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Research ArticleDemazure Descent and Representations of Reductive Groups
Sergey Arkhipov1 and Tina Kanstrup2
1 Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark2 Centre for Quantum Geometry of Moduli Spaces Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark
Correspondence should be addressed to Tina Kanstrup tinaqgmaudk
Received 1 November 2013 Accepted 21 January 2014 Published 25 May 2014
Academic Editor Sorin Dascalescu
Copyright copy 2014 S Arkhipov and T KanstrupThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We introduce the notion of Demazure descent data on a triangulated categoryC and define the descent category for such data Weillustrate the definition by our basic example Let 119866 be a reductive algebraic group with a Borel subgroup 119861 Demazure functorsform Demazure descent data on 119863
119887(Rep (119861)) and the descent category is equivalent to 119863
119887(Rep (119866))
1 Motivation
Thepresent paper is the first one in a series devoted to variouscases of categorical descent Philosophically our interest inthe subject grew out of attempts to understand the mainconstruction from the recent paper by Ben-Zvi andNadler [1]in plain terms that would not involve higher category theory
11 Beilinson-Bernstein Localization and Derived DescentLet 119866 be a reductive algebraic group with the Lie algebrag Denote the Flag variety of 119866 by Fl A major part ofGeometric Representation Theory originated in the seminalwork of Beılinson and Bernstein [2] devoted to investigationof the globalization functor D-mod(Fl) rarr 119880(g)-mod Thisfunctor turns out to be fully faithful and provides geometricand topological tools to investigate a wide class of 119880(g)-modules in particular the ones from the famous category OVarious generalizations of this result lead to the investigationof the categories of twistedD-modules on the Flag variety andon the base affine space for 119866 and of their derived categories
Ben-Zvi and Nadler define a certain comonad acting ona higher categorical version for the derived category of D-modules on the base affine space In fact the functor is builtinto the higher categorical treatment of Beilinson-Bernsteinlocalization-globalization construction
Using the heavy machinery of Barr-Beck-Lurie descentthe authors argue that the derived category of 119880(g)-modulesis equivalent to the category of D-modules equivariant with
respect to this comonad Thus the global sections functorbecomes equivariant with respect to the actionThe comonadis called theHecke comonad It provides a categorification forthe classical action of theWeyl group on various homologicaland K-theoretic invariants of the Flag variety
Notice that the descent construction fails to work on thelevel of the usual triangulated categories Ideally one wouldlike to replace it by a categorical action of the Weyl group orrather of the Braid group on categories of D-modules relatedto the Flag variety One would need to define a notion ofldquoinvariantsrdquo with respect to such action
12 Descent in Equivariant K-Theory Another source ofinspiration for the present paper which is in a way closerto our work is a recent article of Harada et al [3] Given acompact space 119883 with an action of a compact reductive Liegroup 119866 the authors express the 119866-equivariant K-theory of119883 via the 119879-equivariant one Here 119879 denotes a fixed maximaltorus in 119866 Harada et al show that the natural action of theWeyl group119882 on119870119879(119883) extends to an action of a degenerateHecke ring generated by divided difference operators whichwas introduced earlier in the context of Schubert calculus byDemazure The operators are called Demazure operators
Themain result in the paper [3] states that the ring119870119866(119883)
is isomorphic to the subring of 119870119879(119883) annihilated by theaugmentation ideal in the degenerate Hecke algebra In otherwords a 119879-equivariant class is 119866-equivariant if and only if itis killed by the Demazure operators
Hindawi Publishing CorporationAlgebraVolume 2014 Article ID 823467 6 pageshttpdxdoiorg1011552014823467
2 Algebra
In the present paper we define a notion of Demazuredescent on a triangulated category C Thus Demazure oper-ators are replaced by Demazure functors These functorssatisfy a categorified version of degenerate Hecke algebrarelations and form a Demazure descent data onC We definethe descent category for such data Demazure descent issupposed to be a technique replacing the naive notion ofWeylgroup invariants on the categorical level
We provide the first example of Demazure descentConsider a reductive algebraic group 119866 and fix a Borelsubgroup 119861 sub 119866 Categorifying the construction form [3]we consider Demazure functors 119863119904119894
acting on the derivedcategory of 119861-modules We prove that the functors form aDemazure descent data and identify the descent categorywith the derived category of 119866-modules
2 The Setting
21 Root Data Let 119866 be a reductive algebraic group over analgebraically closed field 119896 of characteristic zero Let 119879 be aCartan subgroup of 119866 and let (119868 119883 119884) be the correspondingroot data where 119868 is the set of vertices of theDynkin diagram119883 is the weight lattice of 119866 and 119884 is the coroot lattice of119866 Choose a Borel subgroup 119879 sub 119861 sub 119866 Denote the setof roots for 119866 by Φ = Φ
+⊔ Φminus Let 1205721 120572119899 be the set
of simple roots The Weyl group 119882 = Norm(119879)119879 of thefixed maximal torus acts naturally on the lattices 119883 and 119884
and on the R-vector spaces spanned by them by reflectionsin root hyperplanes The simple reflection corresponding toan 120572119894 is denoted by 119904119894 The elements 1199041 119904119899 form a set ofgenerators for 119882 For 119908 isin 119882 denote the length of a minimalexpression of 119908 via the generators by ℓ(119908) We have a partialordering on 119882 called the Bruhat ordering 1199081015840 le 119908 if thereexists a reduced expression for 119908
1015840 that can be obtained froma reduced expression for 119908 by deleting a number of simplereflections
The monoid Br+ with generators 119879119908 119908 isin 119882 andrelations
22 Categories of Representations For an algebraic group 119867we denote the Hopf algebra of polynomial functions on119867 byO(119867) Let Rep(119867) be the category of O(119867)-comodules Thisis an Abelian tensor category
Let 119875119894 be the parabolic subgroup of119866 containing 119861whoseLevi subgroup has the root system 120572119894 minus120572119894 Using the naturalHopf algebramapsO(119866) rarr O(119861) andO(119875119894) rarr O(119861)we canget restriction functors
The restriction functors are exact and naturally commutewith taking tensor product of representations Let 119867 be asubgroup of 119866 and119872 isin Rep(119866) Define the119867-invariant part
of 119872 to be 119872119867
= HomRep(119867)(119896119872) Consider the inductionfunctors
Ind Rep (119861) 997888rarr Rep (119866) 119872 997891997888rarr (O (119866) otimes 119872)119861
(3)
Set Δ 119894 = Res119894 ∘ Ind119894 Rep(119861) and Δ = Res ∘ Ind
Rep(119861) Notice thatΔ 119894 andΔ are left exact since the inductionfunctors are left exact
23 The Derived Categories For an algebraic group 119867 theregular comodule O(119867) is injective in Rep(119867) moreover forany119872 isin Rep(119867) the coactionmap119872 rarr O(119867)otimes119872 providesan embedding of 119872 into an injective object In particularRep(119867) has enough injectives The algebraic De RhamcomplexΩ
∙(119867) provides an injective resolution for the trivial
comodule of the length equal to the dimension of119867 For any119872 isin Rep(119867) the complex Ω
∙(119867) otimes 119872 provides an injective
resolution for 119872 of the same lengthConsider now the bounded derived categories
119863119887(Rep(119861)) 119863
119887(Rep(119875119894)) and 119863
119887(Rep(119866)) Let 119871 119894 and 119871
be the derived functors of Res119894 and Res respectively Denotethe right derived functors of Ind119894 and Ind by 119868119894 and 119868respectively Let119863119894 = 119871 119894 ∘ 119868119894 and119863 = 119871∘119868 be the right derivedfunctors of Δ 119894 and Δ respectively
Proposition 1 (a) The functors 119871 119894 and 119871 are left adjoint to 119868119894
and 119868 respectively(b) For119872 isin 119863
119887(Rep (119861)) and119873 isin 119863
119887(Rep (119875119894)) (resp for
119872 isin 119863119887(Rep (119861)) and 119873 isin 119863
(c) The functors 119868119894 and 119868 take the trivial O(119861)-comoduleto the trivial O(119875119894)-comodule (resp to the trivial O(119866)-comodule)
(d) 119863119894 and 119863 are comonads for which the comonad maps119863119894 rarr 119863
2
119894and 119863 rarr 119863
2 are isomorphisms
Proof The statements corresponding to (a) and (b) for Resand Ind (resp Res119894 and Ind119894) are Propositions 34 and 36in [4] The derived functors of a pair of adjoint functors areadjoint (b) also follows from these statement for the non-derived functors since tensoring over a field is exact
119868119894 (Id otimes 119871 119894) ≃ 119877 (Ind119894 (Id otimes Res119894)) ≃ 119877 (Ind119894 otimes Id) ≃ 119868119894 otimes Id(5)
By (a) 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868 are comonads (see [5Section VI1]) (b) and (c) imply that 119868119894 ∘ 119871 119894(119873) ≃ 119873 for119873 isin 119863
119887(Rep(119875119894)) and 119868 ∘ 119871(119873) ≃ 119873 for 119873 isin 119863
119887(Rep(119866))
Thus Id rarr 119868119894 ∘ 119871 119894 (resp Id rarr 119868 ∘ 119871) and from this we get thedesired isomorphism
Notice that we neither fix the braid relations isomor-phisms nor impose any additional relations on them
Definition 4 Demazure descent data on the category C is aweak braidmonoid action 119863119908 such that for each simple root119904119894 the corresponding functor 119863119904119894
is a comonad for which thecomonad map 119863119904119894
rarr 1198632
119904119894is an isomorphism
Here is the central construction of the paper Consider atriangulated category C with a fixed Demazure descent data119863119908 119908 isin 119882 of the type (119868 119883 119884)
Definition 5 The descent category Desc(C 119863119908 119908 isin 119882) isthe full subcategory in C consisting of objects 119872 such thatfor all 119894 the cones of the counit maps 119863119904119894
(119872)120598
997888rarr 119872 areisomorphic to 0
Remark 6 Suppose that C has functorial cones ThenDesc(C 119863119908 119908 isin 119882) a full triangulated subcategory inC being the intersection of kernels of Cone(119863119904119894 rarr Id)However one can prove this statement not using functorialityof cones
Lemma 7 An object 119872 isin Desc(C 119863119908 119908 isin 119882) is naturallya comodule over each 119863119904119894
Proof By definition the comonad maps
120578 119863119904119894997888rarr 119863
2
119904119894 120598 119863119904119894
997888rarr Id (9)
make the following diagram commutative
Ds119894
D2s119894
120578
120598 ∘ Ds119894
Id ∘ Ds119894
(10)
For Demazure descent data we require that 120578 is an iso-morphism so 120598 ∘ 119863119904119894
is also an isomorphism Let 119872 isin
Desc(C 119863119908 119908 isin 119882) That Cone(119863119904119894(119872)120598
997888rarr 119872) is
isomorphic to 0 is equivalent to saying that 119863119904119894(119872)
120598
997888rarr 119872
is an isomorphism This gives the commutative diagramConsider
120578
120598minus1
120598minus1
Ds119894(M)
Ds119894(M) D
2s119894(M)
(120598 ∘ Ds119894)minus1
M
(11)
Thus 120598minus1 satisfies the axiom for the coaction
Remark 8 Recall that in the usual descent setting either inAlgebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functorsand their composition which is a comonad By definition thedescent category for such data is the category of comodulesover this comonad Our definition of Desc(119862119863119908 119908 isin 119882)
for Demazure descent data formally is not about comodulesyet the previous Lemma demonstrates that every object ofDesc(119862119863119908 119908 isin 119882) is naturally equipped with struc-tures of a comodule over each 119863119894 and any morphism inDesc(119862119863119908 119908 isin 119882) is a morphism of 119863119894-comodules
4 Main Theorem
We now go back to considering 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868
Proposition 9 Let 119908 isin 119882 and let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reducedexpressionThen119863119908 = 1198631198941
∘sdot sdot sdot∘119863119894119899is independent of the choice
of reduced expression and the 119863119908rsquos form Demazure descentdata onC = 119863
119887(Rep (119861))
Lemma 10 Let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reduced expression Then
1198751198941sdot sdot sdot 119875119894119899
= ⋃
1199081015840le119908
1198611199081015840119861 (12)
where the union is over all 1199081015840 isin 119882 which is le 119908 in the Bruhatorder
Proof The proof goes by induction on 119899 = ℓ(119908) It is truefor 119899 = 1 by definition of 119875119894 Set V = 1199041198941
If 119895119896 = 119899 then 11990810158401015840
le V Since 11990810158401015840119904119894119899
le 11990810158401015840 by assumption we
get 11990810158401015840119904119894119899 le VThis completes the proof of the claimIf1199081015840 le V in the first union satisfies that1199081015840119904119894119899 le 119908
1015840 then itis also contained in the second union Using the claim we get
be two reduced expressions for119908 By Lemma 10 thisimplies that 1198751198941 sdot sdot sdot 119875119894119899 = 1198751198951
sdot sdot sdot 119875119895119899 By [8 Theorem 31] the 119861-
module structure ofΔ 1198941 ∘ sdot sdot sdot∘Δ 119894119899 is determined upto a naturalisomorphism by the set 1198751198941 sdot sdot sdot 119875119894119899 Hence
Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899
≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899
(20)
Hence for any choice of reduced expression we can define
Proof Let119872 isin 119863119887(Rep(119861)) Being able to extend119872 to an ele-
ment in119863119887(Rep(119866) is equivalent to119872 being in the image of119871
Assume119872 = 119871(119873) for some119873 isin 119863119887(Rep(119866))Then119863(119872) =
119871 ∘ 119868 ∘ 119871(119873) ≃ 119871(119873) = 119872 If119863(119872) rarr119872 then119872 ≃ 119871(119868(119872))so 119872 is in the image of 119871 Thus being in the image of 119871 isequivalent to 119863(119872) rarr 119872 being an isomorphism which isagain equivalent to119872 isin ker(119862) where 119862 = Cone(119863 rarr Id)Set 119862119894 = Cone(119863119894 rarr Id)
Claim ker(119862) = ⋂119894ker(119862119894)
Proof of Claim Assume that 119872 isin ker(119862) Then 119872 = 119871(119873)
for some 119873 isin 119863119887(Rep(119866)) But then 119872 = 119871 119894(119873|119875119894
) for all 119894so 119863119894(119872) rarr 119872 is an isomorphism for all 119894 Hence 119872 isin
⋂119894ker(119862119894) Assume that 119872 isin ⋂
119894ker(119862119894) Then all 119863119894(119872) rarr
119872 are isomorphisms Choose a reduced expression 1199041198941sdot sdot sdot 119904119894119873
Algebra 5
for the longest element in the Weyl group Then 1198751198941sdot sdot sdot 119875119894119873
=
119866 By [8] we have 119863 = 1198631198941∘ sdot sdot sdot ∘ 119863119894119873
By definition of a comonad we have the following commuta-tive diagram
Id ∘ D
D
D2
120598D
120578
(27)
Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)
This completes the proof of the claimFrom the claim we get that
119863119887(Rep (119866)) = ⋂
119894
ker (119862119894) (28)
which is exactly the descent category
5 Further Directions
51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A
Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863
119887(Rep(UA)) 119863119887(Rep(BA)) and 119863
119887(Rep(P119894A))
Like in the reductive algebraic group case the restrictionfunctors
are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863
119887(Rep(BA)) One can easily
prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)
is equivalent to 119863119887(Rep(UA))
52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863
119887(119876Coh119866(119883)) (resp by
119863119887(119876Coh119861(119883)) resp by 119863
119887(119876Coh119875119894(119883))) We have the
natural functors provided by restriction of equivariance119871 119863
The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871
+119866 Let 1198750 119875119899 be the standard minimal
parahoric subgroups in 119871+119866 One considers the adjoint pairs
of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863
119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894
by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863
119887(Rep(119868119908)) We conjecture that
the descent category is equivalent to 119863119887(Rep(119871119866)) (direct
sum of the categories over all positive integral levels)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo
References
[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188
6 Algebra
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991
In the present paper we define a notion of Demazuredescent on a triangulated category C Thus Demazure oper-ators are replaced by Demazure functors These functorssatisfy a categorified version of degenerate Hecke algebrarelations and form a Demazure descent data onC We definethe descent category for such data Demazure descent issupposed to be a technique replacing the naive notion ofWeylgroup invariants on the categorical level
We provide the first example of Demazure descentConsider a reductive algebraic group 119866 and fix a Borelsubgroup 119861 sub 119866 Categorifying the construction form [3]we consider Demazure functors 119863119904119894
acting on the derivedcategory of 119861-modules We prove that the functors form aDemazure descent data and identify the descent categorywith the derived category of 119866-modules
2 The Setting
21 Root Data Let 119866 be a reductive algebraic group over analgebraically closed field 119896 of characteristic zero Let 119879 be aCartan subgroup of 119866 and let (119868 119883 119884) be the correspondingroot data where 119868 is the set of vertices of theDynkin diagram119883 is the weight lattice of 119866 and 119884 is the coroot lattice of119866 Choose a Borel subgroup 119879 sub 119861 sub 119866 Denote the setof roots for 119866 by Φ = Φ
+⊔ Φminus Let 1205721 120572119899 be the set
of simple roots The Weyl group 119882 = Norm(119879)119879 of thefixed maximal torus acts naturally on the lattices 119883 and 119884
and on the R-vector spaces spanned by them by reflectionsin root hyperplanes The simple reflection corresponding toan 120572119894 is denoted by 119904119894 The elements 1199041 119904119899 form a set ofgenerators for 119882 For 119908 isin 119882 denote the length of a minimalexpression of 119908 via the generators by ℓ(119908) We have a partialordering on 119882 called the Bruhat ordering 1199081015840 le 119908 if thereexists a reduced expression for 119908
1015840 that can be obtained froma reduced expression for 119908 by deleting a number of simplereflections
The monoid Br+ with generators 119879119908 119908 isin 119882 andrelations
22 Categories of Representations For an algebraic group 119867we denote the Hopf algebra of polynomial functions on119867 byO(119867) Let Rep(119867) be the category of O(119867)-comodules Thisis an Abelian tensor category
Let 119875119894 be the parabolic subgroup of119866 containing 119861whoseLevi subgroup has the root system 120572119894 minus120572119894 Using the naturalHopf algebramapsO(119866) rarr O(119861) andO(119875119894) rarr O(119861)we canget restriction functors
The restriction functors are exact and naturally commutewith taking tensor product of representations Let 119867 be asubgroup of 119866 and119872 isin Rep(119866) Define the119867-invariant part
of 119872 to be 119872119867
= HomRep(119867)(119896119872) Consider the inductionfunctors
Ind Rep (119861) 997888rarr Rep (119866) 119872 997891997888rarr (O (119866) otimes 119872)119861
(3)
Set Δ 119894 = Res119894 ∘ Ind119894 Rep(119861) and Δ = Res ∘ Ind
Rep(119861) Notice thatΔ 119894 andΔ are left exact since the inductionfunctors are left exact
23 The Derived Categories For an algebraic group 119867 theregular comodule O(119867) is injective in Rep(119867) moreover forany119872 isin Rep(119867) the coactionmap119872 rarr O(119867)otimes119872 providesan embedding of 119872 into an injective object In particularRep(119867) has enough injectives The algebraic De RhamcomplexΩ
∙(119867) provides an injective resolution for the trivial
comodule of the length equal to the dimension of119867 For any119872 isin Rep(119867) the complex Ω
∙(119867) otimes 119872 provides an injective
resolution for 119872 of the same lengthConsider now the bounded derived categories
119863119887(Rep(119861)) 119863
119887(Rep(119875119894)) and 119863
119887(Rep(119866)) Let 119871 119894 and 119871
be the derived functors of Res119894 and Res respectively Denotethe right derived functors of Ind119894 and Ind by 119868119894 and 119868respectively Let119863119894 = 119871 119894 ∘ 119868119894 and119863 = 119871∘119868 be the right derivedfunctors of Δ 119894 and Δ respectively
Proposition 1 (a) The functors 119871 119894 and 119871 are left adjoint to 119868119894
and 119868 respectively(b) For119872 isin 119863
119887(Rep (119861)) and119873 isin 119863
119887(Rep (119875119894)) (resp for
119872 isin 119863119887(Rep (119861)) and 119873 isin 119863
(c) The functors 119868119894 and 119868 take the trivial O(119861)-comoduleto the trivial O(119875119894)-comodule (resp to the trivial O(119866)-comodule)
(d) 119863119894 and 119863 are comonads for which the comonad maps119863119894 rarr 119863
2
119894and 119863 rarr 119863
2 are isomorphisms
Proof The statements corresponding to (a) and (b) for Resand Ind (resp Res119894 and Ind119894) are Propositions 34 and 36in [4] The derived functors of a pair of adjoint functors areadjoint (b) also follows from these statement for the non-derived functors since tensoring over a field is exact
119868119894 (Id otimes 119871 119894) ≃ 119877 (Ind119894 (Id otimes Res119894)) ≃ 119877 (Ind119894 otimes Id) ≃ 119868119894 otimes Id(5)
By (a) 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868 are comonads (see [5Section VI1]) (b) and (c) imply that 119868119894 ∘ 119871 119894(119873) ≃ 119873 for119873 isin 119863
119887(Rep(119875119894)) and 119868 ∘ 119871(119873) ≃ 119873 for 119873 isin 119863
119887(Rep(119866))
Thus Id rarr 119868119894 ∘ 119871 119894 (resp Id rarr 119868 ∘ 119871) and from this we get thedesired isomorphism
Notice that we neither fix the braid relations isomor-phisms nor impose any additional relations on them
Definition 4 Demazure descent data on the category C is aweak braidmonoid action 119863119908 such that for each simple root119904119894 the corresponding functor 119863119904119894
is a comonad for which thecomonad map 119863119904119894
rarr 1198632
119904119894is an isomorphism
Here is the central construction of the paper Consider atriangulated category C with a fixed Demazure descent data119863119908 119908 isin 119882 of the type (119868 119883 119884)
Definition 5 The descent category Desc(C 119863119908 119908 isin 119882) isthe full subcategory in C consisting of objects 119872 such thatfor all 119894 the cones of the counit maps 119863119904119894
(119872)120598
997888rarr 119872 areisomorphic to 0
Remark 6 Suppose that C has functorial cones ThenDesc(C 119863119908 119908 isin 119882) a full triangulated subcategory inC being the intersection of kernels of Cone(119863119904119894 rarr Id)However one can prove this statement not using functorialityof cones
Lemma 7 An object 119872 isin Desc(C 119863119908 119908 isin 119882) is naturallya comodule over each 119863119904119894
Proof By definition the comonad maps
120578 119863119904119894997888rarr 119863
2
119904119894 120598 119863119904119894
997888rarr Id (9)
make the following diagram commutative
Ds119894
D2s119894
120578
120598 ∘ Ds119894
Id ∘ Ds119894
(10)
For Demazure descent data we require that 120578 is an iso-morphism so 120598 ∘ 119863119904119894
is also an isomorphism Let 119872 isin
Desc(C 119863119908 119908 isin 119882) That Cone(119863119904119894(119872)120598
997888rarr 119872) is
isomorphic to 0 is equivalent to saying that 119863119904119894(119872)
120598
997888rarr 119872
is an isomorphism This gives the commutative diagramConsider
120578
120598minus1
120598minus1
Ds119894(M)
Ds119894(M) D
2s119894(M)
(120598 ∘ Ds119894)minus1
M
(11)
Thus 120598minus1 satisfies the axiom for the coaction
Remark 8 Recall that in the usual descent setting either inAlgebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functorsand their composition which is a comonad By definition thedescent category for such data is the category of comodulesover this comonad Our definition of Desc(119862119863119908 119908 isin 119882)
for Demazure descent data formally is not about comodulesyet the previous Lemma demonstrates that every object ofDesc(119862119863119908 119908 isin 119882) is naturally equipped with struc-tures of a comodule over each 119863119894 and any morphism inDesc(119862119863119908 119908 isin 119882) is a morphism of 119863119894-comodules
4 Main Theorem
We now go back to considering 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868
Proposition 9 Let 119908 isin 119882 and let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reducedexpressionThen119863119908 = 1198631198941
∘sdot sdot sdot∘119863119894119899is independent of the choice
of reduced expression and the 119863119908rsquos form Demazure descentdata onC = 119863
119887(Rep (119861))
Lemma 10 Let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reduced expression Then
1198751198941sdot sdot sdot 119875119894119899
= ⋃
1199081015840le119908
1198611199081015840119861 (12)
where the union is over all 1199081015840 isin 119882 which is le 119908 in the Bruhatorder
Proof The proof goes by induction on 119899 = ℓ(119908) It is truefor 119899 = 1 by definition of 119875119894 Set V = 1199041198941
If 119895119896 = 119899 then 11990810158401015840
le V Since 11990810158401015840119904119894119899
le 11990810158401015840 by assumption we
get 11990810158401015840119904119894119899 le VThis completes the proof of the claimIf1199081015840 le V in the first union satisfies that1199081015840119904119894119899 le 119908
1015840 then itis also contained in the second union Using the claim we get
be two reduced expressions for119908 By Lemma 10 thisimplies that 1198751198941 sdot sdot sdot 119875119894119899 = 1198751198951
sdot sdot sdot 119875119895119899 By [8 Theorem 31] the 119861-
module structure ofΔ 1198941 ∘ sdot sdot sdot∘Δ 119894119899 is determined upto a naturalisomorphism by the set 1198751198941 sdot sdot sdot 119875119894119899 Hence
Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899
≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899
(20)
Hence for any choice of reduced expression we can define
Proof Let119872 isin 119863119887(Rep(119861)) Being able to extend119872 to an ele-
ment in119863119887(Rep(119866) is equivalent to119872 being in the image of119871
Assume119872 = 119871(119873) for some119873 isin 119863119887(Rep(119866))Then119863(119872) =
119871 ∘ 119868 ∘ 119871(119873) ≃ 119871(119873) = 119872 If119863(119872) rarr119872 then119872 ≃ 119871(119868(119872))so 119872 is in the image of 119871 Thus being in the image of 119871 isequivalent to 119863(119872) rarr 119872 being an isomorphism which isagain equivalent to119872 isin ker(119862) where 119862 = Cone(119863 rarr Id)Set 119862119894 = Cone(119863119894 rarr Id)
Claim ker(119862) = ⋂119894ker(119862119894)
Proof of Claim Assume that 119872 isin ker(119862) Then 119872 = 119871(119873)
for some 119873 isin 119863119887(Rep(119866)) But then 119872 = 119871 119894(119873|119875119894
) for all 119894so 119863119894(119872) rarr 119872 is an isomorphism for all 119894 Hence 119872 isin
⋂119894ker(119862119894) Assume that 119872 isin ⋂
119894ker(119862119894) Then all 119863119894(119872) rarr
119872 are isomorphisms Choose a reduced expression 1199041198941sdot sdot sdot 119904119894119873
Algebra 5
for the longest element in the Weyl group Then 1198751198941sdot sdot sdot 119875119894119873
=
119866 By [8] we have 119863 = 1198631198941∘ sdot sdot sdot ∘ 119863119894119873
By definition of a comonad we have the following commuta-tive diagram
Id ∘ D
D
D2
120598D
120578
(27)
Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)
This completes the proof of the claimFrom the claim we get that
119863119887(Rep (119866)) = ⋂
119894
ker (119862119894) (28)
which is exactly the descent category
5 Further Directions
51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A
Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863
119887(Rep(UA)) 119863119887(Rep(BA)) and 119863
119887(Rep(P119894A))
Like in the reductive algebraic group case the restrictionfunctors
are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863
119887(Rep(BA)) One can easily
prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)
is equivalent to 119863119887(Rep(UA))
52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863
119887(119876Coh119866(119883)) (resp by
119863119887(119876Coh119861(119883)) resp by 119863
119887(119876Coh119875119894(119883))) We have the
natural functors provided by restriction of equivariance119871 119863
The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871
+119866 Let 1198750 119875119899 be the standard minimal
parahoric subgroups in 119871+119866 One considers the adjoint pairs
of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863
119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894
by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863
119887(Rep(119868119908)) We conjecture that
the descent category is equivalent to 119863119887(Rep(119871119866)) (direct
sum of the categories over all positive integral levels)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo
References
[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188
6 Algebra
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991
Notice that we neither fix the braid relations isomor-phisms nor impose any additional relations on them
Definition 4 Demazure descent data on the category C is aweak braidmonoid action 119863119908 such that for each simple root119904119894 the corresponding functor 119863119904119894
is a comonad for which thecomonad map 119863119904119894
rarr 1198632
119904119894is an isomorphism
Here is the central construction of the paper Consider atriangulated category C with a fixed Demazure descent data119863119908 119908 isin 119882 of the type (119868 119883 119884)
Definition 5 The descent category Desc(C 119863119908 119908 isin 119882) isthe full subcategory in C consisting of objects 119872 such thatfor all 119894 the cones of the counit maps 119863119904119894
(119872)120598
997888rarr 119872 areisomorphic to 0
Remark 6 Suppose that C has functorial cones ThenDesc(C 119863119908 119908 isin 119882) a full triangulated subcategory inC being the intersection of kernels of Cone(119863119904119894 rarr Id)However one can prove this statement not using functorialityof cones
Lemma 7 An object 119872 isin Desc(C 119863119908 119908 isin 119882) is naturallya comodule over each 119863119904119894
Proof By definition the comonad maps
120578 119863119904119894997888rarr 119863
2
119904119894 120598 119863119904119894
997888rarr Id (9)
make the following diagram commutative
Ds119894
D2s119894
120578
120598 ∘ Ds119894
Id ∘ Ds119894
(10)
For Demazure descent data we require that 120578 is an iso-morphism so 120598 ∘ 119863119904119894
is also an isomorphism Let 119872 isin
Desc(C 119863119908 119908 isin 119882) That Cone(119863119904119894(119872)120598
997888rarr 119872) is
isomorphic to 0 is equivalent to saying that 119863119904119894(119872)
120598
997888rarr 119872
is an isomorphism This gives the commutative diagramConsider
120578
120598minus1
120598minus1
Ds119894(M)
Ds119894(M) D
2s119894(M)
(120598 ∘ Ds119894)minus1
M
(11)
Thus 120598minus1 satisfies the axiom for the coaction
Remark 8 Recall that in the usual descent setting either inAlgebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functorsand their composition which is a comonad By definition thedescent category for such data is the category of comodulesover this comonad Our definition of Desc(119862119863119908 119908 isin 119882)
for Demazure descent data formally is not about comodulesyet the previous Lemma demonstrates that every object ofDesc(119862119863119908 119908 isin 119882) is naturally equipped with struc-tures of a comodule over each 119863119894 and any morphism inDesc(119862119863119908 119908 isin 119882) is a morphism of 119863119894-comodules
4 Main Theorem
We now go back to considering 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868
Proposition 9 Let 119908 isin 119882 and let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reducedexpressionThen119863119908 = 1198631198941
∘sdot sdot sdot∘119863119894119899is independent of the choice
of reduced expression and the 119863119908rsquos form Demazure descentdata onC = 119863
119887(Rep (119861))
Lemma 10 Let 119908 = 1199041198941sdot sdot sdot 119904119894119899
be a reduced expression Then
1198751198941sdot sdot sdot 119875119894119899
= ⋃
1199081015840le119908
1198611199081015840119861 (12)
where the union is over all 1199081015840 isin 119882 which is le 119908 in the Bruhatorder
Proof The proof goes by induction on 119899 = ℓ(119908) It is truefor 119899 = 1 by definition of 119875119894 Set V = 1199041198941
If 119895119896 = 119899 then 11990810158401015840
le V Since 11990810158401015840119904119894119899
le 11990810158401015840 by assumption we
get 11990810158401015840119904119894119899 le VThis completes the proof of the claimIf1199081015840 le V in the first union satisfies that1199081015840119904119894119899 le 119908
1015840 then itis also contained in the second union Using the claim we get
be two reduced expressions for119908 By Lemma 10 thisimplies that 1198751198941 sdot sdot sdot 119875119894119899 = 1198751198951
sdot sdot sdot 119875119895119899 By [8 Theorem 31] the 119861-
module structure ofΔ 1198941 ∘ sdot sdot sdot∘Δ 119894119899 is determined upto a naturalisomorphism by the set 1198751198941 sdot sdot sdot 119875119894119899 Hence
Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899
≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899
(20)
Hence for any choice of reduced expression we can define
Proof Let119872 isin 119863119887(Rep(119861)) Being able to extend119872 to an ele-
ment in119863119887(Rep(119866) is equivalent to119872 being in the image of119871
Assume119872 = 119871(119873) for some119873 isin 119863119887(Rep(119866))Then119863(119872) =
119871 ∘ 119868 ∘ 119871(119873) ≃ 119871(119873) = 119872 If119863(119872) rarr119872 then119872 ≃ 119871(119868(119872))so 119872 is in the image of 119871 Thus being in the image of 119871 isequivalent to 119863(119872) rarr 119872 being an isomorphism which isagain equivalent to119872 isin ker(119862) where 119862 = Cone(119863 rarr Id)Set 119862119894 = Cone(119863119894 rarr Id)
Claim ker(119862) = ⋂119894ker(119862119894)
Proof of Claim Assume that 119872 isin ker(119862) Then 119872 = 119871(119873)
for some 119873 isin 119863119887(Rep(119866)) But then 119872 = 119871 119894(119873|119875119894
) for all 119894so 119863119894(119872) rarr 119872 is an isomorphism for all 119894 Hence 119872 isin
⋂119894ker(119862119894) Assume that 119872 isin ⋂
119894ker(119862119894) Then all 119863119894(119872) rarr
119872 are isomorphisms Choose a reduced expression 1199041198941sdot sdot sdot 119904119894119873
Algebra 5
for the longest element in the Weyl group Then 1198751198941sdot sdot sdot 119875119894119873
=
119866 By [8] we have 119863 = 1198631198941∘ sdot sdot sdot ∘ 119863119894119873
By definition of a comonad we have the following commuta-tive diagram
Id ∘ D
D
D2
120598D
120578
(27)
Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)
This completes the proof of the claimFrom the claim we get that
119863119887(Rep (119866)) = ⋂
119894
ker (119862119894) (28)
which is exactly the descent category
5 Further Directions
51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A
Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863
119887(Rep(UA)) 119863119887(Rep(BA)) and 119863
119887(Rep(P119894A))
Like in the reductive algebraic group case the restrictionfunctors
are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863
119887(Rep(BA)) One can easily
prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)
is equivalent to 119863119887(Rep(UA))
52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863
119887(119876Coh119866(119883)) (resp by
119863119887(119876Coh119861(119883)) resp by 119863
119887(119876Coh119875119894(119883))) We have the
natural functors provided by restriction of equivariance119871 119863
The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871
+119866 Let 1198750 119875119899 be the standard minimal
parahoric subgroups in 119871+119866 One considers the adjoint pairs
of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863
119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894
by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863
119887(Rep(119868119908)) We conjecture that
the descent category is equivalent to 119863119887(Rep(119871119866)) (direct
sum of the categories over all positive integral levels)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo
References
[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188
6 Algebra
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991
If 119895119896 = 119899 then 11990810158401015840
le V Since 11990810158401015840119904119894119899
le 11990810158401015840 by assumption we
get 11990810158401015840119904119894119899 le VThis completes the proof of the claimIf1199081015840 le V in the first union satisfies that1199081015840119904119894119899 le 119908
1015840 then itis also contained in the second union Using the claim we get
be two reduced expressions for119908 By Lemma 10 thisimplies that 1198751198941 sdot sdot sdot 119875119894119899 = 1198751198951
sdot sdot sdot 119875119895119899 By [8 Theorem 31] the 119861-
module structure ofΔ 1198941 ∘ sdot sdot sdot∘Δ 119894119899 is determined upto a naturalisomorphism by the set 1198751198941 sdot sdot sdot 119875119894119899 Hence
Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899
≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899
(20)
Hence for any choice of reduced expression we can define
Proof Let119872 isin 119863119887(Rep(119861)) Being able to extend119872 to an ele-
ment in119863119887(Rep(119866) is equivalent to119872 being in the image of119871
Assume119872 = 119871(119873) for some119873 isin 119863119887(Rep(119866))Then119863(119872) =
119871 ∘ 119868 ∘ 119871(119873) ≃ 119871(119873) = 119872 If119863(119872) rarr119872 then119872 ≃ 119871(119868(119872))so 119872 is in the image of 119871 Thus being in the image of 119871 isequivalent to 119863(119872) rarr 119872 being an isomorphism which isagain equivalent to119872 isin ker(119862) where 119862 = Cone(119863 rarr Id)Set 119862119894 = Cone(119863119894 rarr Id)
Claim ker(119862) = ⋂119894ker(119862119894)
Proof of Claim Assume that 119872 isin ker(119862) Then 119872 = 119871(119873)
for some 119873 isin 119863119887(Rep(119866)) But then 119872 = 119871 119894(119873|119875119894
) for all 119894so 119863119894(119872) rarr 119872 is an isomorphism for all 119894 Hence 119872 isin
⋂119894ker(119862119894) Assume that 119872 isin ⋂
119894ker(119862119894) Then all 119863119894(119872) rarr
119872 are isomorphisms Choose a reduced expression 1199041198941sdot sdot sdot 119904119894119873
Algebra 5
for the longest element in the Weyl group Then 1198751198941sdot sdot sdot 119875119894119873
=
119866 By [8] we have 119863 = 1198631198941∘ sdot sdot sdot ∘ 119863119894119873
By definition of a comonad we have the following commuta-tive diagram
Id ∘ D
D
D2
120598D
120578
(27)
Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)
This completes the proof of the claimFrom the claim we get that
119863119887(Rep (119866)) = ⋂
119894
ker (119862119894) (28)
which is exactly the descent category
5 Further Directions
51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A
Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863
119887(Rep(UA)) 119863119887(Rep(BA)) and 119863
119887(Rep(P119894A))
Like in the reductive algebraic group case the restrictionfunctors
are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863
119887(Rep(BA)) One can easily
prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)
is equivalent to 119863119887(Rep(UA))
52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863
119887(119876Coh119866(119883)) (resp by
119863119887(119876Coh119861(119883)) resp by 119863
119887(119876Coh119875119894(119883))) We have the
natural functors provided by restriction of equivariance119871 119863
The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871
+119866 Let 1198750 119875119899 be the standard minimal
parahoric subgroups in 119871+119866 One considers the adjoint pairs
of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863
119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894
by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863
119887(Rep(119868119908)) We conjecture that
the descent category is equivalent to 119863119887(Rep(119871119866)) (direct
sum of the categories over all positive integral levels)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo
References
[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188
6 Algebra
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991
By definition of a comonad we have the following commuta-tive diagram
Id ∘ D
D
D2
120598D
120578
(27)
Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)
This completes the proof of the claimFrom the claim we get that
119863119887(Rep (119866)) = ⋂
119894
ker (119862119894) (28)
which is exactly the descent category
5 Further Directions
51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A
Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863
119887(Rep(UA)) 119863119887(Rep(BA)) and 119863
119887(Rep(P119894A))
Like in the reductive algebraic group case the restrictionfunctors
are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863
119887(Rep(BA)) One can easily
prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)
is equivalent to 119863119887(Rep(UA))
52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863
119887(119876Coh119866(119883)) (resp by
119863119887(119876Coh119861(119883)) resp by 119863
119887(119876Coh119875119894(119883))) We have the
natural functors provided by restriction of equivariance119871 119863
The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871
+119866 Let 1198750 119875119899 be the standard minimal
parahoric subgroups in 119871+119866 One considers the adjoint pairs
of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863
119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894
by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863
119887(Rep(119868119908)) We conjecture that
the descent category is equivalent to 119863119887(Rep(119871119866)) (direct
sum of the categories over all positive integral levels)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo
References
[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188
6 Algebra
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991
[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981
[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010
[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987
[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998
[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975
[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997
[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978
[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991