4.3.
At the same time, we can analogously define the graded -category as having objects for left cells , which we now identify with the corresponding graded cell -representations of (which are graded semisimple), and with morphism categories given by the corresponding morphisms of -representations. Again, this is graded biequivalent to the bicategory with the same objects, and the graded morphism category from to given by . Since the are idempotents with the trivial coalgebra structure, this implies that is graded biequivalent to and hence there is a graded biequivalence
[TABLE]
In Theorem 7.8 we will prove that is locally graded semisimple, which is a generalization of Theorem 6.11. The reason for including the latter theorem here is that the proof above, which does not depend on the results in Section 7, shows that both -categories in (6.33) are graded biequivalent to , which is a nice result in its own right. The analogous result for is more involved and, since we do not need it in this paper, we have omitted it.
7. The main theorem
In this section we will prove our main theorem:
Theorem 7.1**.**
There are biequivalences of graded -categories
[TABLE]
Remark 7.1*.*
To avoid confusion about the gradings, recall that by Proposition 6.9. The last biequivalence in Theorem 7.1 then means that, for every , there is an such that in , and this correspondence establishes a bijection between the equivalence classes of objects in and . Moreover, for every pair , there is an equivalence of graded categories .
Note that this does not mean that and are biequivalent as graded -categories, because the morphism categories of the former -category are non-trivially graded, whereas those of the latter -category are trivially graded (meaning that all -morphisms live in degree zero), c.f. Remark 2.6.
The proof of Theorem 7.1 is the content of Propositions 7.4 and 7.7 in the next two subsections, each of which proves one of the two biequivalences above, but let us first state and prove a very important corollary.
Corollary 7.2**.**
For any finite Coxeter type, there are finitely many equivalence classes of graded simple transitive -representations of S .
Proof.
Let be any finite Coxeter group and . By the composite of the biequivalences in Theorem 7.1, there is a bijection between the set of equivalence classes of graded simple transitive -representations of with apex and the set of equivalence classes of simple transitive birepresentations of , for any diagonal -cell of S . Since every is pivotal fusion, the corollary now follows from [EGNO, Corollary 9.1.6 and Proposition 3.4.6], strong -reduction, see [MMMZ, Theorem 15] and [MMMTZ, Theorem 4.32]), and the fact that S has only finitely many two-sided cells. ∎
7.1. The right biequivalence in Theorem 7.1
Lemma 7.3**.**
Every graded simple transitive -representation of is graded semisimple.
Proof.
Let be a graded simple transitive -representation of . We want to show that the radical of is trivial, which implies the claim. Recall that the radical is the unique maximal nilpotent ideal of , which can be defined as the union of the radicals of all full subcategories of with finitely many objects. Since such a subcategory can also be seen as a finite dimensional algebra, its radical can be defined as in [DoKi, Section 3.1]. By [DoKi, Theorem 3.1.1], this also implies that is semisimple. We first consider the -representation , and note that this does not contain any proper ideals, see the last paragraph of Subsection 2.7.
We then use Proposition 6.9 to restrict to a -representation of , and let denote the transitive -subrepresentation of generated by a non-zero indecomposable object in .
Recall that is pivotal fusion and therefore only has one two-sided cell, namely , which contains the identity -morphism . Thus, Theorem 2.1 implies that acts projectively, so any simple transitive -representation of is semisimple. This in turn implies that the radical is the unique maximal ideal of . By Proposition 6.9, the direct sums
[TABLE]
are both -invariant, and hence so is their sum, which equals . Since does not contain any proper ideals, the latter radical is zero. Thus is semisimple, and hence is graded semisimple. ∎
Proposition 7.4**.**
There is a biequivalence of graded -categories
[TABLE]
Proof.
Given a simple transitive -representation of , we can choose a simple object to obtain \mathbf{N}\simeq\mathbf{inj}_{\scc\mbox{A}\hskip 1.0pt_{\mathcal{H}}}\big{(}[\mathrm{X},\mathrm{X}]\big{)}. By Proposition 6.9 and [MMMT, Theorem 9] and [MMMZ, Corollary 12], \mathbf{M}:=\mathbf{inj}_{\scc\mbox{B}_{\mathcal{H}}\hskip 1.0pt}\big{(}[\mathrm{X},\mathrm{X}]\big{)}\simeq\mathbf{N}^{\prime} is a graded simple transitive -representation of . (To simplify notation, we use the identification [\mathrm{X},\mathrm{X}]=\Theta\big{(}[\mathrm{X},\mathrm{X}]\big{)}.) This provides an assignment on objects, and we claim that it is essentially surjective.
Indeed, let be a graded simple transitive -representation of . As before, we consider the -representation , which again does not contain any proper ideals, see Subsection 2.7, and restrict it to . Letting be a non-zero indecomposable object in , we let denote the transitive -subrepresentation of generated by . By virtue of being pivotal fusion, is simple transitive and, moreover, the subcategory of is also semisimple by Lemma 7.3.
Also by Lemma 7.3, we have
[TABLE]
as -representations of . By [MMMT, Theorem 4.7] and [MMMZ, Corollary 12] (which in the context of finitary semisimple categories are the dual versions of [EGNO, Corollary 7.10.5 and Exercise 7.10.6]) and local semisimplicity of , we have
[TABLE]
where is the cosimple coalgebra given by the internal cohom construction, see Subsection 2.4. Proposition 6.9 implies that it is also cosimple as a coalgebra in and from (7.1) and (7.2) it follows that
[TABLE]
as -representations of . Moreover, and, by semisimplicity, we have
[TABLE]
as graded finitary -representations, and our assignment is essentially surjective
We further claim that any morphism of -representations of between and extends in a unique way to a morphism of graded -representations of between and . To see this, note that any object of is a direct sum of objects of the form for some . Since , we obtain that
[TABLE]
Similarly, all components of a morphism in are scalar multiples of for indecomposable , again by semisimplicity. Since , we obtain
[TABLE]
which completes the proof of our claim. Recall that -morphisms in are of the form , where is a morphism of -representations of between and and . Extending as above, we send to . Modifications extend similarly, and we obtain a graded pseudofunctor .
To check that the above graded functor
[TABLE]
is an equivalence, let be a morphism of graded -representations for . Suppose we have defined \mathbf{N}_{1}\simeq\mathbf{inj}_{\scc\mbox{A}_{\mathcal{H}}\hskip 1.0pt}\big{(}[\mathrm{X}_{1},\mathrm{X}_{1}]\big{)},\mathbf{N}_{2}\simeq\mathbf{inj}_{\scc\mbox{A}_{\mathcal{H}}\hskip 1.0pt}\big{(}[\mathrm{X}_{2},\mathrm{X}_{2}]\big{)} via a choice of objects . If , it is easy to see that restricts to a morphism of -representations . Modifications restrict naturally. It is immediate that these assignments provide a quasi-inverse to the functor . ∎
7.2. The left biequivalence in Theorem 7.1
A functor between two finitary categories is called injective if its extension to the injective abelianizations of those categories is an injective object in the category of left exact functors between the said abelianizations. Further, a morphism of finitary -representations is injective if its underlying functors are injective functors. For a finitary -category C and two finitary -representations , of C , we denote by the category of injective morphisms of -representations. Moreover, considering injective endomorphisms of , we obtain the -semicategory . We further denote by the additive closure of the injective endomorphisms and the identity morphism of . This category also defines the -category whose single object is .
For future use, we record the following lemma.
Lemma 7.5**.**
Let be an injective functor and assume that both and are finitary Frobenius categories. Then is isomorphic to tensoring with a projective bimodule and the collection of such injective functor is closed under adjunctions.
Proof.
Assume without loss of generality that and for self-injective algebras and , respectively. Any left exact functor from to is isomorphic to for some --bimodules . The morphism space between such functors and is given by , thus is injective if and only if is a projective bimodule. In this case, using together with additivity of and self-injectivity of , we see that injective functors correspond to tensoring with projective bimodules, and that these are, moreover, closed under taking adjoints. ∎
Before we can prove the left biequivalence in Theorem 7.1, we need to recall the appropriate application of the double centralizer theorem [MMMTZ, Theorem 5.2] to the case of the cell -representation of . The -actions of and on commute by definition, this implies that there is the canonical (graded) -functor
[TABLE]
The following theorem is a special, but graded, case of [MMMTZ, Theorem 5.2].
Theorem 7.6**.**
The canonical -functor is fully faithful on -morphisms and essentially surjective on -morphisms when restricted to and corestricted to .
We are now ready to prove the existence of the left graded biequivalence in Theorem 7.1.
Proposition 7.7**.**
There is a biequivalence of graded -categories
[TABLE]
Proof.
Consider the following graded fiat -category with two objects, denoted and . We identify the object with the graded category , i.e. the underlying graded category of the cell -representation of with apex , and the object with the graded category , i.e. the graded category underlying the principal birepresentation of . Recall that, due to the graded equivalence in Corollary 4.7, the -category acts, by graded endofunctors and homogeneous degree-zero natural transformations, on both and .
The graded morphism categories of are defined as
[TABLE]
Note that
[TABLE]
since is graded semisimple by Proposition 6.9. The graded -categorical structure on is given by composition in the same way as in e.g. [MM3, Subsection 2.3], and the fiat structure is defined by taking adjoints, which preserves injective functors, since the underlying categories are Frobenius (see Lemma 7.5).
Next, note that the regular birepresentation of defines a natural graded embedding
[TABLE]
as it clearly commutes with the right -action of on . Locally graded semisimplicity of implies that this graded embedding is a graded biequivalence.
By Theorem 7.6, the natural graded embedding of into is fully faithful on -morphisms and essentially surjective on -morphisms when restricted to and corestricted to . In particular, the endomorphism -category of the object has two -cells: that containing the identity and that consisting of indecomposable objects in , which we can identify with . Theorem 7.6 then implies that there is a biequivalence of graded -categories
[TABLE]
The statement now follows by strong -reduction, see Theorem 2.6. To see this, note that has only one non-trivial two-sided cell (provided is not trivial, of course), which we denote by and is formed by the isomorphism classes of all indecomposable -morphisms that are not isomorphic to . This two-sided cell contains two diagonal -cells, the one formed by the isomorphism classes of the indecomposable -morphisms in and the other formed by the isomorphism classes of the indecomposable -morphisms in , which can both be identified with due to Proposition 6.9 and Theorem 7.6. By the above and Theorem 2.6, there are biequivalences of graded -categories
[TABLE]
where the middle biequivalence is induced by in (7.3) and the last equality uses that only has one -cell, which follows from Remark 2.3 and Proposition 6.9. This completes the proof. ∎
Remark 7.7*.*
The construction of the -category in the proof of Proposition 7.7 is in the spirit of what is known as a Morita context in the tensor category literature, see [Mu, Remark 3.18 and Section 4].
7.3. Local graded semisimplicity
In this subsection we generalize Theorem 6.11.
Theorem 7.8**.**
The -category is locally graded semisimple.
Proof.
By Theorem 2.6 and Theorem 7.1, it suffices to prove that is locally semisimple. Noting that is pivotal fusion and that, therefore, its simple transitive -representations coincide with its so-called indecomposable exact module categories, this is a consequence of [ENO, Theorems 2.15 and 2.16]. ∎
7.4. The ungraded case
In this section, let be the ungraded category of Soergel bimodules for a finite Coxeter system . This category is defined just as its graded counterpart, but ignores the grading. The indecomposables in S are still indexed by and will be denoted by as well. This follows from the fact that for any object , the endomorphism ring is local in the graded setting if and only if it is local in the ungraded setting. The character isomorphism in the Soergel–Elias–Williamson categorification theorem then has a -linear counterpart in the ungraded setting which sends to , where is the image of the Kazhdan–Lusztig basis element under the -linear map which sends the standard basis element to , for , and to .
Recall that denotes the -category of all simple transitive -representations of with apex without any grading assumptions. Any simple transitive graded -representation of the graded with apex remains simple transitive in the ungraded setting, but there is no a priori reason why the forgetful -functor
[TABLE]
should be essentially surjective. In principle, there could be ungradable simple transitive -representations of with apex . This was actually an open question until now, e.g. for finite dihedral Coxeter type it was unknown if the bicolored ADE classification of the graded simple transitive -representations of S with subregular apex in [KMMZ] and [MT] remained valid in the ungraded case. We can now answer that question affirmatively, because is indeed a biequivalence for any finite Coxeter type.
This follows from the fact that in the ungraded case the double centralizer property from Theorem 7.6 still holds (note that [MMMTZ, Theorem 5.2] was formulated and proved in the ungraded setting) and that there is a biequivalence
[TABLE]
The proof of Theorem 7.6 does not use the grading in an essential way, so that theorem remains true in the ungraded setup. The arguments which prove the existence of the biequivalence in (7.4) in the ungraded case, are exactly the same as the ones which prove Proposition 6.9. The only difference is that there is no translation in the ungraded setting, so the trivial -cover of “collapses” to itself. Just as in the proof of Theorem 7.1, the double centralizer theorem and the biequivalence in (7.4) imply that there is a biequivalence
[TABLE]
by Theorem 2.6 (which also holds both in the graded and the ungraded setting). This shows that the forgetful -functor is indeed a biequivalence.
As in Corollary 7.2, the biequivalence in (7.5) implies that, for any finite Coxeter type, there are finitely many equivalence classes of simple transitive -representations of S .
8. Classification results
The asymptotic bicategory and its -representations
Recall that, by Proposition 3.3, is a pivotal fusion bicategory and thus, is its only cell and all of its simple transitive -representations are semisimple. Further, the notion of a simple transitive -representation agrees with the notion of an indecomposable exact module category, which is often used as the terminology in [BFO], [EGNO], [KO], [Os2], [Os3], [Os4] below.
Up to a handful of exceptions, the asymptotic bicategory comes in three flavors and for all of them a classification of simple transitive -representations is known, as we will summarize now (giving more details below). Recall that is the pivotal fusion -category of finite dimensional -modules. Let denote the pivotal fusion -category of finite dimensional -graded -vector spaces ( are plain finite dimensional vector spaces), and the pivotal fusion -category of complex, finite dimensional representations of quantum semisimplified at level and without twists, see e.g. [EGNO, Examples 2.3.4 and 8.18.5].
- (a)
Finite Weyl type (excluding ): generic case. Up to three exceptions in types and , see b, for each two-sided cell there exists a diagonal -cell such that \sc\mbox{A}_{\mathcal{H}}\hskip 1.0pt\simeq\sc\mbox{V}\hskip 1.0pt\mathrm{ect}\big{(}(\mathbb{Z}/2\mathbb{Z})^{k}\big{)} for some , or for being , or . This follows from (the arguments in the proof of) [BFO, Theorem 4].
For all of these, the classification of the associated simple transitive -representation works as follows. Let be a finite group and let denote the set of subgroups of up to conjugacy, a choice of representative of , and the second group cohomology of with values in , whose non-trivial generators are called Schur multipliers. By e.g. [EGNO, Example 7.4.10 and Corollary 7.12.20], we have
[TABLE]
The simple transitive -representations of have rank and the ones for are the -twisted representation categories (in particular, their rank is equal to the rank of the character ring of for trivial ). Hence, we need to analyze the simple transitive -representations of or , which are given by (conjugacy classes of) subgroups of , their number , and the Schur multipliers in of these subgroups. We additionally list their rank . Listing the data that we need is easy (calculating the subgroups and their numbers for is a pleasant exercise, while the Schur multipliers of these subgroups were already determined by Schur, see e.g. [Ber, Theorem 4] for a more modern reference; the data for the other three cases, , and , can be calculated by computer). In the row in (8.20), two entries correspond to two different simple transitive 2-representations, one for each Schur multiplier with the one for the trivial Schur multiplier listed first.
[TABLE]
(Here is the general affine group of rank one over .) 2. (b)
Finite Weyl type: exceptional case. Type contains one and type two so-called exceptional cells. For these, by [Os4, Theorem 1.1], we have , having its -structure twisted by the non-trivial element in the third group cohomology group . In this case has only one associated simple transitive -representation, which is the regular -representation of rank , see e.g. [Os2, Theorem 3.1]. 3. (c)
Finite dihedral type (including ). We have for the cells containing the identity element or the longest element, or for the middle cell, by [El, Theorem 2.15]. By e.g. [KO, Theorem 6.1] and [Os3, Theorem 6.1], we have
[TABLE]
The corresponding simple transitive -representations have rank equal to the number of vertices of the associated ADE diagram. 4. (d)
Types and . We do not know what is in general. For details see below.
What the main theorem covers
Recall that any (graded) simple transitive -representation of S has an apex in . Then, Theorem 7.1 together with Theorem 2.6, implies that
[TABLE]
holds for any diagonal -cell . Thus, we get
[TABLE]
(Note that Corollary 3.9 also gives us the rank of the simple transitive -representations of associated to the ones from . However, the corresponding simple transitive -representations for S will have greater rank in general.) Thus, the above shows that only certain cells in Coxeter types and – most prominently, the cell (8.56) in type given below – would remain open with respect to a complete classification of (graded) simple transitive -representations of S . For all other cases, Theorem 7.1 gives a complete classification and parametrization of the (graded) simple transitive -representations of S , as we will summarize now.
What we cover
Let us give some details of what is covered by the results in this paper. We say that a type is “done” if we can identify and parameterize its simple transitive -representations for at least one diagonal -cell .
For this purpose, we use what we call a full cell matrix:
[TABLE]
Here we indicate the number of elements in left or right cells, where e.g. is to be understood as a -by- matrix containing only the entry (thus, having elements), i.e. it is , where is the -by- identity matrix. The shaded boxes are (matrices of) diagonal -cells.
The full cell matrices are block matrices, but we also view them as matrices containing only the scalars in and call them cell matrices. The difference between the two is that the cell matrix actually encodes equivalence classes of cell -representation: each column of the full cell matrix corresponds to a (left) cell -representation, while the columns of the cell matrix itself correspond to (left) cell -representation up to equivalence.
Here are a few more examples, which have appeared under various names in the literature; the first and second cases are respectively below.
[TABLE]
where , , or , respectively, is the size of the cell in question. (Knowing the size of the cells, one can recover since there is always a unique solution in positive integers.) Note for example that in the nice case there are only two columns in the cell matrix, thus, only two cell -representations up to equivalence. However, there are cell -representations as encoded in the columns of the full cell matrix.
Type
This type is done for all , since all cells are strongly regular. In this case there is one equivalence class of simple transitive -representation per two-sided cell, all of which are cell -representations.
Type
This type is done for all :
- (i)
For all , we have for some with . The classification in this case is given by (8.20). 2. (ii)
The diagonal of the cell matrix is , all other entries are for . 3. (iii)
Type is the first case not covered by previous classification results. 4. (iv)
is the smallest example (i.e. the example of smallest rank) in classical type where we have a non-cell simple transitive -representation; see below.
Type is:
[TABLE]
Here and throughout: from left to right, we have listed the numbered cells, paired (with [math] being the minimal containing and the maximal cell containing ). From top to bottom, we have listed their sizes, the -values, and , where the number means that for .
Type is:
[TABLE]
The cell is displayed in (8.31). In this case, we have , which has the (non-conjugate) subgroups . The subgroups and all have trivial second group cohomology, but . Thus, we have six equivalence classes of (graded) simple transitive -representations of ranks , , , , and , respectively, see (8.20). It follows from Theorem 7.1 that this case gives a non-cell simple transitive -representation. The same happens repeatedly for higher ranks.
Type
This type is done for all :
- i,ii
As for type , but with . 2. iii
Type is the first case not covered by previous results. 3. iv
In type the first non-cell simple transitive -representations appear.
Type
This type is done for all :
- (i)
Every contains a ; there are only three two-sided cells. 2. (ii)
The bottom and top cell are strongly regular. 3. (iii)
The middle cell has a dihedral cell matrix, see (8.32), and the classification is given in (8.25). This is the smallest example with non-cell simple transitive -representations, starting from type .
Type
This type is done:
[TABLE]
Here we write for with . The corresponding cells are strongly regular or nice, except
[TABLE]
Again, we get non-cell simple transitive -representations, see (8.20).
Type
This type is done, and it is quite similar to type :
[TABLE]
[TABLE]
Cell is exceptional with , see [Os4, Theorem 1.1]. The cells and are as in (8.36) (with diagonals , and ), giving non-cell simple transitive -representations. All other cells are strongly regular or nice.
Type
This type is done, and it is similar to type :
[TABLE]
[TABLE]
As before, the majority of cells are strongly regular or nice. We also have exceptional cells with , see [Os4, Theorem 1.1], and also cells as in (8.36) (with diagonals , and , or , and ), giving non-cell, simple transitive -representations. There is one remaining cell with for appropriate , giving again non-cell simple transitive -representations. Its cell matrix is
[TABLE]
The simple transitive -representations of can be obtained from (8.20).
Type
This type is done:
[TABLE]
where we write for with , as before, and the cell matrices are strongly regular or nice, see (8.32). In the remaining case we have (for appropriate ):
[TABLE]
For the list of equivalence classes of simple transitive -representations of , see (8.20). This is the second smallest example in Weyl type with a non-cell simple transitive -representation.
Type
This type needs more work:
[TABLE]
- (a)
These cases are strongly regular two-sided cells. 2. (b)
In these cases, the cell is , and the Grothendieck rings of and coincide. 3. (c)
Cell is , and the Grothendieck rings of and coincide.
By [Os1, Subsection 2.5], the only two possibilities for (b) are or (in the notation of Ostrik). Similarly, by [Os1, Subsection 2.4], the only two possibilities for (c) are or . However, only in the case of the cell do we know which option it is, namely , since this case is covered by [KMMZ, Theorem 28]. In this case the classification is as in (8.25).
Type
This type needs much more work:
[TABLE]
(a),(b),(c) are similar to (a),(b),(c) in type , and the same remark as in (d) holds. In the remaining case we have:
[TABLE]
We were not able to find in the literature. In fact, for none of the diagonal -cells do we know what is; we only know the multiplication tables of their Grothendieck rings with respect to the asymptotic Kazhdan–Lusztig bases , see also [Al]. For example, if is in the -by- block, then the Grothendieck ring of is not commutative, has Perron–Frobenius dimension and a simple generating -morphism of Perron–Frobenius dimension and fusion graph
[TABLE]
Remark 8.0*.*
Note that the two-sided cell in type , whose structure is detailed in (8.56), contains a diagonal -cell for which the fusion algebra is non-commutative. This implies, of course, that the pivotal fusion category , whose precise structure is unknown, is not braided. For completeness, we list all non-commutative in all finite Coxeter types below:
- •
In types , the of order 6 in the two-sided cell . 2. •
In type , the of order 6 in the two-sided cells and . 3. •
In type , the of order 6 in the two-sided cells , and the of order in the two-sided cell . 4. •
In type , the of order 9 in the two-sided cell . 5. •
In type , all in the two-sided cell .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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