Demazure descent and representations of reductive groups

We introduce the notion of Demazure descent data on a triangulated category C and define the descent category for such data. We illustrate the definition by our basic example. Let G be a reductive algebraic group with a Borel subgroup B. Demazure functors form Demazure descent data on the derived category of Rep(B) and the descent category is equivalent to the derived category of Rep(G).


Motivation
The present paper is the first one in 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 [BN] in plain terms that would not involve higher category theory.
1.1. Beilinson-Bernstein localization and derived descent. Let G be a reductive algebraic group with the Lie algebra g. Denote the Flag variety of G by Fl. A major part of Geometric Representation Theory originated in the seminal work of Beilinson and Bernstein [BB] devoted to investigation of the globalization functor Dmod(Fl) → U (g)mod. This functor turns out to be fully faithful and provides geometric and topological tools to investigate a wide class of U (g)-modules, in particular the ones from the famous category O. Various generalizations of this result lead to investigation of the categories of twisted D-modules on the Flag variety and on the base affine space for G, 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 treating of Beilinson-Bernstein localization-globalization construction.
Using the heavy machinery of Barr-Beck-Lurie descent, the authors argue that the derived category of U (g)-modules is equivalent to the category of D-modules equivariant with respect to this comonad. Thus the global sections functor becomes equivariantization 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 K-theory. Another source of inspiration for the present paper, which is in a way closer to our work, is a recent article of Harada, Landweber and Sjamaar [HLS]. Given a compact space X with an action of a compact reductive Lie group G, the authors express the G-equivariant K-theory of X via the T -equivariant one. Here T denotes a fixed maximal torus in G. Harada et al. show that the natural action of the Weyl group W on K T (X) 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 [HLS] states that the ring K G (X) is isomorphic to the subring of K T (X) annihilated by the augmentation ideal in the degenerate Hecke algebra. In other words, a T -equivariant class is G-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 C. 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 C. 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 G, fix a Borel subgroup B ⊂ G. Categorifying the construction form [HLS], we consider Demazure functors D s i acting on the derived category of B-modules. We prove that the functors form a Demazure descent data and identify the descent category with the derived category of G-modules.
1.3. Acknowledgements. 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. S.A. 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".
2.1. Root data. Let G be a reductive algebraic group over an algebraically closed field k of characteristic zero. Let T be a Cartan subgroup of G and let (I, X, Y ) be the corresponding root data, where I is the set of vertices of the Dynkin diagram, X is the weight lattice of G and Y is the coroot lattice of G. Choose a Borel subgroup T ⊂ B ⊂ G. Denote the set of roots for G by Φ = Φ + ⊔ Φ − . Let {α 1 , . . . , α n } be the set of simple roots. The Weyl group W = Norm(T )/T of the fixed maximal torus acts naturally on the lattices X and Y and on the R-vector spaces spanned by them, by reflections in root hyperplanes. The simple reflection corresponding to an α i is denoted by s i . The elements s 1 , . . . , s n form a set of generators for W . For w ∈ W denote the length of a minimal expression of w via the generators by ℓ(w). We have a partial ordering on W called the Bruhat ordering. w ′ ≤ w if there exists a reduced expression for w ′ that can be obtained from a reduced expression for w by deleting a number of simple reflections.
The monoid Br + with generators {T w , w ∈ W } and relations is called the braid monoid of G.
we have the tensor identities:  Proof. The statements corresponding to (a) and (b) for Res and Ind (resp. Res i and Ind i ) are proposition 3.4 and 3.6 in [Jant]. The derived functors of a pair of adjoint functors are adjoint. (b) also follows from thes statement for the non-derived functors since tensoring over a field is exact.
and from this we get the desired isomorphism i . and likewise for D.
Remark 2.2. It follows that the restriction functors L i and L are fully faithful.
Fix a root data (I, X, Y ) of the finite type, with the Weyl group W and the braid monoid Br + . Consider a triangulated category C.
Definition 3.1. A weak braid monoid action on the category C is a collection of triangulated functors D w : C → C, w ∈ W satisfying braid monoid relations, i.e. for all w 1 , w 2 ∈ W there exist isomorphisms of functors Notice that we neither fix the braid relations isomorphisms nor impose any additional relations on them.
Definition 3.2. Demazure descent data on the category C is a weak braid monoid action {D w } such that for each simple root s i the corresponding functor D s i is a comonad for which the comonad map D s i → D 2 s i is an isomorphism. Here is the central construction of the paper. Consider a triangulated category C with a fixed Demazure descent data {D w , w ∈ W } of the type (I, X, Y ). Remark 3.4. Suppose that C has functorial cones. Then Desc(C, D w , w ∈ W ) a full triangulated subcategory in C being the intersection of kernels of Cone(D s i → Id). However, one can prove this statement not using functoriality of cones.
Lemma 3.5. An object M ∈ Desc(C, D w , w ∈ W ) is naturally a comodule over each D s i .

s s s s s s s s s s s s s s s s s s s
Thus, ǫ −1 satisfies the axiom for the coaction.
Remark 3.6. 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 Desc(C, D w , w ∈ W ) for Demazure descent data formally is not about comodules, yet the previous Lemma demonstrates that every object of Desc(C, D w , w ∈ W ) is naturally equipped with structures of a comodule over each D i and any morphism in Desc(C, D w , w ∈ W ) is a morphism of D i -comodules.

Main Theorem
We now go back to considering D i = L i • I i and D = L • I.
Proposition 4.1. Let w ∈ W and let w = s i 1 · · · s in be a reduced expression. Then D w := D i 1 •· · ·•D in is independent of the choice of reduced expression and the D w 's form Demazure descent data on C = D b (Rep(B)).
Lemma 4.2. Let w = s i 1 · · · s in be a reduced expression. Then where the union is over all w ′ ∈ W which is ≤ w in the Bruhat order.
Proof. The proof goes by induction on n = ℓ(w). It is true for n = 1 by definition of P i . Set v = s i 1 · · · s i n−1 . Using the hypotheses we get Let w ′ be any element in W and s a simple reflection. Then by [Hum,Cor. 28 [Hum,Lemma 29.3A and section 29.1]. Thus, the product can be written as Proof of the claim. Assume that w ′′ s in ≤ v. By [Hum2,Prop. 5.9] this implies that w ′′ ≤ v or w ′′ ≤ vs in = w. In both cases we get w ′′ ≤ w since v ≤ w. Assume now that w ′′ ≤ w and w ′′ s in ≤ w ′′ . w ′′ has a reduced expression of the form where theˆindicates that the term has been removed from the product. If j k = n then w ′′ s in = s i 1 · · ·ŝ i j 1 · · ·ŝ i j 2 · · ·ŝ i j k · · · s i n−1 ≤ s i 1 · · · s i n−1 = v.
If j k = n then w ′′ ≤ v. Since w ′′ s in ≤ w ′′ by assumption we get w ′′ s in ≤ v.
If w ′ ≤ v in the first union satisfies that w ′ s in ≤ w ′ then it is also contained in the second union. Using the claim we get Assume that w ′ ≤ w and w ′ ≤ w ′ s in . Then w ′ has a reduced expression of the form w ′ = s i 1 · · ·ŝ i j 1 · · ·ŝ i j 2 · · ·ŝ i j k · · · s in .
Hence, the conditions w ′ ≤ v and w ′ ≤ w ′ s in can be replaced by w ′ ≤ w and w ′ ≤ w ′ s in . Thus, This finishes the induction step.
Proof of the proposition. Let w ∈ W and let s i 1 · · · s in = s j i · · · s jn be two reduced expressions for w. By lemma 4.2 this implies that P i 1 · · · P in = P j 1 · · · P jn . By [CPS,Thm. 3.1] the B-module structure of ∆ i 1 • · · · • ∆ in is determined up to a natural isomorphism by the set P i 1 · · · P in . Hence Hence, for any choice of reduced expression we can define Let w 1 and w 2 be elements in W such that ℓ(w 1 w 2 ) = ℓ(w 1 ) + ℓ(w 2 ). Pick reduced expressions s i 1 · · · s ir and s j 1 · · · s jt for w 1 and w 2 respectively. Then s i 1 · · · s ir s j 1 · · · s jt is a reduced expression for w 1 w 2 and we get braid relations for the ∆ w The braid relations for D w now follows from the braid relations for ∆ w  Choose a reduced expression s i 1 · · · s i N for the longest element in the Weyl group. Then P i 1 · · · P i N = G. By [CPS] is also an isomorphism. Hence, M ∈ ker(C).
From the claim we get that which is exactly the descent category.

Further directions
5.1. Quantum groups. Fix a root data (I, X, Y ) of the finite type. Let U A be the Lusztig quantum group over the ring of quantum integers A = Z[v, v −1 ]. Denote the quantum Borel subalgebra by B A . For a simple root α i the corresponding quantum parabolic sub algebra is denoted by P i,A . Following [APK] we consider the categories of locally finite weight modules over U A (resp. over B A , resp. over P i,A ) denoted by Rep(U A ) (resp. by Rep(B A ), resp. by Rep(P i,A )). We consider the corresponding derived categories Like in the reductive algebraic group case, the restriction functors are fully faithful and possess right adjoint functors denoted by I (resp. by I i ). Denote the comonad L i • I i by D i . Andersen, Polo and Wen proved that the functors D i define a weak braid monoid action on the category D b (Rep(B A )). One can easily prove that the functors form Demazure descent data. The corresponding descent category

Equivariant sheaves.
Let X be an affine scheme equipped with an action of a reductive algebraic group G. Fix a Borel subgroup B ⊂ G. Like in the main body of the present paper, consider the minimal parabolic subgroups in G denoted by P 1 , . . . P n . Denote the derived categories of quasicoherent sheaves on X equivariant with respect to G (resp., B, resp., P i ) by D b (QCoh G (X)) (resp., by D b (QCoh B (X)), resp. by D b (QCoh P i (X))). We have the natural functors provided by restriction of equivariance L : D b (QCoh G (X)) → D b (QCoh B (X)) and L i : D b (QCoh P i (X)) → D b (QCoh B (X)). These functors have the right adjoint ones I, resp. I 1 , . . . I n . The comonads D 1 , . . . D n given by the compositions of extension and restriction of equivariance define a Demazure descent data on the category D b (QCoh B (X)). The corresponding descent category is equivalent to D b (QCoh G (X)).

Algebraic loop group.
For a simple algebraic group G consider the algebraic loop group LG = Map( • D, G) (resp. the formal arcs group L + G = Map(D, G)). Here D (resp. • D) denotes the formal disc (resp. the formal punctured disc). Consider the affine Kac-Moody central extension 1 → G m → LG → LG → 1. The affine analog of the Borel subgroup B ⊂ G is the Iwahori subgroup Iw ⊂ L + G. Let P 0 , . . . , P n be the standard minimal parahoric subgroups in L + G. One considers the adjoint pairs of coinduction-restriction functors I 0 , L 0 , . . . , I n , L n between D b (Rep(Iw)) and D b (Rep(P i )). Denote the comonads L i •I i by D i for i = 0, . . . , n. We claim that D 0 , . . . , D n form affine Demazure descent data on D b (Rep(Iw)). We conjecture that the descent category is equivalent to D b (Rep( LG)) (direct sum of the categories over all positive integral levels).