On the Langlands retraction

0.12 MB·English

by  Vladimir Drinfeld

ON THE LANGLANDS RETRACTION
V.DRINFELD
3
1
Abstract. GivenarootsysteminavectorspaceV,Langlandsdefinedin1973acanonical
0
retraction L : V ։ V+, where V+ ⊂ V is the dominant chamber. In this note we give
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a short review of the material on this retraction (which is well known under the name of

n “Langlands’geometriclemmas”).
a Themainpurposeofthisreviewistoprovideaconvenientreferenceforthework[DrGa],
J inwhichtheLanglands retractionisusedtodefineacoarseningoftheHarder-Narasimhan-
9 ShatzstratificationofthestackofG-bundlesonasmoothprojectivecurve.

]
T
R
1. Introduction
.
h
Given a root system in a Euclidean space V, Langlands defined in [La2, Sect. 4] a certain
t
a retractionL:V →V+,whereV+ isthedominantchamber. Laterthisretractionwasdiscussed
m
in [BoWa, Ch. IV, Subsect. 3.3] and [C, Sect. 1].
[ InthisnotewebrieflyrecallthedefinitionandpropertiesofL. Ithasnonewresultscompared

with [La2] and [C]; my goal is only to provide a convenient reference for the work [DrGa] and
1
v possibly for some future works.
0 FollowingJ. Carmona,we begin in Sect. 2 with the most naivedefinition ofL (which makes
4 sense for a Euclidean space equipped with any basis {α }): namely, L(x) is the point of V+
9 i
1 closest to x.
1. Starting with Section 3, we assume that hαi,αji ≤ 0 for i 6= j. The key point is that
under this assumption L can be characterized in terms of the usual ordering on V: namely,
0
3 Corollary 3.2 says that L(x) is the least element of the set
1 (1.1) {y ∈V+|y ≥x}.
:
v It is this characterization of L that is important for most applications (in particular, it is
i
X used in [DrGa, Appendix B]). One can consider it as a definition of L and Corollary 3.2 as a
r way to prove the existence of the least element of the set (1.1). In Section 4 we give another
a
proofofthis fact, whichisindependent ofSections2-3;closelyrelatedtoitareRemark4.2and
Example 4.3.
InSection5wedefinetheLanglandsretractionasamapfromthespaceofrationalcoweights
of a reductive group to the dominant cone.
In Section 6 we make some historical remarks.
I thank R. Bezrukavnikov and R. Kottwitz for drawing my attention to Langlands’ articles
[La1,La2]. IthankS.SchiederandA.Zelevinskyforvaluablecomments. Theauthor’sresearch
was partially supported by NSF grant DMS-1001660.
2. The retraction defined by the metric
Let V be a finite-dimensional vector space over R with a positive definite scalar product
h , i. Let {αi}i∈Γ be an arbitrary basis in V and {ωi}i∈Γ the dual basis. Let V+ ⊂ V denote
the closed convex cone generated by the ω ’s, i∈Γ.
i
1
2 V.DRINFELD
FollowingJ.Carmona[C,Sect.1],wedefinetheLanglands retraction L:V →V+ asfollows:
L(x) is the point of V+ closest to x (such point exists and is unique because V+ is closed and
convex). It is easy to see that the map L is continuous.
Let us give another description of L. For a subset J ⊂ Γ let K denote the closed convex
J
cone generatedby ω for j ∈Γ−J and by −α for i∈J. Clearly, each K is a simplicial cone
j i J
of full dimension in V. Let V denote the linear span of α , j ∈ J (so V⊥ is spanned by ω ,
J j J i
i6∈J). Let pr :V →V denote the orthogonalprojection onto V⊥, so ker(pr )=V .
J J J J
Proposition 2.1. (a) The map L is piecewise linear. The cones K are exactly the linearity
J
domains of L. For x∈K one has L(x)=pr (x).
J J
(b) The cones K and their faces form a complete simplicial fan1 in V, combinatorially equiv-
J
alent to the coordinate fan2.
Remark 2.2. The wording in the above proposition was suggested to us by A. Zelevinsky.
The proposition immediately follows from the next lemma, whose proof is straightforward.
Lemma 2.3. Let x ∈ V and y ∈ V+. Set J := {j ∈ Γ|hα ,yi = 0}. Then the following are
j
equivalent:
(a) y =L(x).
(b) x−y belongs to the the closed convex cone generated by −α for j ∈J. (cid:3)
j
3. The key statements
Let Vpos denote the cone dual to V+, i.e., the closed convex cone generated by the α ’s,
i
i ∈ Γ. Equip V and V+ with the following partial ordering: x ≤ y if y −x ∈ Vpos. By
Lemma 2.3, the retraction L:V →V+ from Section 2 has the following property:
(3.1) L(x)≥x, x∈V.
Theorem 3.1. Assume that
(3.2) hα ,α i≤0 for i6=j.
i j
Then the retraction L:V →V+ is order-preserving.
By (3.1),Theorem3.1implies thefollowingstatement,whichcharacterizesLintermsofthe
order relation.
Corollary 3.2. If (3.2) holds then L(x) is the least element in {y ∈V+|y ≥x}. (cid:3)
LetusproveTheorem3.1. Toshowthatapiecewiselinearmapisorder-preservingitsuffices
to check that this is true on eachof its linearity domains. So Theorem 3.1 follows from Propo-
sition 2.1(a) and the next proposition, which I learned from S. Schieder [Sch, Prop.3.1.2(a)].
Proposition 3.3. Assume (3.2). Then for each subset J ⊂Γ the map pr :V →V defined in
J
Section 2 is order-preserving.
To prove the proposition, we need the following lemma.
Lemma 3.4. Let J ⊂Γ. Suppose that x∈V and hx,α i≥0 for all j ∈J. Then x≥0.
J j
Proof of the lemma. We can assume that J = Γ (otherwise replace V by V and Γ by Γ ).
J J
Then the lemma just says that V+ ⊂Vpos. This is a well known consequence of (3.2). (cid:3)
1Thismeansthattheseconescover V andeachintersectionKJ∩KJ′ isafaceinbothKJ andKJ′.
2Thecoordinatefaniswhatonegets whenthebasis{αi}isorthogonal.
ON THE LANGLANDS RETRACTION 3
Proof of Proposition 3.3. We have to show that pr (α ) ≥ 0 for any i ∈ Γ. If i ∈ J then
J i
pr (α ) = 0. Now suppose that i 6∈ J. By the definition of pr , we have pr (α ) = α +x,
J i J J i i
where x is the element of V such that hx,α i = −hα ,α i for all j ∈ J. By (3.2) and
J j i j
Lemma 3.4, x≥0, so pr (α )=α +x≥0. (cid:3)
J i i
4. Another approach to the Langlands retraction
Supposethat(3.2)holds. ThenonecouldtakeCorollary3.2asthedefinition oftheLanglands
retraction L : V → V+, i.e., one could define L(x) to be the least element of the set {y ∈
V+|y ≥ x}. This set is closed and non-empty (because (3.2) implies that V+ ⊂Vpos), so the
existence of the least element in it follows from the next proposition.
Proposition 4.1. Suppose that hα ,α i ≤ 0 for i 6= j. Then the infinum of any non-empty
i j
subset of V+ belongs to V+.
Here “infinum” is understood in terms of the partial ordering defined by Vpos. In other
words, given a family of vectors
(4.1) xt ∈V, xt =Xxi,t·αi,
i
its infinum equals y ·α , where y := infx . Note that if x ∈ V+ then x ∈ Vpos, so
P i i i i,t t t
i t
x ≥0 and infx exists.
i,t i,t
t
Proof of Proposition 4.1. Suppose that we have a family of vectors x ∈ V+ and y = infx .
t t
t
The assumption x ∈ V+ means that hx ,α i ≥ 0 for all i. We have to show that hy,α i ≥ 0
t t i i
for all i.
Fix i. Write x =x′ +x′′, y =y′+y′′, where
t t t
x′t,y′ ∈Rαi, x′t′,y′′ ∈MRαj.
j6=i
Clearly y′ =infx′, y′′ =infx′′. Then for every t one has
t t
t t
hx′,α i=hx ,α i−hx′′,α i≥−hx′′,α i≥−hy′′,α i
t i t i t i t i i
(the second inequality holds because −hα ,α i≥0 for j 6=i). So
j i
hy′,α i=infhx′,α i≥−hy′′,α i,
i t i i
t
i.e., hy,α i≥0. (cid:3)
i
Remark 4.2. InthesituationofthefollowingexampleProposition4.1justsaysthattheinfinum
of any family of concave functions is concave. In fact, the above proof of Proposition 4.1 is
identical to the proof of this classical statement.
Example 4.3. Considerthe rootsystemof SL(n). In this case V is the orthogonalcomplement
of the vector ε1+...+εn in the Euclidean space with orthonormal basis ε1,...εn, and αi =
εi − εi+1, 1 ≤ i ≤ n − 1. Let ωi ∈ V be the basis dual to αi. For each v ∈ V define
f :{0,...,n}→R by
v
f (0)=f (n)=0, f (i)=hv,ω i for 0<i<n.
v v v i
Then the map v 7→ f identifies V with the space of functions f : {0,...,n} → R such that
v
f(0)=f(n)=0. Moreover,Vpos identifieswiththesubsetofnon-negativefunctionsf andV+
with the subset of concave functions f. Thus the Langlands retraction assigns to a function
f :{0,...,n}→R the smallest concave function which is ≥f.
4 V.DRINFELD
5. Reductive groups
5.1. A remark on rationaility. SupposethatinthesituationofSect.2onehashα ,α i∈Q
i j
for all i,j ∈Γ. Then the Q-linear span of the α′s equals the Q-linear span of the ω′s. Denote
i i
it by VQ. Then V = VQ⊗R. The cones K , the subspaces V , and the operators pr from
J J J
Section 2 are clearly defined over Q. So by Proposition 2.1, one has
(5.1) L(VQ)⊂VQ.
5.2. The Langalnds retraction for coweights. Now let G be a connected reductive group
overanalgebraicallyclosedfield. LetΛ beitscoweightlattice,i.e.,Λ =Hom(G ,T),where
G G m
T is the maximal torus of G. Set ΛQ :=Λ ⊗Q. We have the simple coroots αˇ ∈Λ and the
G G i G
simple roots α ∈ Hom(Λ ,Z). Let Λ+,Q ⊂ ΛQ denote the dominant cone. Equip Λ+,Q with
i G G G G
the following partial ordering: λ1 ≤λ2 if λ2−λ1 is a linear combination of the simple coroots
G
with non-negative coefficients.
Now define the Langlands retraction L :ΛQ →Λ+,Q as follows: L (λ) is the least element
G G G G
of the set
(5.2) {µ∈Λ+,Q |µ≥λ}
G
G
with respect to the ≤ ordering.
G
Corollary 5.3. (i) L (λ) exists.
G
(ii) L (λ) is the element of Λ+,Q closest to λ with respect to any positive scalar product on
G G
Λ+,Q⊗R which is invariant with respect to the Weyl group.
G
(iii) L (λ)istheuniqueelementoftheset (5.2)withthefollowing property: hL (λ),α i=0
G G i
for any simple root α such that the coefficient of αˇ in L (λ)−λ is nonzero.
i i G
Proof. Combine Lemma 2.3, Corollary 3.2, and the inclusion (5.1). (cid:3)
5.4. Example: G = GL(n). In this case, just as in Example 4.3, one identifies ΛQ with the
G
space of functions f : {0,...,n} → Q such that f(0) = 0 (while f(n) is arbitrary). Then the
subsetΛQ ⊂Λ+,Qidentifieswiththesubsetofconcave functionsf :{0,...,n}→Qwithf(0)=
G G
0. Just as in Example 4.3, the Langlands retraction assigns to a function f : {0,...,n} → Q
the smallest concave function which is ≥f.
6. Some historical remarks
In [La2] R. Langlands defined the retraction L and formulated his “geometric lemmas” (see
[La2,Lemmas4.4-4.5andCorollary4.6])forthe purposeofthe classificationofrepresentations
ofrealreductivegroupsin terms oftempered ones. However,muchearlierhe hadformulateda
closelyrelated(andmorecomplicated)combinatoriallemma3in his theoryofEisensteinseries,
see [La1, Sect. 8]. In this work Langlands considers Eisenstein series on quotients of the form
G(R)/Γ, where G is a reductive group over Q and Γ is an arithmetic subgroup, but the same
technique applies to quotients of the form G(A)/G(Q). Note that the stack Bun considered
G
in [DrGa] is not far away from G(A)/G(Q), so the fact that the Langlands retraction is used
in [DrGa, Appendix B] is not surprising.
3
Anelementaryintroductiontothislemmacanbefoundin[Cas1,Cas2].
ON THE LANGLANDS RETRACTION 5
References
[BoWa] A.BorelandN.Wallach,Continuous cohomology, discretesubgroups, andrepresentations ofreductive
groups, Mathematical SurveysandMonographs67,AMS,(2000).
[C] J. Carmona, Sur la classification des modules admissibles irreductibles, in: Noncommutative harmonic
analysis and Lie groups (Marseille, 1982), 11–34, Lecture Notes in Math., 1020, Springer-Verlag, Berlin,
1983.
[Cas1] W.Casselman,Truncation exercises,in: Functional analysisVIII,84–104, VariousPubl.Ser.(Aarhus),
47,AarhusUniv.,Aarhus,2004;alsoavailableathttp://www.math.ubc.ca/∼cass/research.html
[Cas2] W. Casselman, Chipping away at convex sets (slides of a 2004 talk in Sydney), available at
http://www.math.ubc.ca/∼cass/sydney/
[DrGa] V. Drinfeld and D. Gaitsgory, Compact generation of the category of D-modules on the stack of G-
bundles on a curve,arXiv:1112.2402.
[La1] R. P. Langlands, Eisenstein series, in: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos.
PureMath.,Boulder,Colo.,1965), 235–252, Amer.Math.Soc.,Providence,R.I.,1966.
[La2] R.Langlands,Ontheclassificationofirreduciblerepresentationsofrealalgebraicgroups,in: Representa-
tiontheoryandharmonicanalysisonsemi-simpleLiegroups,editedbyP.SallyandD.Vogan,AMS1989,
101–170.
[Sch] S.Schieder,Onthe Harder-Narasimhan stratification of BunG viaDrinfeld’s compactifications,
arXiv:1212.6814.
Vladimir Drinfeld,Departmentof Mathematics,University of Chicago.
E-mail address: [email protected]