Abstract
We introduce the notion of Demazure descent data on a triangulated category and define the descent category for such data. We illustrate the definition by our basic example. Let be a reductive algebraic group with a Borel subgroup . Demazure functors form Demazure descent data on and the descent category is equivalent to .
1. Motivation
The present paper is the first one in a series devoted to various cases of categorical descent. Philosophically, our interest in the subject grew out of attempts to understand the main construction from the recent paper by Ben-Zvi and Nadler [1] in plain terms that would not involve higher category theory.
1.1. Beilinson-Bernstein Localization and Derived Descent
Let be a reductive algebraic group with the Lie algebra . Denote the Flag variety of by . A major part of Geometric Representation Theory originated in the seminal work of Beĭlinson and Bernstein [2] devoted to investigation of the globalization functor . This functor turns out to be fully faithful and provides geometric and topological tools to investigate a wide class of -modules, in particular the ones from the famous category . Various generalizations of this result lead to the investigation of the categories of twisted D-modules on the Flag variety and on the base affine space for and of their derived categories.
Ben-Zvi and Nadler define a certain comonad acting on a higher categorical version for the derived category of D-modules on the base affine space. In fact, the functor is built into the higher categorical treatment of Beilinson-Bernstein localization-globalization construction.
Using the heavy machinery of Barr-Beck-Lurie descent, the authors argue that the derived category of -modules is equivalent to the category of D-modules equivariant with respect to this comonad. Thus the global sections functor becomes equivariant with respect to the action. The comonad is called the Hecke comonad. It provides a categorification for the classical action of the Weyl group on various homological and K-theoretic invariants of the Flag variety.
Notice that the descent construction fails to work on the level of the usual triangulated categories. Ideally one would like to replace it by a categorical action of the Weyl group or rather of the Braid group on categories of D-modules related to the Flag variety. One would need to define a notion of “invariants” with respect to such action.
1.2. Descent in Equivariant -Theory
Another source of inspiration for the present paper, which is in a way closer to our work, is a recent article of Harada et al. [3]. Given a compact space with an action of a compact reductive Lie group , the authors express the -equivariant K-theory of via the -equivariant one. Here denotes a fixed maximal torus in . Harada et al. show that the natural action of the Weyl group on extends to an action of a degenerate Hecke ring generated by divided difference operators which was introduced earlier in the context of Schubert calculus by Demazure. The operators are called Demazure operators.
The main result in the paper [3] states that the ring is isomorphic to the subring of annihilated by the augmentation ideal in the degenerate Hecke algebra. In other words, a -equivariant class is -equivariant if and only if it is killed by the Demazure operators.
In the present paper, we define a notion of Demazure descent on a triangulated category . Thus Demazure operators are replaced by Demazure functors. These functors satisfy a categorified version of degenerate Hecke algebra relations and form a Demazure descent data on . We define the descent category for such data. Demazure descent is supposed to be a technique replacing the naive notion of Weyl group invariants, on the categorical level.
We provide the first example of Demazure descent. Consider a reductive algebraic group and fix a Borel subgroup . Categorifying the construction form [3], we consider Demazure functors acting on the derived category of -modules. We prove that the functors form a Demazure descent data and identify the descent category with the derived category of -modules.
2. The Setting
2.1. Root Data
Let be a reductive algebraic group over an algebraically closed field of characteristic zero. Let be a Cartan subgroup of and let be the corresponding root data, where is the set of vertices of the Dynkin diagram, is the weight lattice of , and is the coroot lattice of . Choose a Borel subgroup . Denote the set of roots for by . Let be the set of simple roots. The Weyl group of the fixed maximal torus acts naturally on the lattices and and on the -vector spaces spanned by them, by reflections in root hyperplanes. The simple reflection corresponding to an is denoted by . The elements form a set of generators for . For denote the length of a minimal expression of via the generators by . We have a partial ordering on called the Bruhat ordering. , if there exists a reduced expression for that can be obtained from a reduced expression for by deleting a number of simple reflections.
The monoid with generators and relations is called the braid monoid of .
2.2. Categories of Representations
For an algebraic group , we denote the Hopf algebra of polynomial functions on by . Let be the category of -comodules. This is an Abelian tensor category.
Let be the parabolic subgroup of containing whose Levi subgroup has the root system . Using the natural Hopf algebra maps and we can get restriction functors The restriction functors are exact and naturally commute with taking tensor product of representations. Let be a subgroup of and . Define the -invariant part of to be . Consider the induction functors Set and . Notice that and are left exact, since the induction functors are left exact.
2.3. The Derived Categories
For an algebraic group , the regular comodule is injective in ; moreover, for any the coaction map provides an embedding of into an injective object. In particular, has enough injectives. The algebraic De Rham complex provides an injective resolution for the trivial comodule, of the length equal to the dimension of . For any the complex provides an injective resolution for of the same length.
Consider now the bounded derived categories , , and . Let and be the derived functors of and , respectively. Denote the right derived functors of and by and , respectively. Let and be the right derived functors of and , respectively.
Proposition 1. (a) The functors and are left adjoint to and , respectively.
(b) For and (resp., for and ) we have the tensor identities:
(c) The functors and take the trivial -comodule to the trivial -comodule (resp., to the trivial -comodule).
(d) and are comonads for which the comonad maps and are isomorphisms.
Proof. The statements corresponding to (a) and (b) for Res and Ind (resp., and ) are Propositions 3.4 and 3.6 in [4]. The derived functors of a pair of adjoint functors are adjoint. (b) also follows from these statement for the non-derived functors since tensoring over a field is exact. By (a) and are comonads (see [5, Section VI.1]). (b) and (c) imply that for and for . Thus, (resp., ) and from this we get the desired isomorphism and likewise for .
Remark 2. It follows that the restriction functors and are fully faithful.
3. Demazure Descent
Fix a root data of the finite type, with the Weyl group and the braid monoid . Consider a triangulated category .
Definition 3. A weak braid monoid action on the category is a collection of triangulated functors satisfying braid monoid relations; that is, for all there exist isomorphisms of functors
Notice that we neither fix the braid relations isomorphisms nor impose any additional relations on them.
Definition 4. Demazure descent data on the category is a weak braid monoid action such that for each simple root the corresponding functor is a comonad for which the comonad map is an isomorphism.
Here is the central construction of the paper. Consider a triangulated category with a fixed Demazure descent data of the type .
Definition 5. The descent category is the full subcategory in consisting of objects such that for all the cones of the counit maps are isomorphic to .
Remark 6. Suppose that has functorial cones. Then a full triangulated subcategory in being the intersection of kernels of . However, one can prove this statement not using functoriality of cones.
Lemma 7. An object is naturally a comodule over each .
Proof. By definition the comonad maps
make the following diagram commutative:
(10)
For Demazure descent data we require that is an isomorphism, so is also an isomorphism. Let . That is isomorphic to 0 is equivalent to saying that is an isomorphism. This gives the commutative diagram.
Consider
(11)
Thus, satisfies the axiom for the coaction.
Remark 8. Recall that in the usual descent setting either in Algebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functors and their composition which is a comonad. By definition, the descent category for such data is the category of comodules over this comonad. Our definition of for Demazure descent data formally is not about comodules, yet the previous Lemma demonstrates that every object of is naturally equipped with structures of a comodule over each and any morphism in is a morphism of -comodules.
4. Main Theorem
We now go back to considering and .
Proposition 9. Let and let be a reduced expression. Then is independent of the choice of reduced expression and the 's form Demazure descent data on .
Lemma 10. Let be a reduced expression. Then where the union is over all which is in the Bruhat order.
Proof. The proof goes by induction on . It is true for by definition of . Set . Using the hypotheses we get
Let be any element in and a simple reflection. Then by [6, Corollary 28.3] we have . Thus, if , then is contained in the first union. If , then we have by [6, Lemma 29.3A and section 29.1]. Thus, the product can be written as
Claim. The conditions and are equivalent to the conditions and .
Proof of the Claim. Assume that . By [7, Proposition 5.9] this implies that or . In both cases we get since . Assume now that and . has a reduced expression of the form
where the indicates that the term has been removed from the product. If , then
If , then . Since by assumption we get .
This completes the proof of the claim.
If in the first union satisfies that , then it is also contained in the second union. Using the claim we get
Assume that and . Then has a reduced expression of the form
If , then . If , then , but since we get . Hence, the conditions and can be replaced by and . Thus,
This finishes the induction step.
Proof of the Proposition. Let and let be two reduced expressions for . By Lemma 10 this implies that . By [8, Theorem 3.1] the -module structure of is determined upto a natural isomorphism by the set . Hence Hence, for any choice of reduced expression we can define Let and be elements in such that . Pick reduced expressions and for and , respectively. Then is a reduced expression for and we get braid relations for the Define The braid relations for now follows from the braid relations for :
Theorem 11. is equivalent to .
Proof. Let . Being able to extend to an element in is equivalent to being in the image of . Assume for some . Then . If , then , so is in the image of . Thus, being in the image of is equivalent to being an isomorphism which is again equivalent to , where . Set .
Claim. .
Proof of Claim. Assume that . Then for some . But then for all , so is an isomorphism for all . Hence, . Assume that . Then all are isomorphisms. Choose a reduced expression for the longest element in the Weyl group. Then . By [8] we have .
Hence,
By definition of a comonad we have the following commutative diagram:
(27)
Since is an isomorphism so is and thus . This shows that .
This completes the proof of the claim.
From the claim we get that
which is exactly the descent category.
5. Further Directions
5.1. Quantum Groups
Fix a root data of the finite type. Let be the Lusztig quantum group over the ring of quantum integers . Denote the quantum Borel subalgebra by . For a simple root the corresponding quantum parabolic subalgebra is denoted by .
Following [9] we consider the categories of locally finite weight modules over (resp., over , resp., over ) denoted by (resp., by , resp., by ). We consider the corresponding derived categories , , and .
Like in the reductive algebraic group case, the restriction functors are fully faithful and possess right adjoint functors denoted by (resp., by ). Denote the comonad by . Andersen, Polo, and Wen proved that the functors define a weak braid monoid action on the category . One can easily prove that the functors form Demazure descent data. The corresponding descent category is equivalent to Rep.
5.2. Equivariant Sheaves
Let be an affine scheme equipped with an action of a reductive algebraic group . Fix a Borel subgroup . Like in the main body of the present paper, consider the minimal parabolic subgroups in denoted by . Denote the derived categories of quasicoherent sheaves on equivariant with respect to (resp., , resp., ) by (resp., by , resp., by ). We have the natural functors provided by restriction of equivariance and . These functors have the right adjoint ones , resp. . The comonads given by the compositions of extension and restriction of equivariance define a Demazure descent data on the category . The corresponding descent category is equivalent to .
5.3. Algebraic Loop Group
For a simple algebraic group consider the algebraic loop group (resp., the formal arcs group ). Here (resp., ) denotes the formal disc (resp., the formal punctured disc). Consider the affine Kac-Moody central extension The affine analog of the Borel subgroup is the Iwahori subgroup . Let be the standard minimal parahoric subgroups in . One considers the adjoint pairs of coinduction-restriction functors between and . Denote the comonads by for . We claim that form affine Demazure descent data on . We conjecture that the descent category is equivalent to (direct sum of the categories over all positive integral levels).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to H. H. Andersen, C. Dodd, V. Ginzburg, M. Harada, and R. Rouquier for many stimulating discussions. The project started in the summer of 2012 when the first named author visited IHES. Sergey Arkhipov is grateful to IHES for perfect working conditions. Both authors’ research was supported in part by center of excellence grants “Centre for Quantum Geometry of Moduli Spaces” and by FNU grant “Algebraic Groups and Applications.”