Double MV Cycles, Affine PBW
Bases, and Crystal Combinatorics
by
Dinakar Muthiah
B. S., Stanford University, 2007
M. S., Stanford University, 2008
A Dissertation submitted in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy
in the Department of Mathematics at Brown University
Providence, Rhode Island
May 2013
c Copyright 2013 by Dinakar Muthiah
This dissertation by Dinakar Muthiah is accepted in its present form
by the Department of Mathematics as satisfying the
dissertation requirement for the degree of Doctor of Philosophy.
Date
Alexander Braverman, Director
Recommended to the Graduate Council
Date
Dan Abramovich, Reader
Date
Bruno Harris, Reader
Approved by the Graduate Council
Date
Peter Weber, Dean of the Graduate School
iii
Vitæ
Dinakar Muthiah was born in New York City in 1985 but soon after moved to
South Florida, where he spent all of his childhood and adolescence. He survived
Hurricane Andrew, and he fell in love with the automobile. He begrudgingly went to
Boyd H. Anderson high school, where he was forced to make friends with the most
open and wonderful people in the world. He completed degrees in Mathematics and
Electrical Engineering at Stanford University in 2007 and 2008, after being subjected
to further openness and wonderfulness. The final straw was graduate school in Mathematics at Brown, which began in the Fall of 2008. Given the chance to do things
differently, he would change everything.
iv
Dedicated to my grandfather S. R. Sivanandy
v
Acknowledgments
For chapter 2, I thank Michael Finkelberg, Joel Kamnitzer, and Peter Tingley for
many useful conversations leading up to the completion of this work. I would like
to particularly thank Joel Kamnitzer for pointing me to the work of Baumann and
Gaussent, which proved to be the key tool in completing this work. I thank Peter
Tingley for letting me modify his pictures of Maya diagrams and charged partitions
from his expository notes on Fock space [Ting].
Much of the work was completed during the semester program “Langlands Duality
in Representation Theory and Gauge Theory” at the Institute for Advanced Studies
at the Hebrew University of Jerusalem in the Fall of 2010. I would like to thank the
organizers of the program and the staff of IAS for their hospitality.
For chapter 3, I thank my collaborator Peter Tingley, and we thank Pierre Baumann, Jonathan Beck, Alexander Braverman, Joel Kamnitzer and Hiraku Nakajima
for many useful discussions.
For both chapters, I thank the referees for useful comments and suggestions that
have improved the works in this thesis.
On a more general note, I would like to especially like to thank my advisor,
Alexander Braverman, for his guidance and for countless conversations throughout
the course my graduate career. Without his steady patience and encouragement none
of this would be possible. He has a tremendous aesthetic that I will always strive to
emulate. I will certainly never be done learning from him, and I hope many more
students insist on his supervision.
I thank Joel Kamnitzer for his generosity in sharing his ideas and his consistent
support of my career. I eagerly look forward to working with him during the next
three years in Toronto.
vi
I thank Ben Wieland for his guidance during the formative years of my career. So
much of who I am as a mathematician is due to conversations over beers with Ben.
I thank my mother and father, Mallika and Muthiah, for their unconditional and
unending support. I thank my sister, Anju, for always being the source of rationality
in our family, for always reminding me how awesome it is that I get to learn for a
living, and for always accompanying me on cross-country road trips.
I thank my aunt’s family, Mala, Jeyakumar, and Ajay, for their support during
these past five years. It was wonderful having family living so close, and I’m sad to
move away. I especially thank Ajay for being such a precocious light of optimism and
empathy. You are a good kid, and I know you will do so much good in this world.
I thank the city of Providence for hosting me during these past five years. The
praise I have for Ajay applies verbatim to you as well.
I thank all the friends who have been a constant source of comfort and support
during the years I have spend at Brown. You are numerous, and you know who you
are. I hesitate to name names because I fear omission.
Finally, I thank my grandfather S. R. Sivanandy, the first math teacher in my
family. I dedicate this thesis to you.
vii
Contents
Vitæ
iv
Dedication
v
Acknowledgments
vi
List of Figures
xi
Publication References
1
Chapter 1. Introduction
2
Double MV Cycles and the Naito-Sagaki-Saito Crystal
3
Affine PBW Bases and MV Polytopes in Rank 2
4
Chapter 2. Double MV Cycles and the Naito-Sagaki-Saito Crystal
6
1. Overview
6
1.1. MV Cycles in the Finite Dimensional Case
6
1.2. MV Cycles in the Affine Case
8
1.3. The Naito-Sagaki-Saito Crystal
9
1.4. Main Results
10
1.5. Heuristic Motivation
10
1.6. Remarks on Open Problems
11
1.7. Organization of the Chapter
12
2. Preliminary Notions
13
2.1. Terminology
13
2.2. Kac-Moody Lie Algebras
13
2.3. The Dual Lie Algebra
14
2.4. Kac-Moody Groups
14
viii
2.5. Lusztig’s Canonical Basis
14
2.6. Definition of Crystals
15
2.7. The Affine Grassmannian
16
2.8. MV Cycles and Quasimap Spaces
16
3. The Braverman-Finkelberg-Gaitsgory Crystal Structure
18
3.1. Recalling the BFG Construction
20
3.2.
23
4. Maya Diagrams and Fock Space
27
4.1. Valuations
30
5. The Naito-Sagaki-Saito Crystal
31
6. Geometric Realization of NSS Data
34
Chapter 3. Affine PBW Bases and MV Polytopes in rank 2
40
1. Overview
40
2. Background
43
2.1. Quantum Affine Algebras and Canonical Basis
43
2.2. Crystals
44
3. Rank 2 Affine MV polytopes
46
3.1. Rank 2 affine root systems
46
3.2. Lusztig data and pseudo-Weyl polytopes
47
3.3. Definition and characterization of MV polytopes
48
4. Rank-2 affine MV polytopes from PBW bases
53
4.1. Definition and basic properties of rank-2 affine PBW bases
53
4.2. Relationship of PBW bases with the canonical basis
56
4.3. Relationship with MV Polytopes
59
b 2 MV polytopes
5. Comparing combinatorial and geometric sl
61
b 2 quiver variety
5.1. The sl
61
5.2. Reflection functors and Harder-Narasimhan filtrations
63
5.3. MV polytopes from quiver varieties
67
5.4. Characterization of symmetric affine MV polytopes
68
ix
Bibliography
70
x
List of Figures
1
b 2 MV polytope. The partitions labeling the vertical edges are
An sl
indicated by including extra vertices on the vertical edges, such
that the edge is cut into the pieces indicated by the partition. Here:
crα1 = 2, crα1 +δ = 1, crα1 +2δ = 1, crδ = (9, 2, 1, 1), crα0 +2δ = 1, crα0 = 1,
c`α0 = 1, c`α0 +δ = 2, c`α0 +2δ = 1, c`α0 +3δ = 1, c`δ = (2, 1, 1), c`α1 +3δ = 1, c`α1 +δ = 1, c`α1 = 5.
49
2
The representations from Definition 5.4. In each case, the number of 0 is j.
Here the vertices represent basis elements, the dotted arrows represent matrix
elements of 1 for tα , and solid arrows represent matrix elements of 1 for tβ ,
and all other matrix elements are 0.
64
xi
Abstract of “Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics”
by Dinakar Muthiah, Ph.D., Brown University, May 2013
The theory of Mirković-Vilonen (MV) cycles and polytopes associated to a complex reductive group G has proven to be a rich source of structures related to representation theory. MV polytopes have proven to be a useful tool in understanding and
unifying many constructions of crystals for finite-type Kac-Moody algebras. These
polytopes arise naturally in many places, including the affine Grassmannian, preprojective algebras, PBW bases, and KLR algebras. This thesis is part of an ongoing
program to extend this theory to the affine Kac-Moody algebras
In the first part of this thesis, we investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group. We prove an explicit
formula for the Braverman-Finkelberg-Gaitsgory [BFG] crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent
[BauGau]. As an application, we give a geometric construction of the Naito-Sagakic n on Fermionic Fock space. In particular, this
Saito [NSS] crystal via the action of SL
construction gives rise to an isomorphism of crystals between the set of double MV
cycles and the Naito-Sagaki-Saito crystal. As a result, we can independently prove
that the Naito-Sagaki-Saito crystal is the B(∞) crystal. In particular, our geometric
b 2.
proof works in the previously unknown case of sl
In the second part of this thesis, which is joint work with Peter Tingley, we investigate the recently proposed notions of affine MV polytopes. A definition of MV
polytopes in symmetric affine cases has been proposed using pre-projective algebras.
In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in
rank-2, we show that this can also be used to define MV polytopes. The main result
of this paper is that these three notions of MV polytope all agree in the relevant
rank-2 cases. As a corollary, we can give a complete combinatorial characterization
of the affine MV polytopes arising from pre-projective algebras. Our main tool is a
new characterization of rank-2 affine MV polytopes.
xiii
Publication References
This thesis comprises two related works in the field of affine Lie algebras. The first
result is the subject of chapter 2, which will appear in published form in Advances in
Mathematics as the paper “Double MV Cycles and the Naito-Sagaki-Saito Crystal”.
The second work, which is joint with Peter Tingley, comprises chapter 3 and will
appear in published form in Selecta Mathematica, N.S. as the paper “Affine PBW
bases and MV polytopes in affine rank 2”.
1
CHAPTER 1
Introduction
The philosophical starting point of this thesis is the Mirković-Vilonen proof of the
geometric Satake correspondence [MV]. This is an equivalence between the category
of spherical perverse sheaves on the affine Grassmannian for a reductive group G
and the category of algebraic representations of the Langlands dual group G∨ . In
the course of their proof, Mirković and Vilonen construct explicit cycles, the MV
(Mirković-Vilonen) cycles, that give rise to a basis in each irreducible representation of G∨ . A natural problem is to understand MV cycles and the structures
associated with them. A major step toward understanding these cycles comes from
the work of Anderson and Kamnitzer on MV polytopes.
The affine Grassmannian is canonically projective, and therefore the underlying
real space carries a natural symplectic form. A question, first considered by Anderson
[And], is to study the image of MV cycles under the moment map for the torus action.
These moment map images are called MV polytopes. Kamnitzer [Kam1] explicitly
computes the MV polytopes and showed that they satisfy exactly the same equations
that Lusztig discovered when comparing different PBW (Poincaré-Birkhoff-Witt)
bases [Lus3]. As a consequence, one can provide a purely algebraic definition of MV
polytopes using PBW bases. Moreover, as PBW bases are in natural bijection with
the canonical basis, one gets an explicit bijection between MV cycles and the canonical
basis.
Braverman-Finkelberg-Gaitsgory [BFG] also construct a bijection between MV
cycles and the canonical basis. They geometrically construct a crystal structure on the
set of MV cycles and prove that this crystal is B(∞), which is in unique bijection with
the canonical basis. In [Kam2], Kamnitzer proves that the bijection coming from
MV polytopes and the bijection coming from the crystal structure agree. Finally,
2
Baumann-Gaussent [BauGau] give explicit formulas for the crystal structure on MV
cycles and reprove Kamnitzer’s results using their new formulas.
MV polytopes also arise in work by Baumann-Kamnitzer [BK] on quiver varieties.
They extract MV polytopes from irreducible components of a certain moduli space of
preprojective algebra representations. In this way, they construct an explicit bijection
between the semicanonical basis and MV polytopes.
This thesis is part of a currently ongoing program to investigate how much of
the above picture extends to the affine case, i.e. the case where we replace finitedimensional reductive groups with infinite-dimensional affine Kac-Moody groups.
Double MV Cycles and the Naito-Sagaki-Saito Crystal
In the first part of this thesis, chapter 2, we investigate double MV cycles, which
are the analogue of MV cycles for the double affine Grassmannian (the affine Grassmannian of an affine Kac-Moody group). The double affine Grassmannian is severely
infinite-dimensional, but work of Braverman-Finkelberg-Gaitsgory [BFG] provides a
definition of double MV cycles as certain finite-dimensional schemes. Moreover, as
in the reductive case, they define a crystal structure on the set of double MV cycles
that gives rise to the crystal B(∞).
An open problem is to construct some analogue of MV polytope theory for double
MV cycles. Specifically, one would like to provide a geometric bijection between
double MV cycles and a combinatorial model for the crystal B(∞).
In type A, we provide such a bijection between double MV cycles and elements
of the Naito-Sagaki-Saito crystal. The Naito-Sagaki-Saito crystal [NSS] arises by a
careful analysis of the combinatorics of MV polytopes in finite-type A. By letting the
rank go to infinity, they construct combinatorial data that behave like MV polytopes
]
for GL
∞ , the universal central extension of GL∞ . After a certain folding operation,
c n and prove that for n ≥ 3 this crystal is B(∞).
they construct a crystal for SL
c n → GL
]
Geometrically, this folding corresponds to looking at the map SL
∞ and
3
studying the induced map on affine Grassmannians. Unfortunately, with current
geometric technology this is only a heuristic picture.
What is precise, however, is to understand this folding procedure through reprec
]
sentation theory, i.e. to study the restrictions of GL
∞ -representations to SLn . One
c n on the Fermionic Fock space .
particularly interesting example is the action of SL
We show that looking at generic valuations of double MV cycles on the basis vectors
in Fermionic Fock space provides a bijection between the double MV cycles and the
Naito-Saito-Sagaki crystal. In fact, this provides an independent geometric proof that
c 2 , this was the only proof known at
the Naito-Sagaki-Saito crystal is B(∞); for SL
the time.
The main technical tool needed for this result is a generalization of the BaumannGaussent formula [BauGau] for the crystal operators. We prove this formula for all
untwisted affine types, and we expect this formula should prove to be useful in further
investigations.
Affine PBW Bases and MV Polytopes in Rank 2
The second part of this thesis, chapter 3, is joint work with Peter Tingley. We
consider PBW bases and the preprojective quiver construction in affine type. Generalizing PBW bases to the affine cases is not at all straightforward. The main
difficulty is constructing imaginary root vectors in quantum affine algebras. In the
case of symmetric affine algebras, Beck-Chari-Pressley [BCP] solve this problem and
define PBW bases. Work of Akasaka [Aka] and Beck-Nakajima [BN] extend these
b 2 and A(2) , these definitions
constructions to all affine types. In the rank-2 cases, sl
2
can be used to produced a candidate definition of affine MV polytopes
Generalizing the quiver variety construction of Baumann-Kamnitzer [BK], BaumannKamnitzer-Tingley [BKT] produce a second definition of affine MV polytopes, one
for every symmetric affine algebra. As in the PBW basis construction, the new phenomenon is a construction of imaginary root vectors. Moreover, their construction
4
has the property that understanding the quiver definition of affine MV polytopes
b 2 MV polytopes.
reduces to understanding sl
Finally, Baumann-Dunlap-Kamnitzer-Tingley [BDKT] give a third definition of
affine MV polytopes in the two rank-2 cases. Their definition is purely combinatorial.
We prove that all three definitions of affine MV polytope coincide in rank 2. In
particular, because the Baumann-Dunlap-Kamnitzer-Tingley is an explicit combinatorial formula, we now know explicit equations for these polytopes. In the case of
PBW bases, this generalizes Lusztig’s calculations that compare the various PBW
bases [Lus3]. In particular, this answers a problem posed by Beck-Nakajima [BN,
Remark 3.29] for affine rank 2.
Furthermore, because the quiver-variety construction of affine MV polytopes is
combinatorially determined by the combinatorics in rank-2, our result provides explicit combinatorial formulas for all affine MV polytopes coming from quiver varieties.
Our main tool is a new theorem that uniquely characterizes rank-2 MV polytopes
by a short list of conditions: a condition related to understanding how crystal operators act, a condition related to Saito reflections, and a condition related to imaginary
root vectors. Moreover, our theorem can be interpreted in finite-type cases to give
a new characterization of finite-type MV polytopes. An application of this theorem
appears in Tingley-Websters’s work on affine MV polytopes and KLR algebras [TW].
5
CHAPTER 2
Double MV Cycles and the Naito-Sagaki-Saito Crystal
1. Overview
1.1. MV Cycles in the Finite Dimensional Case. Let G be a complex reductive group. The geometric Satake equivalence of Lusztig [Lus1], Beilinson-Drinfeld
[BD], Ginzburg [Ginz], and Mirkovic-Vilonen [MV] relates the geometry of the affine
Grassmannian with the representation theory of the dual group G∨ . The most recent proof, due to Mirković-Vilonen, provided even finer information; they gave an
explicit basis for each irreducible representation of G∨ indexed by certain irreducible
subvarieties of the affine Grassmannian of G. These irreducible subvarieties are called
Mirković-Vilonen (MV) cycles, and they are highly structured.
Remark 1.1. MV cycles come in two flavors: there are those that correspond to
basis vectors in irreducible representations, and there are those that correspond to
basis vectors in Verma modules. In this paper we will focus almost exclusively on MV
cycles corresponding to basis vectors in Verma modules. Unless we specify otherwise,
we will mean this latter variety when we write “MV cycles”.
Let us highlight some key results in the theory of MV cycles, which will be relevant
to our later discussion:
• Braverman, Finkelberg, and Gaitsgory [BravGait, BFG] proved that the
MV cycles corresponding to basis vectors in irreducible representations carry
a natural crystal structure for the dual Lie algebra. More specifically, corresponding to each irreducible representation V (λ) of G∨ with highest weight
λ, they endow the set of MV cycles corresponding to V (λ) with the structure
of the crystal B(λ).
6
As we discussed earlier, there is a natural way to extract a B(∞) crystal
given a suitable family of crystals {B(λ)}. The geometric counterpart is
exactly the process of passing from MV cycles corresponding to irreducible
representations to MV cycles corresponding to Verma modules.
• Kamnitzer [Kam1], following initial work of Anderson [And], studied MV
cycles via their moment polytopes, which he calls MV polytopes. In particular, he gave an explicit combinatorial description of all MV polytopes.
He discovered that MV polytopes are precisely the polytopes considered
by Berenstein and Zelevinsky in their study of the canonical basis (hence
the anachronistic name of the previous section). As a result, he obtained a
natural bijection between MV cycles and the canonical basis using MV polytopes as an intermediary. Consequentially, the set of MV polytopes acquire
two crystal structures: one arises from the Braverman-Finkelberg-Gaitsgory
crystal structure on MV cycles, and the other comes from the crystal structure on Lusztig’s canonical basis. In [Kam2], Kamnitzer proved that these
two crystal structures in fact agree.
• Baumann and Gaussent in [BauGau] gave explicit formulas for the crystal
structure on MV cycles. They gave a recipe to construct a MV cycle from
its crystal-theoretic string parameterization. They were able to reproduce
many of Kamnitzer’s results using their formula, and they compared the
crystal structure on MV cycles with that coming from the theory of LS
galleries.
• Hong in [Hong] investigated how MV cycles and MV polytopes behave
when you pass from G to Gσ , the fixed point group of a Dynkin diagram
automorphism. He noted that the Dynkin diagram automorphism also acts
on MV cycles and MV polytopes, He proves that MV cycles and MV polytopes for Gσ are canonically identified with those of G that are fixed under
the diagram automorphism.
7
1.2. MV Cycles in the Affine Case. Let us consider a general Kac-Moody
group G. In this generality, there is currently no way to speak of the affine Grassmannian as a geometric object, and there is no geometric Satake equivalence. However,
because the representation theory of a general Kac-Moody group is formally very
similar to that of a finite-dimensional reductive group, it is natural to expect that
some of the geometric theory should generalize.
Indeed, we can define a substitute for MV cycles using spaces of maps from the
projective line into flag schemes (we will call them map spaces from now onward).
When G is finite-dimensional reductive group, we have an isomorphism between MV
cycles and irreducible components of certain map spaces [FM]. So for many purposes,
it suffices to study the map spaces. The main benefit of the map spaces is that they
can be defined for any symmetrizable Kac-Moody group.
Thus the starting point for our discussion is replacing MV cycles with their map
space analogues. Unfortunately, unless G is finite dimensional or of untwisted affine
type, there is little that we can say about the map space analogues of MV cycles.
In this untwisted affine case, let us call the map space analogues of MV cycles
double MV cycles. Braverman-Finkelberg-Gaitsgory [BFG] proved that double MV
cycles are finite-dimensional schemes. They proved that all double MV cycles of a
given weight have the same dimension, and they compute that dimension to be exactly
the number predicted from the study of ordinary MV cycles. With this computation,
they were able to define a crystal structure on the set of double MV cycles in a way
completely analogous to their construction when G is finite dimensional. Moreover,
they are able to prove that the resulting crystal is the B(∞) crystal for the dual Lie
algebra.
We should mention at this point that Nakajima has a construction of MV cycles
in the untwisted affine type A case [Nak] via quiver varieties. His construction is the
analogue of MV cycles corresponding to basis vectors in irreducible representations (in
contrast to the Braverman-Finkelberg-Gaitsgory construction, which is the analogue
of MV cycles corresponding to basis vectors in Verma modules).
8
1.3. The Naito-Sagaki-Saito Crystal. It is natural to ask what are the analogues of MV polytopes in the general Kac-Moody case. In the affine case, there is
still the notion of Lusztig’s canonical basis. However, the notion of a PBW parameterization is much more complicated because of the appearance of imaginary roots
[BCP, BN], and the corresponding combinatorics is still unsolved.
b n , Naito, Sagaki, and Saito [NSS] develop an approach to MV
In the case of sl
polytopes as follows. Consider the Dynkin diagram A∞ , which is the type-A Dynkin
diagram with nodes extending infinitely in both directions. They develop a reasonable candidate for A∞ MV polytopes, which they call A∞ -Berenstein-Zelevinsky
(BZ) data. They do this by noticing that the Dynkin diagram A∞ behaves in many
ways like a finite type Dynkin diagram. In particular, all its finite subdiagrams are
finite-type Dynkin diagrams. Using this observation, they are able to generalize the
combinatorial characterization of MV polytopes in the finite-type case to the case of
A∞ .
b n is obtained by “folding” the
They then observe that the Dynkin diagram for sl
Dynkin diagram A∞ by an automorphism; namely, the automorphism is given by
shifting the nodes of A∞ right by n positions. This automorphism also acts naturally
on the set of A∞ -BZ data. Naito-Sagaki-Saito then consider the set of all A∞ -BZ data
fixed by this automorphism, and define crystal operators on this set in a natural way.
b n , which they prove is isomorphic
Passing to a subset, they construct a crystal for sl
to the B(∞)-crystal when n > 2. They call the resulting crystal the set of “affine
Berenstein-Zelevinsky data”. In this paper, we will refer to their construction as the
Naito-Sagaki-Saito (NSS) crystal. Their methods are purely combinatorial, relying
crucially on a result of Stembridge that characterizes the B(∞)-crystal in the simplylaced affine case. In particular, they do not apply when n = 2.
It is worth noting that the result of their construction is very similar in flavor
to the results of [Hong] that describe how finite-type MV polytopes behave under
Dynkin diagram automorphisms.
9
1.4. Main Results. The first result of this paper is a generalization of the crystal
operator formula of Baumann-Gaussent [BauGau] to the case of double MV cycles.
This formula is valid in all untwisted affine cases. We hope that it will be useful for
further investigations of double MV cycles.
The second result of this paper is an application of this formula to the case of
type A double MV cycles. Here we show that we can extract the Naito-Sagaki-Saito
crystal structure from double MV cycles using the action of the Kac-Moody group
c n on Fermionic Fock space. This gives rise to an explicit isomorphism of crystals
SL
between the set of double MV cycles and the Naito-Sagaki-Saito crystal. Using this
isomorphism, we offer an independent proof of the fact that the Naito-Sagaki-Saito
crystal structure gives rise to the B(∞) crystal. Furthermore, our proof includes the
case of n = 2, which is new.
1.5. Heuristic Motivation. As we mentioned before, Naito-Sagaki-Saito’s construction gives a combinatorial generalization of the work of Hong [Hong] describing
how MV polytopes behave under diagram automorphism. Our result is a partial
geometric generalization of his work.
Kamnitzer proved [Kam1] that MV cycles are determined by computing valuations when acting on extremal weight vectors of fundamental representations of G.
]
Let us denote by GL
∞ the Kac-Moody group corresponding to the Dynkin diagram
A∞ . Since Fermionic Fock space is the direct sum of all fundamental representations
]
of GL
∞ , it is natural to expect that computing valuations on this representation
c n ⊂ GL
]
should determine MV cycles for this group. Via the embedding GL
∞ coming
from fixed points of the diagram automorphism, following Hong, we should expect
c n should be the fixed points for MV cycles of GL
]
that MV cycles of GL
∞ (this inc n , which should be a subset of those for GL
c n ).
cludes the case of MV cycles for SL
In particular, they should be determined by computing valuations on Fermionic Fock
]
space. Unfortunately, we have no good candidate for MV cycles for GL
∞ , so this
argument is only heuristic. However, in this paper we do prove that MV cycles for
c n are determined by computing valuations on Fermionic Fock space, and that these
SL
10
valuations give rise to the Naito-Sagaki-Saito crystal structure. This is exactly what
we should expect from Hong’s results in the finite-dimensional case.
1.6. Remarks on Open Problems. An open problem is to understand double
MV cycles in all types. In particular, we would like to construct an analogue of MV
polytopes for all affine types, i.e. we would like to extract from each double MV cycle a
combinatorial gadget such that the crystal structure on double MV cycles corresponds
to a combinatorial-defined crystal structure on the combinatorial gadgets. An even
more ambitious goal would be to connect any analogue of MV polytopes to the PBW
parameterization of the affine canonical basis explained in [BCP, BN].
If we follow Kamnitzer’s results in the case of a finite-dimensional group, we
should expect that double MV cycles for an affine Kac-Moody group G should be
determined by computing valuations when acting on extremal weight vectors of func 2 . For SL
c 2,
damental representations of G. However, this is not the case, even for SL
explicit computations indicate that only partial information is obtained by computing
valuations on extremal weight vectors. That partial information seems to correspond
to the “real part” of the PBW parameterization of the affine canonical basis. The
missing part has to do with imaginary roots, and it is currently unclear how to recover
this part using double MV cycles.
c n show that
We should remark that the results of this paper in the case of SL
although computing valuations on extremal weight vectors of fundamental representations do not suffice to determine MV cycles, it does suffice to compute valuations on
a larger set of vectors in fundamental representations. This is true because the action
c n on Fermionic Fock space decomposes as a direct sum of copies of fundamental
of SL
representations. An open question is to see if the analogue of this statement is true in
all types, i.e. are double MV cycles in all types determined by computing valuations
on some set of vectors in fundamental representations? Based on our results in type
A, we expect the answer to be yes.
We hope that the results of this paper can come to bear on this problem in two
different ways. First, our explicit formula for the crystal structure on double MV
11
cycles applies in all untwisted affine types. We expect it to be useful when proving
results about combinatorial data related to double MV cycles. Indeed, it is precisely
the tool we used to prove the connection to the NSS crystal.
Second, we can try to apply our understanding of the connection between double
MV cycles and the NSS crystal to the PBW parameterization of the canonical basis.
In principle, we have a bijection between the NSS crystal and the set of PBW parameterization data coming from transporting the crystal structure. Understanding
this bijection explicitly should be an algebraic/combinatorial problem. If we can understand this bijection, we can compose it with our know bijection between double
MV cycles and the NSS crystal. The result would be an explicit bijection between
double MV cycles and PBW parameterization data. As we mentioned before, our
c 2 seem to indicate how this bijection will work for the
explicit computations for SL
“real part” of PBW parameterization data.
Another open problem is to write double MV cycles of a given weight as a disjoint
decomposition of locally closed subsets. Kamnitzer accomplishes this in the case of
a finite-dimensional group [Kam1] after choosing a reduced decomposition for the
longest element of the Weyl group. This choice of reduced decomposition corresponds
exactly to the choice needed to construct the PBW basis. In the affine case, an
analagous choice is needed to construct the PBW basis [BCP, BN]. We hope that
a construction of MV polytopes in the affine case, along with a bijection to the PBW
parameterization of the canonical basis, will lead to a generalization of Kamnitzer’s
disjoint decomposition. Using such a decomposition, one could count finite-fieldvalued points in double MV cycles. Such a computation could lead to an alternate
proof of the affine Gindikin-Karpelevich formula (c.f. [BFK]) and perhaps explain
its mysterious form.
1.7. Organization of the Chapter. In the section 2, we review the definition
of MV cycles, and we review the theory of quasimaps spaces. In the section 3,
we recall the Braverman-Finkelberg-Gaitsgory crystal structure. We prove a direct
12
generalization of Baumann-Gaussents explicit formula for the crystal structure to the
affine case.
For the remainder of the chapter, we focus on the type-A affine case. In section
4, we review the Fermionic Fock space and recall explicit formulas for the action
b n on this vector space. In section 5, we rephrase the Naito-Sagaki-Saito crystal
of sl
structure using the language of Maya diagrams. In section 6, we show how this crystal
c n on Fermionic Fock space. We prove that this
structure arises from the action of SL
construction gives rise to an isomorphism of crystals between the set of double MV
cycles and the Naito-Sagaki-Saito crystal. Using this isomorphism, we reprove that
the Naito-Sagaki-Saito crystal structure is the B(∞)-crystal.
2. Preliminary Notions
2.1. Terminology. We will work with schemes and ind-schemes over the complex numbers. For us, a subvariety V of a scheme S will mean a locally closed subset
of S with the reduced scheme structure. In particular, a dense subvariety will always
be an open subset of the original variety.
2.2. Kac-Moody Lie Algebras. We briefly review the theory of Kac-Moody
Lie algebras (c.f. [HongKang]). Let us fix a generalized Cartan matrix A, and let
I denote the set of nodes of the corresponding Dynkin diagram. We can form the
associated Kac-Moody Lie algebra g. From A we can canonically identify two dual
lattices: the weight lattice X ∗ (A) and the coweight lattice X∗ (A). We also have a
set of simple coroots {αi } ⊂ X∗ (A) and simple roots {α̌i } ⊂ X ∗ (A). Let Λ ⊂ X∗ (A)
be the integral span of the simple coroots, and let Λ+ ⊂ Λ be the positive-integral
span of the simple coroots.
The Lie algebra g is equipped with a natural triangular decomposition g = n− ⊕
t ⊕ n+ , where t is the Cartan subalgebra, and n+ (resp. n− ) is the direct sum of the
positive (resp. negative) root spaces. As usual, n+ is generated as a Lie algebra by
the Chevalley generators {Ei }i∈I . Similarly, we have Chevalley generators {Fi }i∈I for
n− .
13
We can also form the formal Kac-Moody Lie algebra g by completing n+ with
respect to the root grading. Explicitly, g = n− ⊕ t ⊕ n+ , where n+ is the direct product
of the positive root spaces. From now onward, we will suppress the over-bars and
write g and n+ when discussing both the minimal and formal Kac-Moody algebras.
This will simplify the discussion by allowing uniform notation.
2.3. The Dual Lie Algebra. If g is the Kac-Moody Lie algebra corresponding
to a generalized Cartan matrix A, then we define the dual Lie algebra g∨ to be the
Kac-Moody Lie algebra corresponding to transpose generalized Cartan matrix AT .
We remark that X ∗ AT = X∗ (A) and X∗ AT = X ∗ (A), i.e., the coweight and
weight lattices are swapped from those of A. For much of this paper, we will focus
b n , the untwisted affinization of sln . In this case, the generalized
on the case g = sl
b n is self-dual.
Cartan matrix is symmetric, and we see that sl
2.4. Kac-Moody Groups. We will write G for the “simply connected” group
corresponding to a minimal or formal Kac-Moody Lie algebra (c.f. [Kum]); we will
refer to G as either the minimal or formal Kac-Moody group. Let N− , T and N+ be
the subgroups corresponding to n− , t and n+ respectively. Let B+ and B− be the
subgroups corresponding to b− = n− ⊕ t and b+ = t ⊕ n+ respectively. If we fix J ⊂ I,
we can form the corresponding positive parabolic subalgebra pJ+ by adjoining to b+
all the negative root spaces corresponding the nodes in J and taking Lie subalgebra
they jointly generate. Let P+J be the corresponding subgroup of G. Denote by M J
the corresponding Levi factor.
Remark 2.1. In general, G will have the structure of a group ind-scheme. When
A is a finite-type Cartan matrix, G will be a finite dimensional complex reductive
group. Furthermore, in the finite-type case, there is no difference between minimal
and formal versions of Lie algebra and group.
2.5. Lusztig’s Canonical Basis. One particularly nice realization of the B(∞)
crystal comes from Lusztig’s canonical basis [Lus2], which is a basis in the negative
part of the quantum group enjoying many nice properties. When g is finite type, there
14
is a natural parameterization of the canonical basis coming from the various PoincaréBirkhoff-Witt (PBW) bases. An interesting question is to study the combinatorics
that records how to pass between the various parameterizations of the canonical basis
coming from the various PBW bases. Lusztig gave an explicit answer in simply-laced
cases [Lus3]. Berenstein and Zelevinsky [BZ] gave an answer in all types. In addition,
they indicate how the reparameterization data can be arranged into a combinatorial
gadget called an MV polytope (we will explain this name, due to Kamnitzer, in the
next subsection). In particular, they produce a bijection between the canonical basis
and the set of MV polytopes. Finally, they give a combinatorial description of the
crystal structure using only the data coming from MV polytopes.
2.6. Definition of Crystals. We will recall the definition of crystals. For later
convenience, we will recall the definition of a crystal for the Lie algebra g∨ dual to
a given Kac-Moody Lie algebra g (c.f. [BravGait]). A g∨ -crystal is a set B along
with the following data:
• A weight function wt : B → X ∗ (A)
• For each i ∈ I, crystal operators ei , fi : B → B t {0}
• For each i ∈ I, i-string functions εi , φi : B → Z
This data should satisfy the following axioms:
i) For all b ∈ B, φi (b) = εi (b) + hwt (b) , αi i
ii) Let b ∈ B. If ei (b) 6= 0 for some i, then
wt (ei (b)) = wt(b) + α̌i , εi (ei (b)) = εi (b) − 1, ϕi (ei (b)) = ϕi (b) + 1
iii) Similarly, if fi (b) 6= 0 for some i, then
wt (fi (b)) = wt(b) − α̌i , εi (fi (b)) = εi (b) + 1, ϕi (fi (b)) = ϕi (b) − 1
iv) If b, b0 ∈ B, then we have b0 = ei (b) if and only if b = fi (b0 ).
15
As we mentioned in the introduction, to every representation of g∨ in the BGG
category O we can canonically identify a g∨ -crystal (c.f. [HongKang]). We define the
B(∞) crystal to be the crystal associated to the Verma module with highest weight
zero. Similarly, we define B(−∞) to be the crystal associated to the lowest-weight
Verma module with lowest weight zero.
2.7. The Affine Grassmannian. Let O = C[[t]] be the ring of formal Taylor
series in one variable, and let K = C((t)) be the field of formal Laurent series in one
−k
−1
variable. Let C[t−1 ]+
=
a
t
+
·
·
·
a
t
|a
∈
C
, which we view as a subset of K.
k
1
i
k
For any group ind-scheme K (e.g. any of the groups we have defined above), we
define the affine Grassmannian GrK = K(K)/K(O). In general, we can give GrK the
structure of a set. However, when K is a finite-dimensional algebraic group, GrK can
be naturally viewed as the C-points of an ind-scheme of ind-finite type. In particular,
when K = G is a finite-type Kac-Moody group, we can speak of GrG as a geometric
object. Unfortunately, for general Kac-Moody G we do not currently have a good
way of working with GrG as a geometric object. However, for the purposes of this
paper we don’t need the entire affine Grassmannian. Rather we only need certain
subvarieties of GrG called Mirković-Vilonen (MV) cycles. Fortunately, when G is of
untwisted affine type, we have a good geometric substitute for MV cycles coming
from the theory of quasimap spaces.
2.8. MV Cycles and Quasimap Spaces. Let us fix a Kac-Moody group G.
To discuss MV cycles, we need to define certain subsets of GrG . We have a subset
T (K)/T (O) ⊂ G(K)/G(O), which is canonically identified with the coweight lattice
X∗ (A) = Hom (T, Gm ). Let us denote tλ the point of GrG corresponding to the
coweight λ. Consider the subgroups N+ (K) and N− (K). We denote the positive
semi-infinite cells to be the orbits of coweight lattice under N+ (K), i.e.,
S λ = N+ (K) · tλ
16
Similarly, we define the negative semi-infinite cells to be the orbits of the coweight
lattice under N− (K).
T λ = N− (K) · tλ
It is easy to see that the S λ ∩ S γ = ∅ and T λ ∩ T γ = ∅ for λ 6= γ. Let us for a
moment consider only finite-type G. In this case, these sets have the structure of indschemes of infinite dimension and infinite codimension in the affine Grassmannian.
However, for all γ, λ, the intersection S γ ∩ T λ is a finite dimensional algebraic variety.
Moreover the intersection S γ ∩ T λ = ∅ unless γ − λ ∈ Λ+ . In this case, we can identify
the irreducible components of S γ ∩ T λ with a basis in the λ-weight space of the Verma
module with highest weight γ [FFKM]. We call these irreducible components MV
cycles.
However, when G is not finite type we cannot directly give these intersections
the structure of an algebraic variety. However, we have a substitute in the form of
quasimap spaces. Let us now consider only the case when G is a formal Kac-Moody
groups. Associated to G is the Kashiwara flag scheme B, which we can think of as the
quotient G/B− . When G is finite type, this is just the flag variety of G. For general
G, B still retains many of the geometric features of the finite dimensional flag variety.
In particular, it has a Schubert cell decomposition. Let U ⊂ B denote the unique
open Schubert cell, i.e. the “big cell”. Moreover, as in the finite dimensional case the
second homology of B is naturally in bijection with Λ. In particular, any algebraic
map φ : C → B from an algebraic curve into B has a well-defined degree, which we
view as an element of Λ+ . Let us fix λ ∈ Λ+ , and consider the following space (i.e.
◦
functor of points) Fλ , defined in [BFG], that classifies maps φ : P1 → B satisfying
the following conditions:
• deg (φ) = λ
• φ (P1 − 0) ⊂ U
• φ (∞) = 1B (the unit point in B)
17
◦
When G is finite type, we have an isomorphism Fλ ' S 0 ∩T −λ [FM]. In particular,
◦
Fλ is a finite-dimensional scheme. When G is of untwisted affine type, the authors
◦
of [BFG] give a proof that Fλ is a finite-type, finite-dimensional scheme. Moreover,
they prove that it is equidimensional, and they explicitly compute its dimension.
For general type, we always have the following set theoretic bijection [BFK,
Theorem 2.7]:
◦
Fλ (C) ' S 0 ∩ T −λ
(1)
We will give details ofthisbijection in section 3.2.
◦
F
Let L = λ≤0 Irr Fλ denote the set of generalized MV cycles. When G
is finite type, this corresponds exactly with our
notion of MV cycles. In
original
◦
the untwisted affine case, we call elements of Irr Fλ double MV cycles (because
they should be isomorphic to the analogs of MV cycles in the yet-to-be-defined double
affine Grassmannian).
◦
The spaces Fλ have a natural closure Fλ called the quasimap closure. The closures
Fλ are defined in [BFG]. The exact definition will not matter for this paper as we
◦
will use them as an auxiliary tool to study the spaces Fλ . In particular, we will use
F
the fact that we have a canonical identification L = λ≤0 Irr Fλ given by sending
◦
an irreducible component of Fλ to its closure in Fλ .
3. The Braverman-Finkelberg-Gaitsgory Crystal Structure
When G is finite-type or an untwisted (formal) affine Kac-Moody group, we have
the following theorem [BFG] :
Theorem 3.1. The set L has the structure of the B(∞) crystal for the dual Lie
algebra g∨ .
In fact, Braverman-Finkelberg-Gaitsgory define a pair of crystal structures (ẽi , f˜i , εi , φi , wt)
and (ẽ∗i , f˜i∗ , ε∗i , φ∗i , wt) on the set L that give it the structure of the B(∞) crystal in
18
two different ways. In this paper, we will concern ourselves exclusively with the second crystal structure. To reduce clutter we will drop the stars and the tildes, and
write (ei , fi , εi , φi , wt) for what is denoted in [BFG] as (ẽ∗i , f˜i∗ , ε∗i , φ∗i , wt).
In this section, we will recall the definition of the BFG crystal structure and give
explicit formulas for the crystal operators, directly generalizing a finite-dimensional
result of Baumann and Gaussent.
When G is finite type, GrG has a geometric structure, and we have the isomor◦
∼
phism of varieties Fλ → S 0 ∩ T λ . In section 13 of [BFG], the crystal structure is
defined purely in the language of the affine Grassmannian. When G is not finite
type, we only have the quasimaps spaces, and section 14 of [BFG] is a translation of
the construction in the previous section to the language of quasimaps spaces. Thus
sections 13 and 14 provide a dictionary between the affine Grassmannian and the
quasimaps spaces for the purposes of the BFG crystal structure.
When G is finite type, Baumann and Gaussent give a explicit algebraic formula
for the crystal structure on MV cycles. Let us note that in Baumann-Gaussent’s
work, MV cycles are defined to be irreducible components of the closure S 0 ∩ T ν .
0
ν
0
ν
Because Irr (S ∩ T ) = Irr S ∩ T , we can easily translate their results to our
match our convention. To be unambiguous, we will call irreducible components of
S 0 ∩ T ν closed MV cycles. Baumann and Gaussent’s result allow us to compute the
BFG crystal operator explicitly in terms of multiplication by Chevalley subgroups.
Theorem 3.2. [BauGau] Fix i ∈ I. Let Z be a closed MV cycle. Let Z 0 = fik (Z)
be the result of applying the fi operator k times to Z (so that Z 0 is the corresponding
closed MV cycle). Then there exist dense locally closed subvarieties Ż ⊂ Z and
Ż 0 ⊂ Z 0 such that the following holds:
0
φi (Z)
The map f : C[t−1 ]+
z is well-defined and
k × Ż → Ż given by f (p, z) = xi pt
is a homeomorphism.
Here, xi : K → N+ (K) is the one-parameter Chevalley subgroup corresponding to
the simple coroot αi , and the multiplication is the natural action of N+ (K) on the
affine Grassmannian GrG .
19
Remark 3.3. Baumann-Gaussent originally phrase their result in terms of the
B(−∞)-crystal. We have performed the obvious modification to rephrase their result
for the B(∞)-crystal. Also their theorem gives finer information than what we have
given above. In particular, the subvarieties Ż ⊂ Z and Ż 0 ⊂ Z 0 have very explicit
descriptions, which will become clear when we prove the generalization to the affine
case.
The goal of this section is to prove an analog of this result for double MV cycles.
The proof will mostly follow the original proof of Baumann-Gaussent. However, their
proof is written in the language of the affine Grassmannian, and we need to translate
it to the language of quasimap spaces. Fortunately, sections 13 and 14 of [BFG]
provide a sufficient dictionary.
3.1. Recalling the BFG Construction. We will assume familiarity with the
notation and constructions of [BFG]. Fix a node i ∈ I. Let P = P {i} be the
corresponding positive subminimal parabolic subgroup of G, and let M = M {i} be the
corresponding rank one Levi factor. Let B± (M ) be the induced positive and negative
Borel subgroups of M , and let N± (M ) be the corresponding unipotent radicals. Let
us fix µ ∈ Λ. The authors construct an ind-scheme Sµg,b,pi with a natural action by
the group N+ (M )(K) [BFG, Section 14.8]. The exact definition will not be relevant
for this subsection, so we will delay discussing the details of the definition to the next
subsection.
Furthermore, the authors construct a fiber bundle rµ : Sµg,b,pi → GrµB(M ) . This
map rµ is equivariant for an action of the group N+ (M )(K). Since M is rank 1, we
∼
have an isomorphism xi : K → N+ (M )(K) given by the one parameter subgroup
corresponding to exponentiating the unique positive root space.
In summary we have the following diagram where the horizontal arrows are the
action via xi :
20
/
K × Sµg,b,pi
Sµg,b,pi
rµ
rµ
(2)
µ
K × GrB(M
)
/
µ
GrB(M
)
Because of the group K is connected and acts transitively on the base, we can
canonically identify the irreducible components of any two fibers using the group
action. In fact, if X ⊂ GrµB(M ) is any irreducible subvariety, we can canonically
identify the irreducible components of rµ−1 (X) with the irreducible components of
any fiber using the group action.
λ
. Because M is rank
In particular, let λ ∈ Λ+ , and let D = GrµB(M ) ∩ GrB
− (M )
−1
one, D is irreducible. Define Sµ,≤λ
g,b,pi = rµ (D). By the discussion in [BFG], we can
λ
identify Sµ,≤λ
g,b,pi with a locally closed subset of F . Moreover, these subsets are disjoint
for different choices of µ, and we have the following decomposition:
Fλ =
G
Sµ,≤λ
g,b,pi
µ
If we denote by Irrtop Sµ,≤λ
the set of irreducible components of Sµ,≤λ
g,b,pi
g,b,pi whose dimension is the same as that of Fλ , then we have the following decomposition:
G top µ,≤λ Irr Fλ =
Irr
Sg,b,pi
µ
And in particular, we can write
L=
GG
λ
Irr
top
Sµ,≤λ
g,b,pi
µ
Using this decomposition, we can define the [BFG] crystal structure as follows.
Definition 3.4. Let i ∈ I, and let Z ∈ L be a double MV cycle. Then we define
fi (Z) as follows. By the above discussion, Z ∈ Irrtop Sµ,≤λ
g,b,pi for a unique pair µ and
λ. By the discussion about irreducible components of fibers of rµ , we have a canonical
21
µ,≤λ−αi
top
∼
Irr
S
. Define fi (Z) to be the image of Z under
bijection Irrtop Sµ,≤λ
g,b,pi =
g,b,pi
this bijection.
The operator ei is defined to be the standard partial inverse of fi , i.e. ei (Z) = Z 0 if
there exists some (necessarily unique) Z 0 such that Z = fi (Z 0 ). Otherwise ei (Z) = 0.
The auxiliary data of i , φi , wt is obviously defined.
With this definition, we can state the following theorem.
Theorem 3.5. [BFG] The set L ∪ {0} with the operators defined above form a
B(∞) crystal for the dual Lie algebra g∨ .
Moreover, we can now state and prove the following theorem, which is a natural
generalization of Theorem 3.2.
Theorem 3.6. Generalization of the Baumann-Gaussent Formula Fix i ∈
I. Let Z be a irreducible component of Fλ , and let Z 0 = fik (Z) be the result of
applying the fi operator k times to Z. By the above discussion, we can identify Z
µ,≤λ
0
with a irreducible component of Sµ,≤λ
g,b,pi for a unique µ. Let Ż = Z ∩ Sg,b,pi , let Ż =
i
. Then we can define the following map, which is a homeomorphism:
Z 0 ∩ Sµ,≤λ−kα
g,b,pi
0
φi (Z)
f : C[t−1 ]+
) · z, where · denotes the
k × Ż → Ż , given by f (p, z) = xi (pt
restriction of the action defined above.
Proof. This proof proceeds in direct parallel to the proof of [BauGau, Proposition 14]. Let
i
D = GrµB(M ) ∩ GrλB− (M ) , and D0 = GrµB(M ) ∩ Grλ−kα
B− (M )
Note that because M has rank one, both D and D0 are irreducible.
Using an explicit rank-one calculation in Baumann-Gaussent, [BauGau, Propo0
sitions 8], we can conclude that the map g : C[t−1 ]+
k × D → D given by g(p, d) =
xi (ptφi (Z)) · z is a homeomorphism. Restricting diagram (2) to g, we get the following
commutative diagram:
22
f
µ,≤λ
C[t−1 ]+
k × Sg,b,pi
/
i
Sµ,≤λ−kα
g,b,pi
rµ
rµ
g
C[t−1 ]+
k ×D
/
D0
Notice that this square is Cartesian and that the map rµ is faithfully flat. Because
the map g is a homeomorphism, and because the property of being a homeomorphism is preserved under flat base change[EGA4, Proposition 2.6.2], we conclude
the map f is also a homeomorphism. But the map f is precisely how we identify
∼
µ,≤λ
µ,≤λ−αi
top
top
Irr
when defining the crystal structure. So we conSg,b,pi → Irr
Sg,b,pi
clude that the restricted map
0
f : C[t−1 ]+
k × Ż → Ż
is well-defined and a homeomorphism.
◦
3.2.
The bijection Fλ (C) ' S 0 ∩ T −λ .
◦
In this section we will explain in detail how the bijection Fλ (C) ' S 0 ∩T −λ works.
In addition, we will prove a certain equivariance property that will be required for
the remainder of the paper. This section will require manipulations of various moduli
spaces from [BFG] that will not appear later in the paper. We will freely use results
from this paper, giving precise citations when we do.
◦
Recall that Fλ is defined as the moduli space of maps φ : P1 → B satisfying the
following conditions:
• deg (φ) = λ
• φ (P1 − 0) ⊂ U
• φ (∞) = 1B (the unit point in B)
Here B is the Kashiwara flag scheme, and U is the open Schubert cell. By the
Bruhat decompostion, we have a canonical bijection U ' N , where N is the positive
unipotent part of our formal Kac-Moody group G.
23
Thus we have the following sequence of morphisms:
◦
Fλ → Maps(P1 − 0, N ) → Maps(Spec K, N )
Here the first map is given by restriction of a map φ : P1 → B to a map P1 − 0 →
U ' N , and the second map is given by restriction to the formal punctured disk
around 0 in P1 − 0, which we identify with Spec K. Note that the we consider the
second two spaces only as moduli functors and do not concern ourselves with issues
of representability of these functors.
◦
Passing to C-points, we get a map Fλ (C) → N (K). Composing with the quotient
◦
N (K) → N (K)/N (O) ' S 0 , we get a map Fλ (C) → S 0 . We have the following
theorem, due to Braverman-Finkelberg-Kazhdan
◦
Theorem 3.7. [BFK, Theorem 2.7] The image of the map Fλ (C) → S 0 is con◦
tained in S 0 ∩ T −λ , and the resulting map Fλ (C) → S 0 ∩ T −λ is a bijection.
For the remainder of this subsection, we will discuss how this bijection interacts
with the formula from Theorem 3.6. In particular, we will be able to compute the
crystal stucture on the C-points of double MV cycles without having to explicitly
mention the intricate geometry involved.
Recall the N+ (M )(K)-equivariant fiber bundle rµ : Sµg,b,pi → GrµB(M ) used in defin◦
λ
ing the crystal structure on double MV cycles. Let D = GrµB(M ) ∩ GrB
. Define
− (M )
◦
−1
Sµ,λ
D . Then we have the following decomposition into locally closed
g,b,pi = rµ
subsets (c.f the proof of [BFG, Proposition 15.2]):
◦
Fλ =
G
Sµ,λ
g,b,pi
µ
Taking the union over all λ, we have
G◦
G G µ,λ
G µ
Fλ =
Sg,b,pi =
Sg,b,pi
λ
λ
µ
µ
24
Passing to C-points and using the Braverman-Finkelberg-Kazhdan bijection, we have
F
F
a bijection λ S 0 ∩ T −λ = µ Sµg,b,pi (C).
F
Notice that λ S 0 ∩ T −λ is strictly smaller than S 0 . In the terminology of [BFK,
Lemma 2.5], it is precisely the image of “good” elements of N (K) inside of of S 0 .
F
However, it is easy to see that λ S 0 ∩ T −λ is stable under the action of N+ (M )(K)
(essentially because N+ (M )(K) preserves the “good” elements of G(K)).
F
F
Thus we have a bijection λ S 0 ∩ T −λ = µ Sµg,b,pi and actions of N+ (M )(K) on
both sides of this bijection. We can now state the following proposition.
Proposition 3.8. The bijection of sets
F
λ
S 0 ∩ T −λ →
F
µ
Sµg,b,pi is N+ (M )(K)-
equivariant.
Proof. The proof requires nothing more than unwinding the various lengthy
definitions made in [BFG].
◦
First, we will need to use an alternate characterization of Fλ as a closed subscheme
of a Zastava space. Let A1 ⊂ P1 be the complement of ∞ inside P1 . Let λ ∈ Λ+ .
Then the Zastava space Z λ (A1 ) is the moduli of triples (Dθ , FN , κ) where Dθ is a
“colored divisor” of degree λ (c.f. [BFG, Section 2.3]), FN is an N bundle on P1 , and
κ is a “Plücker datum”, which together satisfy a list of conditions detailed in [BFG,
Section 2.12] (the exact conditions are not relevant for our discussion).
◦
Of primary importance to us, is the open subscheme Z λ (A1 ) ⊂ Z λ (A1 ), corresponding to the condition that the Plücker datum κ consist of injective bundle maps.
◦
λ
By [BFG, Proposition 2.21] the closed subscheme of Z (A1 ) consisting of those triples
◦
(Dθ , FN , κ) where Dθ is supported at 0 is naturally isomorphic to Fλ . Hence, for the
◦
remainder of this subsection, we will use this Zastava definition to work with Fλ
◦
From the Zastava perspective, we can also reinterpret the bijection Fλ (C) '
S 0 ∩ T −λ . By [BFG, Lemma 2.13], the N -bundle FN is canonically trivialized away
from 0. Restricting to the formal disk around 0, we get a N bundle on a formal disk
trivialized on the punctured disk, which gives us a point of N (K)/N (O). Thus we see
25
that the action of N+ (M )(K) on
F
◦
λ
λ F (C) corresponds to changing the trivialization
of the N bundle FN in a formal punctured disk centered at 0.
Let us now shift attention to the action of N+ (M )(K) on Sµg,b,pi . The space Sµg,b,pi
is defined to be the moduli space of the following data:
• A B-bundle FB on A1 , such that the induced T -bundle F0T is trivialized, (in
particular, the B bundle has a canonical reduction to N , which we call FN )
• An M -bundle FM on A1 ,
• Regular bundle maps VFν̌B → UFν̌M , and
• Meromorphic maps (regular away from 0) UFν̌M → Lν̌F0 ,
T
such that
• The maps UFν̌M → Lν̌F0 , form a Plücker datum for FM , and the compositions
T
VFν̌B
→
UFν̌M
→
Lν̌F0
T
form a Plücker datum for FB
• The compositions Lν̌F0 → VFν̌B → UFν̌M → Lν̌F0 are the identity maps, and
T
T
• The induced maps
Lν̌F0
T
→
UFν̌M (hµ, ν̌i
· 0) are regular bundle maps.
Here ν̌ varies through all dominant weights, and VFν̌B ,UFν̌M , Lν̌F0 are the associated
T
bundles with fiber equal to the irreducible representation of highest weight ν̌
In particular, we get a pair of opposite Plücker data for both FB and FM that
are related by the maps VFν̌B → UFν̌M . Moreover, because these Plücker data are
“transverse” away from 0, i.e. the meromorphic compositions Lν̌F0 → VFν̌B → UFν̌M →
T
Lν̌F0 are the identity maps, both bundles FB and FM have canonical reductions to
T
the maximal torus away from 0. However, because the T bundle induced from FB
is trivialized, we see that both FB and FM are actually trivialized away from 0. By
[BFG, Proposition 14.3], the action of N+ (M )(K) on Sµg,b,pi corresponds to changing
the trivialization on the bundle FM in a formal punctured disk centered at 0. However,
because this action leaves the maps VFν̌B → UFν̌M fixed, the trivialization on the bundle
FB must change by same amount.
◦
Now, let us study the identification Fλ =
◦
F
µ
Sµ,λ
g,b,pi . The bundle FN in the defini-
tion of Fλ maps to the bundle FB , which we recall has a canonical reduction to N .
26
As the Plücker data are also preserved under this identification, the trivialization of
FN away from 0 corresponds exactly to the trivialization of FB away from 0.
With this equivariance established, we can prove the following corollary to Theorem 3.6.
◦
Corollary 3.9. Fix i ∈ I. Let W be a irreducible component of Fλ , and let
W 0 = fik (Z) be the result of applying the fi operator k times to W (so W 0 is an
◦
irreducible component of Fλ−kαi . Then there exists an dense subvariety U ⊂ W 0 , all
of whose C-points can be written in the form xi (ptφi (Z) ) · z, where p ∈ C[t−1 ]+
k and
z ∈ W . Here we are viewing the C-points of W (resp. W 0 ) as a subset of S 0 ∩ T λ
(resp. S 0 ∩ T λ−kαi ), and the action of xi is given via the left multiplication of N (K)
on S 0 .
Proof. Let Z (resp. Z 0 ) be the closure of W (resp. W 0 ) in Fλ (resp. Fλ−kαi ). We
can find dense subvarieties Ż ⊂ Z and Ż 0 ⊂ Z 0 by intersecting with an appropriate
0
Sµg,b,pi . By Theorem 3.6, we have a homeomorphism f : C[t−1 ]+
k × Ż → Ż . Let
V = Ż 0 ∩ W 0 . Then U = f (f −1 (V ) ∩ C[t−1 ]+
k × Ż) will consist of elements that can
be written as xi (ptφi (Z) ) · z, where p ∈ C[t−1 ]+
k and z ∈ W . By Theorem 3.7, we can
interpret z as a point in S 0 ∩ T λ , and by Proposition 3.8 the action xi (ptφi (Z) ) · z to
be that coming from left multiplication of N (K) on S 0 .
4. Maya Diagrams and Fock Space
For the remainder of the paper, we will let G be a formal affine Kac-Moody group
b n for n ≥ 2. In this case
of type A, i.e. the group corresponding to the Lie algebra sl
we can identify the Dynkin diagram with the integers modulo n, i.e., I = Z/nZ. For
these groups, we can build a very explicit representation called the Fermionic Fock
space. In this section we will discuss the Maya-diagram/charged-partition bases of
Fermionic Fock space, following the notation and diagrams from Tingley’s expository
notes [Ting]. Once we have described this basis, we will be able to write down
27
b n on Fermionic Fock space. To begin, let us define
formulas for an explicit action of sl
Maya diagrams.
Definition 4.1. A left-black Maya diagram is a sequence of white or black
beads indexed by Z+0.5 such that all beads are black in sufficiently positive positions,
and all beads are white in sufficiently negative positions. Here is an example of a leftblack Maya diagram:
. . . u u u u u u u u e u e e u u e e u e e e e e. . .
11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11
Notice we have labeled the positions in increasing order from right to left following
[Ting]. With this convention, all beads sufficiently left of zero must be black; hence
the terminology left-black.
We can identify the set of all left-black Maya diagrams with downward-facing
charged partitions (c.f [Ting] for the recipe). For example, we identify the previous
left-black Maya diagram with the following downward-facing charged partition:
. . . u u u u u u u u e u e e u u e e u e e e e e. . .
11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11
@
@
-1 @
@
0 @-2 @
1@@-1 @-3 @
2@ 0 @-2 @-4 @
@
@
@ @ -1
@-5@
@
@ @
@
@
@
@
R
@
x
y
Following the recipe in [Ting], we can label each box of the partition with an integer
corresponding to which slot of the Maya diagram it lies below.
Let us define the Fermionic Fock space F− to be the formal C-linear span of
all left-black Maya diagrams (equivalently downward-facing charged partitions).
Remark 4.2. Fermionic Fock space is usually defined with a basis consisting of
semi-infinite wedge products. It is easy to see that the above definition is in natural
28
bijection with the usual definition, with every Maya diagram corresponding to a
particular semi-infinite wedge product (c.f. [Ting]).
Similarly, we can define right-black Maya diagrams. Again there is a bijection
between right-black Maya diagrams and upward-facing charged partitions, and we
denote by F+ its formal span. There is a natural bijection between right-black and
left-black Maya diagrams coming from interchanging the roles of white and black (or
equivalently flipping the charged-partition upside-down). This bijection induces a
non-degenerate pairing between F+ and F− .
We will use box-numbering of charged partitions to define a representation of
b n on F+ and F− . For the purposes of defining a representation of sl
b n , we will
sl
only consider the box-numberings modulo n. Because we will think of the affine
Grassmannian as a right quotient, it will be convenient for our Lie algebras to act on
b n on F− by the following formula on Chevalley
the right. We define the action of sl
generators:
b n on Fock Space
Definition 4.3. Action of sl
Let γ be a downward-facing charged partition. Then for all i ∈ Z/nZ,
(3)
hγ|Ei :=
X
hµ|
hγ|Fi :=
X
γ\µ is an
µ\γ is an
i-colored box
i-colored box
hµ|.
Prop 3.5.8 in [Ting] tells us that this definition gives rise to a Lie algebra action.
We then define a dual action on F+ by using the natural pairing with F− . It is easy
to see that these representations are integrable, and they integrate to an action of the
corresponding (minimal) Kac-Moody groups.
Remark 4.4. A natural way to get right actions from left actions is via an antib n , the inverse and the Chevalley
automorphism. There are two natural choices for sl
involution. The above formula comes from applying the Chevalley involution.
29
Remark 4.5. From the definitions, we see that F+ will be a lowest weight representation and F− will be a highest weight representation. So for the formal KacMoody group, only an action on F− will be defined. To define an action on F+ , we
need to complete it so the basis defined above becomes a topological basis.
4.1. Valuations. Let V be a complex vector space with either the discrete topology or the topology of a vector space dual to a discrete vector space. Let us define
ˆ as follows: if V is discrete, then V ⊗K
ˆ = V ⊗ K; if V is dual
the K-vector space V ⊗K
ˆ = HomC (W, K).
to a discrete vector space W , then define V ⊗K
Then we can define a function
ˆ → Z ∪ {±∞}
val : V ⊗K
ˆ by subsets of the form V ⊗t
ˆ `O
as follows. We define a decreasing filtration on V ⊗K
ˆ ` O exactly as above). Let x ∈ V ⊗K.
ˆ
(we define V ⊗t
If x = 0, we set val(x) = ∞.
ˆ ` O; if no such `
Otherwise, define val(x) to be the maximal ` such that x ∈ V ⊗t
exists, we set val(x) = −∞. Note that if V is discrete, the filtration is exhaustive. In
particular, val(x) = −∞ never occurs.
Taking K points of the action of G on F− ,we have an action of G(K) on F− ⊗K. Let
xi : K → N+ (K) be the one parameter subgroup corresponding to the simple coroot
αi . Let us fix p ∈ K with val(p) = `, and let x = xi (p). Let hγ| ∈ F− ⊂ F− ⊗ K
be a basis vector, i.e. γ is a downward-facing charged partition. Then using the
P
k
series expansion of the exponential function, we see that hγ|xi (p) = hγ| ∞
k=0 Ei ⊗
pk
.
k!
Because, the action of the Chevalley generators is locally nilpotent, this sum is
actually a finite sum. Using the explicit formula above, we see that val (hγ|xi (p)) =
min {val(p) · B, 0}, where B is the maximal number of i-colored boxes that can be
added to γ.
Now let x = xi (p) · z, where z ∈ N+ (K). Let us compute val (hγ|xi (p) · z). By
P
what we have just said, hγ|xi (p) consists of a finite sum of terms hµ| ⊗ aµ where µ is
obtained by adding i-colored boxes to γ and val(aµ ) = val(p) · |γ\µ|. So hγ|xi (p) · z =
30
P
hµ|z ⊗ aµ . For a generic choice of p, none of these terms will cancel and we have
proved the following lemma.
Lemma 4.6. Consider all p ∈ K of a fixed valuation. Let z ∈ N+ (K). Then for
hγ| ∈ F− ⊂ F− ⊗ K and generic choices of p, we have:
val (hγ|xi (p) · z) =
(4)
{val (hµ|z) + val(p) · |γ\µ|}
min
µ obtained by removing
i-colored boxes from γ
Note, that for v ∈ F− , val(v · −) makes sense as a function on G (K) /G(O),
because the right action of G(O) preserves the valuation. We will later see that these
functions will pick out double MV cycles when we let hγ| varies through the basis of
F− .
5. The Naito-Sagaki-Saito Crystal
In this section, we will recall the definition of the Naito-Sagaki-Saito (NSS) crystal
defined in [NSS]. Their construction is a natural extension of the theory of MV
polytopes and Berenstein-Zelevinsky (BZ) data in finite-type cases (c.f [Kam1]).
We will present their crystal structure in a way that is most streamlined for our
purposes. In particular, we will rephrase their construction using the language of
Maya diagrams and charged partitions. The translations from their language to
our language is straightforward, but in our presentation the motivation from MV
polytopes is obscured.
Definition 5.1. Let M• be collection of integers indexed by left-black Maya
diagrams satisfying the following conditions. Let τ be any right-black Maya diagram,
and let I be an interval containing its support. We can then form a right-black Maya
diagram γI by inverting all the colors of τ outside of the interval I. We say that M•
is a pre-NSS datum if there is a constant Θ (M )τ such that MγI = Θ (M )τ for all
I sufficiently large, i.e. there exists an interval I0 such that MγI = Θ (M )τ for all
intervals I containing I0 .
31
The main consequence of this condition is that it allows us to define another
collection of integers Θ (M )• indexed by right-black diagrams. Abusing notation, we
will write Mτ := Θ (M )τ for any right-black Maya diagram τ .
Let us define some notation for certain special right-black Maya diagrams. For
every integer i, let us define Λi to be the right-black Maya diagram that has a black
bead in every position less than i and a white bead in every position greater than
i. Let si Λi be the right-black Maya diagram obtained by switching the colors of the
beads in the i + 0.5 and i − 0.5 positions of Λi .
For example, Λ2 is this Maya diagram:
. . . e e e e e e e e e u u u u u u u u u u u u u. . .
11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11
And s2 Λ2 is this one:
. . . e e e e e e e e u e u u u u u u u u u u u u. . .
11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10-11
Remark 5.2. The above notation corresponds to the fact that we can identify
]
{Λi }i∈Z with the fundamental weights for GL
∞ , and {si } with the simple reflections
in its Weyl group.
With this notation in hand, we can define the action of the “A∞ Kashiwara
operators” on pre-NSS data:
Definition 5.3. Let M• be a pre-NSS datum, and let i be an integer. We define
a new pre-NSS datum fi (M ) as follows. Let ci (M ) = MΛi − Msi Λi − 1. Then we
define:
(5)
f˜i M
=
γ
{Mµ + |γ\µ| · ci (M )}
min
µ obtained by removing
i-colored boxes from γ
Notice that there will be at most one removable box of color i, so the min in the
definition is taken over a set of either one or two elements.
32
Remark 5.4. The definition above differs slightly in form from the one in [NSS].
However, an immediate calculation shows that they are equivalent. By the second
part of [NSS, Proposition 3.3.2], we see that the definition makes sense, i.e. fi M
satisfies the pre-NSS condition.
Let us fix an integer n ≥ 2. We have a natural operator σ on the set of left-black
Maya diagrams given by shifting the positions by n. We say a pre-NSS datum M is
n-periodic, if Mσ(γ) = Mγ for all left-black Maya diagrams γ. Restricting ourselves
b n Kashiwara operators as
to the set of n-periodic pre-NSS data, we define the set of sl
follows.
Definition 5.5. Let i ∈ Z/nZ, and let M be an n-periodic pre-NSS datum. We
define
!
(6)
fˆi M =
Y
f˜i+kn (M )
k∈Z
The right hand side of the definition requires some explanation. First we note
that the operators f˜i+kn commute. Furthermore, for every right-black Maya diagram
Q
˜i+kn (M ) =
f
γ, there exists a finite-length interval I0 ⊂ Z + 0.5 such that
k∈I
γ
Q
Q
˜
˜
fi+kn (M )γ for all intervals I containing I0 . We therefore define
k∈Z fi+kn (M )γ =
Qk∈I0
˜
k∈I0 fi+kn (M )γ
Along with these operators, we have the additional data required to define a
crystal. The êi operators are defined in the obvious way. To every n-periodic pren o
P
NSS datum M , we associate a coweight wt(M ) := i∈Z/nZ MΛi · ĥi where ĥi are
b n . We define ε̂i (M ) := −MΛ − Ms Λ + MΛ + MΛi+1 . And
the simple coroots for sl
i
i i
i−1
we define φ̂i (M ) = hwt(M ), hi i + ε̂i (M ). With these definitions in place, we can give
an explicit formula for the fˆi operators.
33
Lemma 5.6. Let M• be a n-periodic pre-NSS datum. Then
n
o
(7)
fˆi M =
min
Mµ + |γ\µ| · φ̂i (M ) − 1
γ
µ obtained by removing
i-colored boxes from γ
Proof. A straightforward calculation shows ci+kn (f˜i+nl N ) = ci+kn (N ) for k 6= l
and any n-periodic pre-NSS datum N . So all the ci+nk (N ) that appear when applying
the various f˜i+nk operators are equal to ci (M ). Each f˜i+nk operator acts by taking
minimum over the two possibilities of either adding a box of color i+nk or not adding
a box. When we take the infinite product, we get a minimum over the possibilities
of removing any number of boxes of color congruent to i modulo n. Because the
numbers ci+kn stay constant at each step, we get the following formula:
(fˆi M )γ = min{M (Z)µ + |γ/µ| · ci (M )}
The final step is to unwind the definition of φ̂i (M ) to see that ci (M ) = φ̂i (M ) − 1.
We end this section by stating the main result of [NSS]. Let O• be the pre-NSS
data that assigns the value zero to each left-black Maya diagram. This clearly satisfies
the pre-NSS condition and is n-periodic for any n. For any n ≥ 2, let us define the
set of NSS data N = Nn to be the set of all pre-NSS data obtained by applying a
sequence of fˆi operators to O.
Theorem 5.7. [NSS] Let n ≥ 3. Then the set N ∪ {0}, along with the data of
b n.
fˆi , êi , wt, ε̂i , φ̂i form a crystal that is isomorphic to the B(∞) crystal for sl
Their proof is purely combinatorial, depending on a result of Stembridge that only
applies for n ≥ 3. In the next section we will give an independent geometric proof of
this result that includes the case n = 2.
6. Geometric Realization of NSS Data
In this section we will show how to construct the set of NSS data from double MV
c n . Moreover, we will give an independent proof that the NSS crystal
cycles for SL
34
b 2 , where the
is the B (∞)-crystal. In particular, our proof will work in the case of sl
combinatorial methods of [NSS] do not apply.
c n . Fix a coweight λ ∈ Λ+ . For
Let us fix an integer n ≥ 2, and let G = SL
◦
each left-black Maya diagram γ, we define a function Dγ : Fλ (C) → Z as follows:
Dγ ([g]) = val(hγ| · g). Here, [g] is an element of S 0 ∩ T −λ , which we view as a subset
of G (K) /G(O).
Proposition 6.1. The functions Dγ are constructible.
◦
Proof. By the discussion of Zastava spaces in section 3.2, Fλ maps into the
moduli space of N bundles on P1 trivialized away from 0 (we view this space only as
a functor, ignoring issues of representablity). In particular, we have a universal N
◦
◦
bundle FN on P1 × Fλ that is trivialized on (P1 − 0) × Fλ . Let us form the associated
vector bundle with fibers equal to the Fock space F− . As N is pro-unipotent, FN
trivializes on any affine open cover, and there is no difficulty in constructing this
◦
associated bundle. Then hγ| ∈ F− determines a section of this bundle on (P1 −0)× Fλ
by means of the trivialization.
◦
Then the function Dγ on Fλ is given by taking the order of the pole of this section
◦
on (P1 − 0) × x for any point x ∈ Fλ . We can check this explicitly by restricting to the
formal punctured disk around 0 in P1 and choosing a trivialization of FN that extends
over the full (non-punctured) formal disk. The difference of the two trivializations on
the punctured disk gives us a lift of the point x to N (K) via the Zastava interpretation
◦
of the bijection Fλ (C) ' S 0 ∩ T −λ (c.f the proof of Proposition 3.8). Using this we
can easily see that the order of the pole of the section of the associated bundle coming
from hγ| is exactly Dγ (x).
Now, if we had a global trivialization of FN , then this section could be viewed as
ˆ
a map from Fλ to F− ⊗K,
which we view as an ind-scheme (here we are identifying
Spec K with the formal punctured disk centered at 0 in P1 ). The function val :
ˆ → Z ∪ +∞ is constructible (in fact, it has locally closed fibers), and the
F− ⊗K
function Dγ would be the pull back of this function to Fλ via this section.
35
We do not have a global trivialization, but this bundle will trivialize on on any
affine cover. So we have proved that Dγ is constructible when restricted to every
◦
affine open, from we easily see that Dγ is in fact constructible on the whole of Fλ . Because the function is constructible, on each double MV cycle Z, there will be a
dense open subset on which it takes a single value, which we call the generic value
of Dγ . To Z we can associate a pre-NSS datum M (Z) defined as follows. For every
left-black Maya diagram γ, we set:
(8)
M (Z)γ = the generic value of Dγ on Z
Abusing notation, we will write M (U )γ = M (Z)γ if U is a dense subvariety of Z.
Remark 6.2. The idea of defining these constructible functions in the case of a
finite-dimensional group is due to Kamnitzer [Kam1], who credits it to a discussion
with D. Speyer.
Proposition 6.3. Geometric Construction of NSS Crystal Operator Let
Z be a double MV cycle, and let M (Z) be its associated pre-NSS datum. Then,
fˆi M (Z)• = M (fi Z)•
(9)
Here fi is the geometrically defined crystal operator on double MV cycles, and fˆi is
the combinatorially defined crystal operator from the previous section.
Proof. By Corollary 3.9, fi Z contains a dense subvariety U whose C-points can
each be written in the form xi (p) · z, where val(p) = φi (Z) − 1 and z ∈ Z. Because
U is dense, we have
M (fi Z)γ = M (U )γ
Using the fact that every point of U is of the form mentioned above, we can use
Lemma 4 to compute:
36
M (U )γ =
n
o
Mµ + |γ\µ| · φ̂i (M ) − 1 ,
min
µ obtained by removing
i-colored boxes from γ
which is precisely equal to f̂i M (Z)• by Lemma 5.6.
This geometric construction also explains the somewhat mysterious operator Θ
from Definition 5.1. Let M = M (Z) be the pre-NSS datum corresponding to a
double MV cycle Z. Then for any right-black Maya diagram τ , we can define Θ (M )τ
as before. For right-black diagrams τ , we can define Dτ analagously as we did for
left-black diagram using the action on F+ . As before, Dτ is constructible:
Proposition 6.4. The functions Dτ are constructible.
Proof. The proof of Proposition 6.1 carries through except for one subtlety. The
ˆ → Z ∪ ±∞ is no longer constructible. However, the
valuation function val : F+ ⊗K
fibers of each point except −∞ are locally closed, and by the discussion of “good”
◦
elements in [BFK], the function Dγ never takes the value −∞ on Fλ . So the argument
still works.
Then a calculation using charged partitions gives us the following very nice formula:
Proposition 6.5. Θ (M )τ = the generic value of Dτ on Z
Proof. Let Z be a double MV cycle as above, and let M = M (Z) be the
associated NSS datum. Let τ be a right-black Maya diagram. Then recall that
Θ (M )τ = MγI , where I is a sufficiently large interval in Z, and γI is the left-black
Maya diagram obtained by inverting all the colors of τ outside the interval I.
Let us now interpret this in terms of charged partitions. We view diagram τ as
an upward-facing charged partition, and γI is a downward-facing charged partition.
Both of these partitions are colored by the integers modulo n + 1. Then it is easy
37
to see that removing any number of boxes of a fixed color from γI corresponds bijectively to adding boxes of the same color to τ . Moreover, there is an integer N such
that for any sequence (i1 , · · · iN ) of colors, the process of removing boxes of color i1 ,
followed by removing boxes of color i2 , and so on up to removing boxes of color iN
from γI corresponds bijectively to a process of adding boxes of the same colors to τ .
Furthermore, by choosing I large enough, we can choose N arbitrarily large.
However, removing boxes from γI is exactly how the Chevalley generators act on
F− , and adding boxes to τ is how the Chevalley generators act on F+ . We thus see
that DγI (z) = Dτ (z) agree for all z ∈ Z that can be written using a product of fewer
than N Chevalley subgroups. But this is always true for z in an open dense subset of
Z by Corollary 3.9 and the fact that every double MV cycle is the result of applying
a finite number of crystal operators to the unique double MV cycle of weight 0.
As before, let L denote the set of double MV cycles, and let N denote the set
of NSS data. The previous lemma implies that the operation M defines a map
M : L → N , i.e. the pre-NSS datum associated to an double MV cycle is in fact
an NSS datum. Moreover, by the definition of NSS data, we see that this map is
surjective. With these observations, we state our main theorem.
Theorem 6.6. The map M is an isomorphism of crystals.
Proof. In light of the previous proposition, we only need to prove that the map
M is a bijection. As we observed above, M is surjective, so we only need to prove
injectivity.
First notice by induction that the i-string functions εi and φi and the weight
function wt also commute with the map M .
We proceed by induction on the weight (more precisely on the height of the
weight). Because there is only one double MV cycle of weight 0 corresponding to
38
the unit point 1 ∈ N (K), we immediately see that this double MV cycle is determined by its NSS datum. Thus we have the base case for our induction.
Let W and Z be double MV cycles of strictly negative weight with M (W ) = M (Z),
and suppose M is a injection for all double MV cycles of larger weight. Because W
and Z are not weight 0, by the general structure of the B(∞) crystal, there exists
an i and j so that ei (Z) 6= 0 and ej (W ) 6= 0 (recall the operators ei and ej are the
partial inverses of fi and fj from Definition 3.4). In the B(∞) crystal, ei (Z) 6= 0
iff εi (Z) 6= 0. Moreover, εi (Z) = εi (W ) because this quantity can be computed
combinatorially from the NSS datum.
So there exists i such that ei (Z) 6= 0 and ei (W ) 6= 0. Thus we can write
W = fi ei (W ) and Z = fi ei (Z). Now we can apply Propostion 6.3 to get M (W ) =
fˆi M (ei (W )) and M (Z) = fˆi M (ei (Z)). Since M (W ) = M (Z), we can apply êi (the
partial inverse to fˆi from Theorem 5.7) to get M (ei (W )) = M (ei (Z)). By induction,
ei (W ) = ei (Z). Applying fi to both sides, we get W = Z
From this theorem, we immediately deduce the following corollary.
b n . In particular,
Corollary 6.7. The NSS crystal N is the B (∞) crystal for sl
b 2.
this holds for the previously unknown case of sl
39
CHAPTER 3
Affine PBW Bases and MV Polytopes in rank 2
This chapter comprises work completed in collaboration with Peter Tingley.
1. Overview
Let g be a symmetrizable Kac-Moody algebra. To each irreducible lowest weight
representation V (−λ) Kashiwara associates a “crystal” B(−λ), which is a combinatorial object that records leading order behavior of the representation. The crystals
B(−λ) form a directed system whose limit B(−∞) can be thought of as the crystal
for U + (g). The foundations of this theory makes heavy use of the quantized universal
enveloping algebra associated with g, but the resulting crystals can often be realized
in other ways.
When g is of finite type, one interesting realization of B(−∞) is via the theory of
MV (Mirković-Vilonen) polytopes. The MV polytope M Vb associated to an element
b ∈ B(−∞) arises in (at least) four natural ways:
(i) The original construction as the moment map image of a certain MV cycle in
the affine Grassmannian, due to Anderson [And] and studied by Kamnitzer
[Kam1, Kam2].
(ii) As the convex hull of the dimension vectors of all subrepresentations of
a certain preprojective algebra representation. This comes from work of
Baumann-Kamnitzer [BK] on quiver varieties.
(iii) As the convex hulls of the paths defined by the PBW monomials corresponding to b with respect to all reduced expressions for the longest word w0 in
the Weyl group. This is a rewording of results in [Kam1], which in turn
makes use of Lusztig’s explicit calculations in [Lus4].
40
(iv) As the character-polytope of an irreducible representation of a KhovanovLauda-Rouquier algebra. This is done by the second author and Webster in
[TW], building on ideas of Kleshchev and Ram [KR].
In addition, MV polytopes can be approached from a purely combinatorial point of
view, using the following result from [Kam1]:
(v) The MV polytopes are exactly those convex polytopes all of whose edges are
parallel to roots, and all of whose 2-faces satisfy a combinatorial condition
called a “tropical Plücker relation.”
The current paper is part of an effort to develop a version of this story in affine
type. There is not yet a true affine analogue of (i), although see [BFG] for a definition
of (open) MV cycles in untwisted affine types, and [Muth, NSS] for some related
combinatorial results. The other constructions are to various extents understood in
the affine situation:
(ii) The first definition of MV polytopes in symmetric affine types was given in
[BKT] using quiver varieties. In that work, an MV “polytope” is actually
a decorated polytope; in addition to the underlying polytope, one must
include the data of a partition (of some N ) associated to each co-dimension
1 face parallel to the imaginary root δ (this is a rewording of [BKT], see
[TW] for this statement). As in finite type the underlying polytope is the
convex hull of the dimension vectors of all subrepresentations of a certain
preprojective algebra representation.
(iii) Affine PBW type crystal bases were defined in [BCP, Aka, BN]. Prior to
the present work, no connection has been made with affine MV polytopes,
although the format of the combinatorics is similar (such as partitions associated to imaginary root directions).
(iv) In [TW], KLR algebras are used to give a construction and combinatorial
characterization of MV polytopes in all affine types. That work makes use
of the current paper to describe how the affine MV polytopes coming from
KLR algebras are related to those coming from quiver varieties in [BKT].
41
(v) It is also shown in [BKT, TW] that affine MV polytopes can be characterized by conditions on 2-faces, which reduces the problem of providing
a combinatorial description to understanding the rank 2 affine cases. See
[BKT] for the symmetric case, and [TW] for the general type affine case.
Finally, in [BDKT] a combinatorial definition of MV polytopes for the two rank-2
affine cases was given, but it was not determined whether this agreed with [BKT] in
b 2.
the relevant case of sl
This leaves several important questions unanswered. First, one would like to show
that the combinatorics in [BDKT] matches the rank 2 case of the construction in
[BKT]. Second, one would like to relate affine MV polytopes with affine PBW bases.
Third, one would like to relate affine MV polytopes to geometry along the geometry
in [BFG]. The current paper addresses the first question and the rank two case of
the second.
Our result showing that the combinatorics in [BDKT] matches the rank 2 case
of the construction in [BKT] is the last step in providing a combinatorial characterization of symmetric affine MV polytopes.
The relationship we describe between rank 2 affine MV polytopes and affine PBW
bases essentially answers the rank 2 case of the problem posed by Beck-Nakajima in
[BN, Remark 3.29]. In rank 2 there are exactly two PBW bases, and each basis
is in bijection with the crystal B(−∞). Hence there is a natural bijection between
these two bases. Answering Beck-Nakajima’s question amounts to giving an explicit
description of this bijection. Our results show that this bijection is given exactly by
the combinatorics in [BDKT]. This should be thought of as a rank-2 affine analogue
of Lusztig’s piecewise-linear bijections from [Lus4]. We believe our methods can be
used to relate affine MV polytopes and PBW bases more generally, and thus to answer
Beck-Nakajima’s question in all affine types. We plan to address this in a future work.
Our main tool is a new characterization of rank-2 affine MV polytopes, or more
precisely of the map that takes an element of B(−∞) to its MV polytope. We show
that this is the unique map to decorated polytopes satisfying some conditions related
42
to the crystal operators and Saito’s crystal reflections, and one other condition to hanb 2 cases, we consider three a priori different maps
dle the imaginary roots. For the sl
from B(−∞) to decorated polytopes: the combinatorial construction from [BDKT],
the geometric construction using quiver varieties from [BKT], and the algebraic construction from PBW bases. We show that all three maps satisfy the conditions of our
(2)
uniqueness theorem. In particular, they all agree. In the A2 case, the first two of
these constructions make sense, and we show that these also agree.
This chapter is organized as follows: In §2 we briefly give some general background
on quantum affine algebras and crystals. In §3 we recall the combinatorial definition
of rank-2 affine MV polytopes from [BDKT] and prove our characterization theorem
(Theorem 3.11). In §4 we present some background on affine PBW bases, define a
map from these to decorated polytopes, and prove that the resulting PBW polytopes
are in fact identical to the combinatorially defined MV polytopes (Theorem 4.17). In
b 2 MV polytopes from [BKT], show that the result
§5 we recall the construction of sl
(after a minor change of conventions) also agrees with the combinatorial construction
(Theorem 5.9), and state the final combinatorial characterization of MV polytopes
in all symmetric affine types (Theorem 5.12). Note that §4 and §5 are completely
independent of one another.
2. Background
2.1. Quantum Affine Algebras and Canonical Basis. Our notation for
quantum affine algebras will mostly follow [BN, §2.2], and we refer the reader there
for more details. Here we only recall the key points, and we will only consider the
b 2 and A(2) , which have rank-2 Dynkin diagrams. We
quantum affine algebras for sl
2
label the nodes of those Dynkin diagram by 0, 1 (thought of as integers mod 2), where
(2)
for A2 the node 0 corresponds to the long root. Throughout:
b 2 or A(2) .
• g is the Kac-Moody algebra sl
2
b 2 or A(2) .
• U is the quantized universal enveloping algebra for sl
2
• E0 , E1 , F0 , F1 are the standard Chevalley generators.
43
• U+ is the subalgebra of U generated by E0 , E1 .
• α0 and α1 are the simple roots for g.
• P is the weight lattice of g.
• Kashiwara’s involution ∗ is the algebra anti-involution of U which fixes all
the Chevalley generators. Notice that ∗ preserves U+ .
b 2 , τ is the algebra involution of U induced by the Dynkin diagram
• For sl
automorphism. τ also preserves U+ .
• T0 , T1 are the generators of the braid group acting on U (which in the rank
two affine cases is just the free group on two generators). By e.g. [Sai,
Corollary 1.3.3], Ti ◦ ∗ = ∗ ◦ Ti−1 .
• B is Lusztig’s canonical basis (equivalently Kashiwara’s global crystal basis)
of U+ .
• A is the ring consisting of all rational functions in C(q) which are regular
at q = ∞, and L = spanA B. Recall that B descends to a crystal basis of
L/q −1 L.
• ẽi , f˜i are Kashiwara’s crystal operators on B (as a crystal basis for L/q −1 L).
2.2. Crystals. We are interested in the crystal B(−∞) associated with U+ (g).
This section contains a brief review of some properties of this object, and we refer
the reader to e.g. [Kas] or [HongKang] for more details. Recall that B(−∞) is a
set along with operators ẽi , f˜i : B(−∞) → B(−∞) t {∅} for each i ∈ I. These satisfy
• B(−∞) has a unique element b0 such that f˜i (b0 ) = 0 for all i.
• Every element of B(−∞) can be obtained from b0 by applying some sequence
of operators ei for various i.
• For b, b0 ∈ B(−∞), ẽi (b) = b0 if and only if f˜i (b0 ) = b.
• There is a unique map wt : B(−∞) → P such that wt(b0 ) = 0 and each
operator ẽi has weight αi , the corresponding simple root.
There are also important functions ϕi , εi : B(−∞) → Z defined by
(10)
ϕi (b) = max{n ∈ Z≥0 : fin b 6= ∅},
44
and εi (b) = ϕi (b) − hαi∨ , wt(b)i.
Finally, there is a weight preserving involution ∗ : B(−∞) → B(−∞) induced by
Kashiwara’s involution on U+ . Define f˜i∗ = ∗f˜i ∗, ẽ∗i = ∗ẽi ∗, ϕ∗i = ϕ ◦ ∗ and ε∗i = ε ◦ ∗.
By [KS, Proposition 3.2.3], for every i and every b ∈ B(−∞), the subset of
B(−∞) generated by b by ẽi , ẽ∗i , f˜i and f˜i∗ looks like
•
•
•
•
•
•
•
•
•
•
•
•
•
•
(11)
•
•
•
•
b
•
•
εi (b)
ϕi (b)
where the solid and dashed arrows show the action of ẽi , and the dotted or dashed
arrows denote the action of ẽ∗i . Here the width of the diagram at the top is εi (b− ),
where b− is the bottom vertex. For any b in this component such that ϕ∗i (b) = 0 (i.e.
such that there are no dotted or dashed arrows pointing to b), ϕi (b) is the longest
path formed by solid arrows that ends at b, and εi (b) is the longest path formed by
solid arrows that starts at b.
The following operation σi comes from [Sai, Corollary 3.4.8]. We will refer to this
as a Saito reflection.
ϕi (b)
Definition 2.1. For b ∈ B(−∞) such that ϕ∗i (b) = 0 , define σi b = (ẽ∗i )i (b) f˜i
b.
As one would expect from the name, σi has the property that wtσi (b) = si wt(b)
where si is the ith simple reflection. Note however that this is only true provided
ϕ∗i (b) = 0. There is also a notion of dual Saito reflection defined by σi∗ b = ∗σi ∗ which
acts as a reflections for those b ∈ B(−∞) such that ϕi (b) = 0. For our purposes, it is
useful to have the following alternative characterization of σi :
Proposition 2.2. Fix b ∈ B(−∞). If ϕ∗i (b) = 0, then σi b = f˜imax (ẽ∗i )N b for any
N ≥ εi (b). Similarly, if ϕi (b) = 0, then σi∗ b = (f˜i∗ )max (ẽi )N b for any N ≥ ε∗i (b).
45
Proof. Consider a component of B(−∞), as shown in (11). The nodes where
ϕ∗i (b) = 0 are exactly those that have no dotted or dashed arrows pointing towards
them, for instance the element b in the diagram in that section. It is clear form the
picture that the two formulas have the same effect on b.
3. Rank 2 Affine MV polytopes
b 2 and A(2) correspond
3.1. Rank 2 affine root systems. The root sysems for sl
2
to the affine Dynkin diagrams
•
•
b 2:
sl
1 ,
0
(2)
A2 :
•
0
•
1 .
The corresponding symmetrized Cartan matrices are




2 −2
8 −4
(2)
b2 : N = 
,
.
sl
A2 : N = 
−2
2
−4
2
b 2 and δ = α0 + 2α1 for
Denote the simple roots by α0 , α1 . Define δ = α0 + α1 for sl
(2)
(2)
A2 . Note that we have chosen α0 to be the long root for A2 , which is opposite
from the conventions in [Kac, BDKT]. We have instead followed the convention in
[Aka, BN] which is more convenient for the theory of affine PBW bases.
The type g weight space is a three dimensional vector space containing α0 , α1 .
This has a standard non-degenerate bilinear form (·, ·) such that (αi , αj ) = Ni,j .
Notice that (α0 , δ) = (α1 , δ) = 0. Fix fundamental coweights ω0 , ω1 which satisfy
(αi , ωj ) = δi,j , where we are identifying coweight space with weight space using (·, ·).
b 2 is
The set of positive roots for sl
(12)
{α0 , α0 + δ, α0 + 2δ, . . .} t {α1 , α1 + δ, α1 + 2δ, . . .} t {δ, 2δ, 3δ . . .},
where the first two families consist of real roots and the third family consists of
(2)
imaginary roots. The set of positive roots for A2 is
(13)
{α1 + kδ, α0 + 2kδ, α1 + α0 + kδ, 2α1 + (2k + 1)δ | k ≥ 0} t {kδ | k ≥ 1},
where the first set consists of real roots and the second set of imaginary roots. We
draw these in the plane as
46
...
..
.
..
.
α0 + 3δ
kδ
.
..
..
.
α0 + 2δ
α1 + 3δ
α0 + 2δ
α1 + 2δ
α0 + δ
α0
α1 + δ
α1
2α1 + 3δ
kδ
..
.
2α1 + δ
α0
α1
(2)
A2
b2
sl
3.2. Lusztig data and pseudo-Weyl polytopes.
Definition 3.1. A Lusztig datum c is a choice of a non-negative integer cβ for
each positive real root β, all but finitely many of which are 0, and a partition cδ . The
P
weight of c is wt(c) = |cδ | · δ + β real cβ · β.
Remark 3.2. Notice that the data of a partition cδ is equivalent to the data of a
number c0kδ for each positive imaginary root kδ, where c0kδ is the number of parts of
cδ of size exactly k. Thus a Lusztig datum is equivalent to a Kostant partition.
If a Lusztig datum c has cβ = 0 for all positive real roots, we call c purely
imaginary. We will sometimes abuse notation and write simply λ to denote the
purely imaginary Lusztig datum c with cδ = λ.
(2)
b 2 or A , a decorated pseudo-Weyl polytope P
Definition 3.3. For either sl
2
consists of a pair of Lusztig data c` (P ) and cr (P ) of the same weight. This weight is
called the weight of P .
Remark 3.4. What we are calling a decorated pseudo-Weyl polytope is called a
decorated GGMS polytope in [BKT] and [BDKT]. Our terminology originates
from [Kam1], where the notion of pseudo-Weyl polytope is defined in finite type.
In order to describe these geometrically (and justify the word polytope) we need
some notation. Label the positive real roots by βk , β k for k ∈ Z>0 as follows:
b 2 : βk = α1 + (k − 1)δ and β k = α0 + (k − 1)δ.
• For sl
47
(2)
• For A2 :


α1 + k−1 δ
2
βk =

2α + (k − 1)δ
1
if k is odd,
βk =


α0 + (k − 1)δ

α + α +
0
1
if k is even,
if k is odd,
k−2
δ
2
if k is even.
To a decorated pseudo-Weyl polytope P , we associate an underlying polytope (up
to translation) in the root lattice whose vertices {µrk , µr,k , µ`k , µr,k } are defined by:
µr0 = µ`0
(14)
µrk − µrk−1 = crβk · βk ,
µr,k−1 − µr,k = crβ k · β k
µ`k − µ`k−1 = c`β k · β k ,
µ`,k−1 − µ`,k = c`βk · βk .
Because Lusztig data take the value zero on all but finitely many roots, the vertices
µrk must all coincide for sufficiently large k, as must the vertices µr,k , µ`k , µ`,k . We
denote
(15)
µr∞ = lim µrk , µr,∞ = lim µr,k , µ`∞ = lim µ`k , µ`,∞ = lim µ`,k .
k→∞
k→∞
k→∞
k→∞
See Figure 1.
3.3. Definition and characterization of MV polytopes. The following definition can be found in [BDKT], although we have changed some noation.
Definition 3.5. An MV polytope is a decorated pseudo-Weyl polytope such
that
(i) For each k ≥ 2, max{(µ`k − µrk−1 , ω1 ), (µrk − µ`k−1 , ω0 )} = 0.
(ii) For each k ≥ 2, min{(µ`,k − µr,k−1 , ω0 ), (µr,k − µ`,k−1 , ω1 )} = 0.
(iii) If the vectors µr∞ −µ`∞ and µr,∞ −µ`,∞ are parallel, then crδ = c`δ . Otherwise,
one is obtained from the other by removing a part of size
(iv) (crδ )1 , (c`δ )1 ≤
|α1 |
(µr∞
2|α0 |
|α1 |
(µr∞ − µ`∞ , α1 ).
2|α0 |
− µ`∞ , α1 ), where e.g. (crδ )1 denotes the largest part
of the partition crδ .
See Figure 1. Let MV denote the set of MV polytopes.
48
µ0 α0
• • µr,1 = µr,2
α1
α0 + 2δ
µ`,1
•
µ
`,3
• µr,3 = µr,4 = · · · = µr,∞
•
•
α1 + δ
`,2 •
=µ
α1 + 3δ
•
µ`,∞ = · · · = µ`,5 = µ`,4 •
•
δ •
δ
µ`∞ = · · · = µ`5 = µ`4 •
α0 + 3δ
µ`3•
• µr3 = µr4 = · · · = µr∞
α0 + 2δ
µ`2
•
α1 + 2δ
•µr2
α0 + δ
•
µ`1 α0 µ•
0
α0
•
α1
• αr 1 + δ
µ1
α1
b 2 MV polytope. The partitions labeling the vertical
Figure 1. An sl
edges are indicated by including extra vertices on the vertical edges,
such that the edge is cut into the pieces indicated by the partition.
Here:
crα1 = 2, crα1 +δ = 1, crα1 +2δ = 1, crδ = (9, 2, 1, 1), crα0 +2δ = 1, crα0 = 1,
c`α0 = 1, c`α0 +δ = 2, c`α0 +2δ = 1, c`α0 +3δ = 1, c`δ = (2, 1, 1), c`α1 +3δ = 1, c`α1 +δ = 1, c`α1 = 5.
Theorem 3.6. [BDKT] For each Lusztig datum c, there is a unique Pcr ∈ MV
such that the right Lusztig data of Pcr is given by c. Similarly, there is a unique
Pc` ∈ MV whose left Lusztig data is given by c.
Definition 3.7. Fix P ∈ MV with right and left Lusztig data cr and c` respectively. Then ẽ0 (P ) is the MV polytope with right Lusztig data ẽ0 (cr ) and ẽ1 (P ) is
the MV polytope with left Lusztig datum ẽ1 (c` ), where ẽ0 (cr ) agrees with cr except
that ẽ0 (cr )α0 = crα0 + 1, and ẽ1 (c` ) agrees with c` except that ẽ1 (c` )α1 = c`α1 + 1.
49
Similarly, f˜0 (P ) is the MV polytope with right data f˜0 (cr ) and f˜1 (P ) is the
MV polytope with left Lusztig data f˜1 (c` ), where f˜0 (cr ) agrees with cr except that
f˜0 (cr )α0 = crα0 − 1 and f˜1 (c` ) agrees with c` except that f˜1 (c` )α1 = c`α1 − 1; if crα0 or
c`α1 are zero then f˜0 or f˜1 sends that polytope to ∅.
Theorem 3.8. [BDKT] MV along with the operators ẽ0 , f˜0 , ẽ1 , f˜1 realizes B(−∞).
The following describes how Saito reflections act on MV polytopes.
Proposition 3.9. For any b ∈ B(−∞) with ϕ0 (b) = 0, we have c`α0 (M Vσ0 (b) ) = 0,
and for all other α, c`α (M Vσ0 (b) ) = crs0 α (M Vb ).
Proof. Fix b ∈ B(−∞). By [BDKT, Remark 4.12], for sufficiently large N and
all k ≥ 1, M VẽN0 b has
(µrk − µ`k−1 , ω0 ) and (µ`,k − µr,k−1 , ω0 ) = 0
b 2 , but the same proof works for A(2) ). Hence each
(this is only stated there for sl
2
diagonal (µrk , µ`k−1 ) and (µ`,k , µr,k−1 ) is parallel to α1 , from which it is clear that, for
all α 6= α0 ,
c`α (M VẽN0 b ) = crs0 (α) (M VẽN0 b ) = crs0 (α) (M Vb ).
The result is then immediate from Proposition 2.2 (which says σi (b) = (fi∗ )max eN
i (b)
for large N ).
Definition 3.10. Given an Lusztig datum c satisfying cαi = 0, we define a new
b 2 , we
Lusztig datum c ◦ si by (c ◦ si )αi = 0, (c ◦ si )β = csi (β) and (c ◦ si )δ = c0 . For sl
define c ◦ τ by (c ◦ τ )αi +δ = cαi+1 +δ and (c ◦ τ )δ = cδ .
b 2 or A(2) . There is a unique map b → Pb
Theorem 3.11. Assume g is of type sl
2
from B(−∞) to type g decorated pseudo-Weyl polytopes (considered up to translation)
such that, for all b ∈ B(−∞) and i = 0 or 1, the following hold.
(W) wt(b) = wt(Pb ).
50
(C1) crα0 (Pẽ0 b ) = crα0 (Pb ) + 1, and for all α 6= α0 , crα (Pẽ0 b ) = crα (Pb ).
(C2) c`α1 (Pẽ1 b ) = c`α1 (Pb ) + 1, and for all α 6= α1 , c`α (Pẽ1 b ) = c`α (Pb ).
(C3) c`α0 (Pẽ∗0 b ) = c`α0 (Pb ) + 1, and for all α 6= α0 , c`α (Pẽ∗0 b ) = c`α (Pb ).
(C4) crα1 (Pẽ∗1 b ) = crα1 (Pb ) + 1, and for all α 6= α1 , crα (Pẽ∗1 b ) = crα (Pb ).
(S1) If ϕ0 (b) = 0, then c` (Pσ0 (b) ) = cr (Pb ) ◦ s0 .
(S2) If ϕ1 (b) = 0, then cr (Pσ1 (b) ) = c` (Pb ) ◦ s1 .
(S3) If ϕ∗0 (b) = 0, then cr (Pσ0∗ (b) ) = c` (Pb ) ◦ s0 .
(S4) If ϕ∗1 (b) = 0, then c` (Pσ1∗ (b) ) = cr (Pb ) ◦ s1 .
(I) If c`β (Pb ) = 0 for all real roots β, and c`δ (Pb ) = λ 6= 0, then crα1 (Pb ) =
|α0 |
λ,
|α1 | 1
˜ +.
crδ (Pb ) = λ\λ1 ; crα0 (Pb ) = λ1 ; and crβ (Pb ) = 0 for all other β ∈ ∆
This map takes each b ∈ B(−∞) to its MV polytope M Vb as defined in §3.1.
Remark 3.12. One can easily see that Theorem 3.11 remains true if (C3), (C4),
(S3) and (S4) are replaced with the single condition
(K) for all b, Pb∗ = −Pb (up to translation) where ∗ is Kashiwara’s involution.
This gives a shorter statement, but the version given here is a-priori stronger.
Remark 3.13. A simplified version of Theorem 3.11 also holds in finite-type rank2 cases, where one only needs conditions (W), (C1), (C2), (S1) and (S2). This was
previously observed in [BK] (see the discussion just before Remark 27). The proof is
also contained in the proof of Theorem 3.11 below.
Remark 3.14. The combinatorics from [BDKT] is only used in the existence
part of the proof below. One could instead prove existence using the construction of
these polytopes describe in §4 or §5, so our characterization does not fundamentally
rely on [BDKT].
b 2;
Proof of Theorem 3.11. We give the details of the proof only for the case sl
(2)
the case of A2 proceeds by the same argument, but the notation gets messier.
We first show that the map b → M Vb has all the required properties. Properties
(W) and (C1)-(C4) are immediate from the definitions of the crystal operators in
§3.1. Property (S1) is Proposition 3.9, and (S2)-(S4) follow by symmetric arguments.
51
To see (I), notice that
r
• c α0 = λ1
•
`
crδ = λ\λ1
cδ = λ
•
• c r = λ1
α1
(16)
is an MV polytopes according to Definition 3.5. Since by Theorem 3.6 there is exactly
one MV polytope with each right (or left) Lusztig datum, (I) follows.
Now suppose we have a map b → Pb that satisfies the conditions above. It suffices
to show that cr (Pb ) = cr (M Vb ) and c` (Pb ) = c` (M Vb ) for all b. We proceed by
induction, the base case where b is the lowest weight element being trivial by (W).
Consider the partial order on Lusztig data where c < c̃ if
• |cδ | < |c̃δ |, or
• |cδ | = |c̃δ | and #{α | cα 6= 0} < #{α | c̃α 6= 0}.
Fix a Lusztig datum c, and make the inductive hypothesis that cr (Pb ) = cr (M Vb )
whenever cr (Pb ) < c and that c` (Pb ) = c` (M Vb ) whenever c` (Pb ) < c. We will show
that, if an element bc ∈ B(−∞) has c` (Pbc ) = c, then c` (M Vbc ) = c as well.
First assume that cα 6= 0 for some real root α. Find k minimal such that cα0 +kδ
or cα1 +kδ is non-zero. We proceed in the case cα1 +kδ 6= 0, the other case following by
a similar argument (using ∗ reflections and operators instead of the unstarred ones).
Let
c
α1 +kδ
σk · · · σ1 bc ,
b0 = f˜k+1
where we read subscripts modulo 2. If k is even, it follows from axioms (S1) and (S2)
and (C2) that
c` (Pbc ) = c` (M Vbc ) if and only if c` (Pb0 ) = c` (M Vb0 ).
But c` (Pb0 ) < c, so by induction c` (Pb0 ) = c` (M Vb0 ), hence c = c` (Pbc ) = c` (M Vbc ).
If k is odd, it follows from axioms (S1) and (S2) and (C1) that
c` (Pbc ) = c` (M Vbc ) if and only if cr (Pb0 ) = cr (M Vb0 ).
52
Again cr (Pb0 ) < c so by induction cr (Pb0 ) = cr (M Vb0 ), hence c = c` (Pbc ) = c` (M Vbc ).
Now assume cα = 0 for every real root α. Let c0 be the Lusztig data defined
by c0δ = cδ \(cδ )1 , c0α0 = c0α1 = (cδ )1 , and cβ = 0 for all other real roots β. By (I),
cr (Pbc ) = c0 , and by induction cr (M Vbc ) = c0 . Since we already know that (16) is the
unique MV polytope such that cr (M Vbc ) = c0 , this implies c` (M Vbc ) = c.
To complete the argument, we must also show that for any element b0c such that
cr (Pb0c ) = c we also have cr (M Vb0c ) = c. This proceeds by an identical argument, with
the only subtlety being that (I) is not symmetric. However, if cα = 0 for every real
root α, then we proceed by noticing that, by (S1), cr (Pb0c ) = cr (M Vb0c ) if and only if
c` (M Vσ0 b0c ) = c` (Pσ0 b0c ), so the previous argument applies.
The following modification of Theorem 3.11 will be needed when we consider the
pseudo-Weyl polytopes arising from PBW bases. It is immediate from the proof
above.
Proposition 3.15. Fix a partition λ. Assume all the conditions of the Theorem
3.11 hold except that (I) is only known to hold for all partitions µ with |µ| < |λ| or
µ = λ. Then the c` (M Vb ) = c` (Pb ) for all b with |c`δ (Pb )| < λ or c`δ (Pb ) = λ. Similarly
cr (M Vb ) = cr (Pb ) for all b with |crδ (Pb )| < λ or crδ (Pb ) = λ.
4. Rank-2 affine MV polytopes from PBW bases
4.1. Definition and basic properties of rank-2 affine PBW bases. For
b 2 and A(2) , we consider the following two PBW bases coming from the work of
sl
2
Beck-Chari-Pressley, Akasaka, and Beck-Nakajima [BCP, Aka, BN]. Following
Beck-Nakajima, for each Lusztig datum c define two elements of U+ by
(cα1 )
T1−1 (E0 )(cs1 (α0 ) ) · · · Scδ · · · T0 (E1 )(cs0 (α1 ) ) E0
(cα0 )
T0−1 (E1 )(cs0 (α1 ) ) · · · T0−1 (Scδ ) · · · T1 (E0 )(cs1 (α0 ) ) E1
(17)
L(c, 0) := E1
(18)
L(c, 1) := E0
(cα0 )
,
(cα1 )
.
All the notation here except Scδ is defined in §2.1. We do not need to define Sλ
exactly; it will be enough to recall that it is a polynomial in the commuting variables
53
ψ̃k for k ≥ 1, and that for all λ, µ, Sλ Sµ =
P
ν
aνλ,µ Sν where aνλ,µ are the Littlewood-
b 2 , the vectors ψ̃k (which is denoted ψ̃k,1 in [BCP]) are
Richardson coefficients. For sl
defined by
ψ̃k := Ekδ−α1 Eα1 − q −2 Eα1 Ekδ−α1 ,
(19)
where by definition Ekδ−α1 = T0 T1 · · · Tk−2 (Ek−1 ) (here subscripts are taken modulo
(2)
2). For A2 , ψ̃k are defined in [Aka, Definition 3.3] by
ψ̃k := Eδ−α1 E(k−1)δ+α1 − q −1 E(k−1)δ+α1 Eδ−α1 ,
(20)
where by definition Eδ−α1 = T0 (E1 ) and E(k−1)δ+α1 = (T1−1 T0−1 )k−1 (E1 ).
Remark 4.1. Our notation regarding Lusztig data is slightly different from that
of Beck-Nakajima. For them, c is a sequence of non-negative integers indexed by the
integers, and what we call cδ is denoted c0 . We have translated their results into our
notation.
It is well known that B is a crystal basis of L/q −1 L, and it is shown in [BCP, Aka,
BN] that {L(c, i) + q −1 L : c is a Lusztig datum} is also a crystal basis of L/q −1 L.
By the uniqueness of crystal basis (see e.g. [Kas, Theorem 8.1]), we see that:
Theorem 4.2. For each i, {L(c, i) + q −1 L : c is a Lusztig datum } = B + q −1 L.
Thus for each i the PBW basis elements index B. Using this indexing some of the
crystal operators are given by explicit formulas:
Proposition 4.3. [BN, Formula 5.3]
(i) ẽ1 L(c, 0) = L(ẽ1 c, 0)
(ii) ẽ∗0 L(c, 0) = L(ẽ0 c, 0)
(iii) ẽ0 L(c, 1) = L(ẽ0 c, 1)
(iv) ẽ∗1 L(c, 1) = L(ẽ1 c, 1)
Here ẽi c is the Lusztig datum with (ẽi c)αi = cαi +1 and otherwise agreeing with c.
54
Beck-Nakajima actually define a whole family of PBW bases {L(·, i)}, one for
each i ∈ Z. For i ≥ 1 the basis vectors are given by
(cα
)
(cαi )
−1
−1
L(c, i) := Ei−1i−1 Ti−1
(Ei−2 )(csi−1 (αi−2 ) ) · · · Ti−1
· · · T0−1 (Scδ ) · · · Ti (Ei+1 )(csi (αi+1 ) ) Ei
,
where all subscripts are taken modulo 2. There is similar formula for i ≤ 0. However,
in the rank-2 cases we are considering, these various bases all coincide with either the
case i = 0 or i = 1 (see Proposition 4.6 below).
b 2 this is also equal to τ ψ̃k . In particular, for
Lemma 4.4. ψ̃k∗ = T1 ψ̃k . In type sl
b 2 this is also equal to τ Sλ .
any partition λ, Sλ∗ = T1 Sλ and in type sl
(2)
b 2 , the statement
Proof. For type A2 this is [Aka, Proposition 3.26 (ii)]. For sl
T1 (ψ̃k ) = τ ψ̃k comes from [Beck, Proposition 2]. To finish it suffices to show that, in
b 2 , ψ̃ ∗ = τ ψ̃k .
type sl
k
Since ∗ is an algebra anti-automorphism one can easily calculate from (19) that
∗
ψ1,1
= τ ψ1,1 . For k > 0, [BCP, Proposition 1.2] gives the following:
ψ̃l+m = Elδ−α1 Emδ+α1 − q −2 Emδ+α1 Elδ−α1 for l > 0, m ≥ 0.
(21)
Here Elδ−α1 = T0 T1 · · · Tl−2 (El−1 ) and Emδ+α1 = T1−1 T2−1 · · · Tm−1 (Em+1 ).
∗
= τ E(l−1)δ+α1 and
Using the relation Ti ◦ ∗ = ∗ ◦ Ti−1 , it follows that Elδ−α
1
∗
Emδ+α
= τ E(m+1)δ−α1 . Replacing the pair (l, m) by (m + 1, l − 1) in (21) gives
1
ψ̃k∗ = τ ψ̃k .
Lemma 4.5. For all k, T0 T1 (ψ̃k ) = ψ̃k . In particular, for any λ, T0 T1 (Sλ ) = Sλ .
b 2 , this follows from Lemma 4.4 since T0 T1 (ψ̃k ) = τ T0 τ T1 (ψ̃k ).
Proof. It type sl
(2)
For A2 this is [Aka, Proposition 3.26 (i)].
Proposition 4.6. Let i, j ∈ Z be congruent modulo 2, and let c be any Lusztig
datum. We then have L(c, i) = L(c, j).
Proof. Suppose both i, j ≥ 1. By the definition of L(c, i), it suffices to prove
that
−1
−1
Ti−1
· · · T1−1 T0−1 (Scδ ) = Tj−1
· · · T1−1 T0−1 (Scδ ).
55
But this follows immediately from Lemma 4.5. The other cases are similar.
Proposition 4.7. Fix i = 0 or 1 and let c be a Lusztig datum. Then L(c, i)∗ =
L(c, i − 1), and τ L(c, i) = L(c ◦ τ, i − 1). Furthermore if cαi = 0 then L(c ◦ si , i) =
Ti L(c, i − 1).
Proof. First assume cα0 = 0.
(cα1 ) −1
(cs1 (α0 ) )
(cs0 (α1 )) (cα0 )
T1 L(c, 0) = T1 E1 T1 (E0 )
· · · Scδ · · · T0 (E1 )
E0
(cα0 )
= (E0 )(cs1 (α0 ) ) T0−1 (E1 )(cs1 s0 (α1 ) ) · · · T1 Scδ · · · T1 T0 (E1 )(cs0 (α1 )) T1 E0
(cα0 )
= (E0 )(cs1 (α0 ) ) T0−1 (E1 )(cs1 s0 (α1 ) ) · · · T0−1 T0 T1 Scδ · · · T1 T0 (E1 )(cs0 (α1 )) T1 E0
(cα0 )
= (E0 )(cs1 (α0 ) ) T0−1 (E1 )(cs1 s0 (α1 ) ) · · · T0−1 Scδ · · · T1 T0 (E1 )(cs0 (α1 )) T1 E0
,
where the last equality is from Lemma 4.5.
The statement that if cα0 = 0 then L(c◦si , 1) = T0 L(c, 0) follows by a similar (and
slightly shorter) argument. The statements that L(c, i)∗ = L(c, i − 1) and τ L(c, i) =
L(c ◦ τ, i − 1) follow by the same logic, but using Lemma 4.4 in place of Lemma 4.5,
and making use of [Sai, Corollary 1.3.3] which says that Ti ◦ ∗ = ∗ ◦ Ti−1 .
4.2. Relationship of PBW bases with the canonical basis.
Definition 4.8. For each i and Lusztig datum c, denote by b(c, i) the unique
element of B that coincides with L(c, i) in L/q −1 L.
Definition 4.9. The partial order ≺0 on Lusztig data is defined as follows: For
any Lusztig datum c, we form two infinite tuples
c+0 = (cα1 , cs1 (α0 ) , · · · ) and c−0 = (cα0 , cs0 (α1 ) , · · · )
We say c ≺0 c0 if
c+0 ≤ c0+0 and c−0 ≤ c0−0
with one of these inequalities strict. Here, ≤ is the left-to-right lexicographic order.
The partial order ≺1 is defined in the same way by swapping the roles of 0 and 1.
56
Lemma 4.10. [BN, Lemma 3.30] Fix i modulo 2. Let c and c0 be Lusztig data.
Write:
L(c, i)L(c0 , i) =
X
00
acc,c0 L(c00 , i)
c00
Then every Lusztig datum c00 that shows up in the righthand side sum satisfies:
c00+i ≥ c+i and c00−i ≥ c0−i .
Theorem 4.11. [BN, Theorem 3.13] The change of basis matrix between {L(c, i) :
c is a Lusztig datum } and {b(c, i) : c is a Lusztig datum } is upper triangular with
1’s on the diagonal with respect to the partial order ≺i on Lusztig data defined above.
The following is immediate from certain geometric constructions of the canonical
basis and B(−∞), but we give an algebraic proof for completeness.
(n)
Proposition 4.12. Fix b ∈ B. Write Ei b =
P
b0
ab,b0 b0 . Then ab,ẽni b 6= 0.
Proof. For notational clarity we present the proof for i = 1; the case i = 0 uses
exactly the same argument. There is a unique Lusztig datum c such that b = b(c, 0).
By Theorem 4.11
(22)
b(c, 0) = L(c, 0) +
X
ac,c0 L(c0 , 0).
c 0 0 c
Multiplying both sides by E1 ,
(23)
(n)
(n)
E1 b(c, 0) = E1 L(c, 0) +
X
(n)
ac,c0 E1 L(c0 , 0).
c0 0 c
(n)
(n)
But we know that up to scaling by a quantum integer E1 L(d, 0) = ẽ1 L(d, 0).
So after scaling, each term on the right hand side lies in the PBW basis. Rewrite
(n)
each term using the canonical basis. Again using Theorem 4.11 only E1 L(c, 0) will
(n)
contribute the the coefficient of ẽ1 b(c, 0). Thus there is no cancellation and the
Proposition holds.
57
Theorem 4.13. [Sai, Proposition 3.4.7] Suppose b is an element of the canonical
basis such that Ti (b) ∈ U+ . Then Ti (b) lies in the crystal lattice, and
Ti (b) ≡ σi b in L/q −1 L.
(24)
Using this formula and our explicit description of braid operators on PBW basis
vectors, we have the following corollary to Saito’s theorem. A version of this formula
is mentioned in [BN, Remark 3.29].
Corollary 4.14. If c is a Lusztig datum with cαi = 0 then σi b(c, i − 1) ∈ L, and
b(c ◦ si , i) ≡ σi b(c, i − 1) in L/q −1 L.
(25)
Proof. Write b(c, i−1) =
P
d
adc L(d, i−1). By Theorem 4.11, every d with adc 6=
0 must have dαi = 0. In particular, Ti L(d, i−1) ∈ U+ . This implies Ti b(c, i−1) ∈ U+ ,
so we can apply Theorem 4.13 to conclude that Ti b(c, i − 1) = σi b(c, i − 1) in L/q −1 L.
By Proposition 4.7, L(c ◦ si , i) = Ti L(c, i − 1), and by definition L(c ◦ si , i) ≡
b(c ◦ si , i) and L(c, i − 1) ≡ b(c, i − 1) in L/q −1 L.
Lemma 4.15. [BCP, Lemma 4.1][Aka, Theorem 8.5, c.f. Proof of Theorem 8.17
]
Fix a positive integer n.
b 2 , let c be the Lusztig datum satisfying c(α0 ) = c(α1 ) = n, and let (n)
(i) For sl
denote the partition consisting of one part of length n.
(2)
(ii) For A2 , let c be the Lusztig datum satisfying c(α0 ) = n, c(α1 ) = 2n, and
let (n) denote the partition consisting of one part of length n.
Then for each i, b(c, i) = b((n), i + 1).
Remark 4.16. We have not stated the most general versions of the results in this
section. With appropriate definitions Lemma 4.10, Theorem 4.11, and Proposition
4.12 hold for all affine Kac-Moody algebras (c.f the discussion immediatedly following
[BN, Theorem 3.13]). Theorem 4.13 holds for all symmetrizable Kac-Moody algebras.
58
4.3. Relationship with MV Polytopes. Recall from Definition 4.8 that, since
B and the two PBW bases agree as crystal bases, we can parameterize B by Lusztig
data. We will use the notation b = b(crb , 1) = b(c`b , 0) to denote the Lusztig data
corresponding to b with respect to the two PBW bases (and will drop the subscripts
of b on cr , c` where it will not cause confusion). Thus by Definition 3.3 each b ∈ B
defines a decorated pseudo-Weyl polytope PBWb whose left Lusztig datum is c`b and
whose right Lusztig datum is crb . Equivalently, this gives a map b 7→ PBWb from
B(−∞) to decorated pseudo-Weyl polytopes.
Theorem 4.17. For each b ∈ B(−∞), P BWb coincides with the affine MV polytope M Vb as defined in [BDKT] (and in §3.1).
Proof. It suffices to show that the map b → P BWb satisfies the conditions of
Theorem 3.11. Condition (W) is immediate since wt(c) = wt(b(c, i)). Conditions
(C1)-(C4) are immediate from Proposition 4.3. Conditions (S1) and (S2) are Corollary 4.14, and (S3)-(S4) follow from these using Proposition 4.7.
All that remains is to check condition (I). That is, to show that, if c` (P BWb ) = λ,
then
(26)
crα1 (P BWb ) =
|α0 |
λ1 , crδ (P BWb ) = λ\λ1 , crα0 (P BWb ) = λ1 ,
|α1 |
and crβ (P BWb ) = 0 for all other β. We proceed by induction on λ, using the total
order where λ < λ0 if
(i) |λ| < |λ0 |, or
(ii) |λ| = |λ0 | and (λ1 , λ2 , . . .) >lex (λ01 , λ02 , . . .).
So, fix λ, and assume that (I) holds for all λ0 < λ. In particular, we can apply
Proposition 3.15.
It is easy to see that there is a unique b ∈ B(−∞) with the following MV polytope:
c`α1 =
|α0 |
λ
|α1 | 1
•
c`δ = λ\λ1
•
crδ = λ\λ1
•
• cr = |α0 | λ
α1
|α1 | 1
59
That is,
(27)
cr (M Vb )δ = c` (M Vb )δ = λ\λ1 ,
cr (M Vb )α1 = c` (M Vb )α1 =
|α0 |
λ1 ,
|α1 |
and all other entries are 0. By Proposition 3.15, P BWb has these same Lusztig data.
That is, b = b(c, 0) = b(c, 1), where c = c` (M Vb ). Let d be the Lusztig datum such
that b(d, 0) = ẽλ0 1 b(c, 0). Using the action of ẽ0 on the 0-PBW basis and the fact that
the map from B(−∞) to PBW basis elements is injective, it suffices to show d = λ.
Using the upper triangularity of the PBW basis,
(
|α0 |
b(c, 0) = E1 |α1 |
(28)
λ1 )
Sλ\λ1 +
X
ac0 ,c L(c0 , 0).
c0 0 c
(λ1 )
Multiplying both sides on the left by E0
(29)
(λ1 )
E0
(λ1 )
b(c, 0) = E0
(
|α0 |
E1 |α1 |
λ1 )
,
Sλ\λ1 +
X
(λ1 )
ac0 ,c E0
L(c0 , 0).
c0 0 c
(λ1 )
For each c0 0 c, rewrite E0
L(c0 , 0) in the 0-PBW basis. Since c0 must have
c0α0 +kδ 6= 0 for some k, by Lemma 4.10, no purely imaginary terms appear. When
we subsequently expand these in B, by Theorem 4.11 we still don’t get any purely
imaginary terms.
(λ1 )
Now expand E0
(
|α0 |
E1 |α1 |
λ1 )
Sλ\λ1 in the basis B. By Theorem 4.11 and Lemma
4.15,
(λ1 )
E0
(
|α0 |
E1 |α1 |
λ1 )
= S(λ1 ) +
X
ac0 ,(λ1 ) L(c0 , 0),
c0 0 (λ1 )
and none of the c0 that appear are purely imaginary. As before, the same remains
true when we expand in the basis B.
By the Pieri rule (see e.g. [BCP, Formula 4.14]),
S(λ1 ) Sλ\λ1 = Sλ +
X
Sµ ,
µ
where the µ that appear all satisfy µ < λ in the order defined above.
Suppose for d 6= λ. By Proposition 4.12 the element b(d, 0) must show up with
(λ1 )
non-zero coefficient when E0
b(c, 0) is written in the basis B. So either d is not
purely imaginary, or d is purely imaginary and equal to µ with µ < λ.
60
If d is not purely imaginary, then |dδ | < |λ|. So by Proposition 3.15, d =
c` (P BWẽλ1 b ) = c` (M Vẽλ1 b ) = λ, which is a contradiction.
0
0
If d = µ with µ < λ, then (I) holds by induction. We then see that cr (P BWẽλ1 b ) =
0
0
c , where
c0δ
=
µ\µ1 , c0α0
=
µ1 , c0α1
=
|α0 |
µ
|α1 | 1
and otherwise zero. But from (27) and
the definition of ẽ0 , we see that cr (P BWẽλ1 b ) = c00 , where c00δ = λ\λ1 , c00α0 = λ1 , c00α1 =
0
|α0 |
λ,
|α1 | 1
which is a contradiction.
b 2 MV polytopes
5. Comparing combinatorial and geometric sl
b 2 quiver variety. We will largely follow the conventions of [BKT,
5.1. The sl
§7.4]. Let Q̃ be the quiver
α
β
0
1
α∗
β∗
Let e0 and e1 denote the lazy paths at the vertices 0 and 1 respectively. The preprojective algebra Π is the quotient of the completed path algebra of Q̃ (completed with
respect to the ideal generated by α, α∗ , β, β ∗ ) by the relations
αα∗ + ββ ∗ = 0,
α∗ α + β ∗ β = 0.
A representation T of Π consists of a {0, 1}-graded vector space V = V0 ⊕ V1 and
a 4-tuple of linear operators (tα , tβ : V0 → V1 , tα∗ , tβ ∗ : V1 → V0 ) that satisfy
tα tα∗ + tβ tβ ∗ = 0
and
tα∗ tα + tβ ∗ tβ = 0,
and which is nilpotent in the sense that, for some N and any path aN · · · a1 in Q̃,
tan · · · ta1 = 0.
b 2 , let Π(µ) be
Given an element µ = nα0 + mα1 in the positive root lattice for sl
variety of Π-representations on a fixed {0, 1}-graded vector space V µ = V0µ ⊕ V1µ with
dim V0µ = n and dim V1µ = m. We refer to µ as the dimension vector of V µ , and
we will drop the superscripts µ when they are clear from context. Let S0 and S1 be
61
the simple modules of dimension vectors α0 and α1 respectively (where all four maps
tα , tβ , tα∗ , tβ ∗ are 0).
Let IrrΠ(µ) denote the set of irreducible components of Π(µ). Kashiwara and
Saito [KS] show that
(30)
a
IrrΠ(µ),
µ
gives a realization of the crystal B(−∞), where the crystal operator ẽi can be defined
as follows:
For each Z ∈ IrrΠ(µ), there is a dense open subset U ⊂ Z such that each T ∈ U
has i-cosocle of the same dimension n. For each T ∈ U , let T 0 = ker(T → Si⊕n ). Let
W be the set of modules T 00 fitting into a short exact sequence as below for some
T ∈ U.
0 → T 0 → T 00 → Si⊕n+1 → 0
It is known that there is a unique irreducible component Z 0 ∈ IrrΠ(µ + αi ) such that
W ∩ Z 0 is dense in Z 0 . Kashiwara and Saito then define ẽi Z = Z 0 .
The map α ↔ α∗ and β ↔ β ∗ extends uniquely to an algebra anti-involution of
Π. Given a Π-module M , the dual module is naturally a right Π module, but we can
twist the action by the above anti-involution to get a new left Π module. We denote
this left Π module by M ∗ . Given Z ∈ IrrΠ(µ),
{S ∈ Π(µ) | S ' T ∗ for some T ∈ Z}
is also an irreducible component of Π(µ), which we denote by Z ∗ . In the above
realization of B(−∞), the map Z → Z ∗ is Kashiwara’s involution as discussed in
§2.2.
We also have an algebra automorphism τ of Π defined on generators by the map
α ↔ α∗ , β ↔ β ∗ , and e0 ↔ e1 . Twisting by τ induces an involutive auto-equivalence
R → Rτ on the category of left Π modules and defines an involution on the set
F
b
µ IrrΠ(µ), inducing the sl2 diagram automorphism on B(−∞).
62
5.2. Reflection functors and Harder-Narasimhan filtrations. Here we review the filtrations given in [BKT, Theorems 5.11 and 5.12]. We must first introduce
the reflection functors Σi and Σ∗i for i ∈ {0, 1} from [BK, BIRS].
Definition 5.1. For i ∈ {0, 1} define the Π − Π bimodule Ii = Π(1 − ei )Π =
Πei+1 Π (subscripts taken modulo 2). If si1 · · · sik is a reduced expression in the Weyl
group, define Isi1 ···sik = Ii1 ⊗Π · · · ⊗Π Iik .
As shown in [BIRS], the bimodule Isi1 ···sik depends only on the Weyl group element w = si1 · · · sik and not on the reduced expression. We need the following two
endofunctors on the category of finite-dimensional Π modules.
Definition 5.2. Σi = HomΠ (Ii , ?) and Σ∗i = Ii ⊗Π ?.
These functors are geometric lifts of Saito’s crystal reflections in the following
sense:
Proposition 5.3. [BK, Theorem 5.3] Fix b ∈ B(−∞) such that f˜i (b) = 0. Let
Zb and Zσi (b) be the irreducible components corresponding to b and σi (b) respectively,
where σi is Saito’s reflection. For generic T ∈ Zb , Σi T is isomorphic to a point in
Zσi (b) , and furthermore this point in generic in the sense that the decorated PseudoWeyl polytope associated to Zσi (b) can be calculated using Σi T .
Similarly if b ∈ B(−∞) is such that f˜i∗ (b) = 0, then for generic T ∈ Zb , Σ∗i T is
isomorphic to a generic point in Zσi∗ (b) .
b 2 , it is convenient to introduce
To make this construction concrete in the case of sl
notation for the following special Π-modules.
Definition 5.4. For k ≥ 0,
(i) R` (α1 + kδ) = Is1 ···sk ⊗Π Sk+1 = Σ∗1 · · · Σ∗k Sk+1
(ii) R` (α0 + kδ) = HomΠ (Isk−1 sk−2 ···s0 , Sk ) = Σ0 · · · Σk−1 Sk .
(iii) Rr (α1 + kδ) = HomΠ (Isk sk−1 ···s1 , Sk+1 ) = Σ1 · · · Σk Sk+1
(iv) Rr (α0 + kδ) = Is0 ···sk−1 ⊗Π Sk = Σ∗0 · · · Σ∗k−1 Sk .
63
That the two definitions given on lines (i) and (iv) agree follows by the definition of
the reflection functors, and for (ii) and (iii) this follow by applying duality to (i) and
(ii).
One can easily verify that these modules are as shown in Figure 2, and in particular
dim R` (α1 + kδ) = dim Rr (α1 + kδ) = α1 + kδ,
dim R` (α0 + kδ) = dim Rr (α0 + kδ) = α0 + kδ,
Rr (α1 + kδ) = R` (α1 + kδ)∗ ,
Rr (α0 + kδ) = R` (α0 + kδ)∗ ,
R` (α0 + kδ) = Rr (α1 + kδ)τ ,
R` (α1 + kδ) = Rr (α0 + kδ)τ .
0 0 ··· 0 0
1 1 ···
0 0 ···
0 0
1 1 ··· 1 1
1 1
R` (α1 + (j − 2)δ)
R` (α0 + (j − 1)δ)
Figure 2. The representations from Definition 5.4. In each case, the
number of 0 is j. Here the vertices represent basis elements, the dotted
arrows represent matrix elements of 1 for tα , and solid arrows represent
matrix elements of 1 for tβ , and all other matrix elements are 0.
By [BKT, Theorems 5.11 and 5.12], any finite dimensional representation T of Π
admits a filtration
(31)
`
T = T `,0 ⊃ T `,1 ⊃ T `,2 ⊃ · · · ⊃ T `,∞ ⊃ T∞
⊃ · · · ⊃ T2` ⊃ T1` ⊃ T0` = 0
given by the following explicit formulas:
(i) T `,k = Σ∗1 · · · Σ∗k Σk · · · Σ1 T
(ii) Tk` = ker(T → Σ0 · · · Σk−1 Σ∗k−1 · · · Σ∗0 T ).
S
T
`
(iii) T∞
= k Tk` and T `,∞ = k T `,k .
These satisfy the following properties:
`
(iv) For all k, Tk+1
/Tk` is a direct sum of copies of R` (α0 + kδ).
(v) For all k, T `,k /T `,k+1 is a direct sums of copies of R` (α1 + kδ).
64
`
(vi) No subrepresentation S ⊂ T∞
has hdim S, α0 i > 0.
(vii) No quotient representation S of T `,∞ has hdim S, α0 i < 0.
There is also a filtration
r
T = T r,0 ⊃ T r,1 ⊃ T r,2 ⊃ · · · ⊃ T r,∞ ⊃ T∞
⊃ · · · ⊃ T2r ⊃ T1r ⊃ T0r = 0
(32)
given by:
(i) T r,k = Σ∗0 · · · Σ∗k−1 Σk−1 · · · Σ0 T
(ii) Tkr = ker(T → Σ1 · · · Σk Σ∗k · · · Σ∗1 T )
S
T
r
(iii) T∞
= k Tkr and T r,∞ = k T r,k .
which has the same properties as the first filtration, except the modules R` (α0 + kδ)
and R` (α1 + kδ) are replaced with Rr (α1 + kδ) and Rr (α0 + kδ) respectively, and α1
and α0 are interchanged in the above list of properties.
Following [BKT, §7.4], let Π(nδ)× be the subvariety of Π(nδ) consisting of those
4-tuples of operators (tα , tβ , tα∗ , tβ ∗ ) where tα is invertible. Define I ` (n) to be the
subvariety of Π(nδ)× where tβ ∗ tα is nilpotent of order n (i.e. (tβ ∗ tα )n = 0, but
(tβ ∗ tα )n−1 6= 0), and notice that I ` (n) consists only of indecomposable modules. By
the discussion in [BKT], I ` (n) is an open subset of an irreducible component of
Π(nδ). Similarly, we define I r (n).
Proposition 5.5. [BKT, Proposition 7.11] Fix an irreducible component Z.
There is a unique partition λ` = λ`1 ≥ · · · ≥ λ`k such that, for all T in some open
L
`
dense subset of Z, T `,∞ /T∞
can be decomposed as i Tλ`i , where Tλ`i ∈ I ` (λ`i ).
r
Similarly there is a partition λr such that, for a generic T ∈ Z, T r,∞ /T∞
can be
L
decomposed as i Tλri , where Tλri ∈ I r (λri ).
Proposition 5.6. Every element T ∈ I ` (n) has socle S1 and cosocle S0 . Furthermore ker(T /S1 → S0 ) is isomorphic to a point in I r (n − 1).
Proof. Let W = tα (ker(tβ ∗ tα )). Since tβ ∗ tα is nilpotent of order n, W is the
socle of T , and it is isomorphic to S1 . Let U = coker(tβ ∗ tα ). Since tβ ∗ tα has order n
and tα is invertible, U is the cosocle of T , and it is isomorphic to S0 . Moreover, we
65
can easily check that the sub quotient ker(T /W → U ) has dimension vector (n − 1)δ,
the induced operator tβ ∗ is invertible, and tα tβ ∗ is nilpotent of order n − 1. So,
ker(T /W → U ) lies in I r (n − 1).
Proposition 5.7. The following hold:
(i) Σ∗0 R` (α1 + kδ) = Rr (α0 + (k + 1)δ)
(ii) Σ∗0 R` (α0 + kδ) = Rr (α1 + (k − 1)δ) for k > 0.
(iii) If T ∈ I ` (n), then Σ∗0 T ∈ I r (n) for n > 0.
(iv) Σ∗1 Rr (α0 + kδ) = R` (α1 + (k + 1)δ)
(v) Σ∗1 Rr (α1 + kδ) = R` (α0 + (k − 1)δ) for k > 0.
(vi) If T ∈ I r (n), then Σ∗1 T ∈ I ` (n) for n > 0.
(vii) Σ0 Rr (α1 + kδ) = R` (α0 + (k + 1)δ)
(viii) Σ0 Rr (α0 + kδ) = R` (α1 + (k − 1)δ) for k > 0.
(ix) If T ∈ I r (n), then Σ0 T ∈ I ` (n) for n > 0.
(x) Σ1 R` (α0 + kδ) = Rr (α1 + (k + 1)δ)
(xi) Σ1 R` (α1 + kδ) = Rr (α0 + (k − 1)δ) for k > 0.
(xii) If T ∈ I ` (n), then Σ1 T ∈ I r (n) for n > 0.
Proof. Statement (i) follows immediately from the definition. Since Rr (α1 + kδ)
has no 0-cosocle, Σ∗0 Σ0 Rr (α1 + kδ) = Rr (α1 + kδ) (see [BKT, Equation 5.2]), which
implies (ii).
For (iii), let T ∈ I ` (n). A short calculation shows that T̃ = Σ∗0 T = (t̃α , t̃β , t̃α∗ , t̃β ∗ )
has the property that t̃β ∗ is invertible and t̃α t̃β ∗ is nilpotent of order n, so T̃ ∈ I r (n).
Statements (iv) - (vi) follow from the first three by applying τ . The remaining six
statements follow from the first six by applying ∗.
Lemma 5.8. The following hold:
(i) If T `,0 = T `,1 , then (Σ1 T )r,k = Σ1 T `,k+1 and (Σ1 T `,k+1 )/(Σ1 T `,k+2 ) =
Σ1 (T `,k+1 /T `,k+2 ).
(ii) If T r,0 = T r,1 , then (Σ0 T )`,k = Σ0 T r,k+1 and (Σ0 T r,k+1 )/(Σ0 T r,k+2 ) =
Σ0 (T r,k+1 /T r,k+2 ).
66
`
`
`
`
`
(iii) If T0` = T1` , then (Σ∗0 T )rk = Σ∗0 Tk+1
and (Σ∗0 Tk+1
)/(Σ∗0 Tk+2
) = Σ∗0 (Tk+1
/Tk+2
).
r
r
r
r
r
).
/Tk+2
) = Σ∗1 (Tk+1
)/(Σ∗1 Tk+2
and (Σ∗1 Tk+1
(iv) If T0r = T1r , then (Σ∗1 T )`k = Σ∗1 Tk+1
Proof. Using the explicit formulas for the filtrations, we have Σ1 T `,k+1 =
Σ1 Σ∗1 (Σ1 T )r,k . We always have a surjective map (Σ1 T )r,k → Σ1 Σ∗1 (Σ1 T )r,k , whose
kernel is precisely the 1-socle of (Σ1 T )r,k . But (Σ1 T )r,k ⊂ Σ1 T , and Σ1 T has vanishing 1-socle because T `,0 = T `,1 . Thus the above map is an isomorphism, giving
(Σ1 T )r,k = Σ1 T `,k+1 .
Because Σ1 is left-exact, we have a injection (Σ1 T `,k+1 )/(Σ1 T `,k+2 ) → Σ1 (T `,k+1 /T `,k+2 ),
and the obstruction to this map being an isomorphism is an element of Ext1 (I1 , T `,k+1 ).
By [BKT, Remark 5.5 (ii)], the essential image of the functor Σ∗1 is precisely those
modules M with Ext1 (I1 , M ) = 0. By construction, T `,k+1 is in the image of Σ∗1 , so
we have Ext1 (I1 , T `,k+1 ) = 0. This completes the proof of (i).
The second statement follows from the first by applying τ , and the remaining two
statements follow by duality using the canonical isomorphisms
(33)
Tk` = (T ∗ /(T ∗ )r,k )∗ and Tkr = (T ∗ /(T ∗ )l,k )∗ .
5.3. MV polytopes from quiver varieties. Associate a Lusztig data c` to
each Π-module T by
`
• c`βk is defined by Tk` /Tk−1
' R` (α0 + (k − 1)δ)
⊕c`β
• c`β k is defined by T `,k−1 /T `,k ' R` (α1 + (k − 1)δ)
k
.
⊕c` k
β
.
• Because the filtration in (31) commutes with direct sums, every indecom`
posable summand of T `,∞ /T∞
must have dimension kδ for some k. Then
c`δ is defined to be the partition whose parts are these k. Notice that, if
`
T `,∞ /T∞
∈ Π(nδ)× , then this agrees with λ` from Proposition 5.5.
Similarly define a Lusztig data cr via the same definition twisted by τ . By Definition
3.3 these are the left and right sides of a decorated pseudo-Weyl polytope PT .
`
The function which associates to every T ∈ v Π(v) the polytope PT is con`
structible. Thus on each irreducible component of v Π(v), PT takes on a unique
generic value, i.e. the constant value it takes on some open dense subset. For
67
b ∈ B(−∞), we write HNb for the generic value of PT on Zb , the irreducible component corresponding to b. This pseudo-Weyl polytope is called the MV polytope for b
in [BKT]. The notation HN stands for “Harder-Narasimhan,” since this polytope is
constructed using Harder-Narasimhan filtrations.
Theorem 5.9. For any b ∈ B(−∞), HNb and the MV polytope M Vb defined
[BDKT] agree except the imaginary parts are transposed partitions.
Proof. For each b ∈ B(−∞), let HNb be the pseudo-Weyl polytope obtained
from HNb be taking the transpose of the partitions decorating each vertical edge. It
suffices to show that b → HNb satisfies the conditions of Theorem 3.11.
Axiom (W) is clear. Axioms (C1) and (C2) follow from the definition of the crystal
operators, and (C3) and (C4) follow by using the star operators.
Conditions (S1), (S2), (S3), and (S4) follow immediately from Proposition 5.7 and
Lemma 5.8.
All that remains is to prove (I). So, fix a component Z such that, for generic
`
T ∈ Z, T = T `,∞ /T∞
. By Proposition 5.5, for generic T ∈ Z, there is a partition
L
`
λ` such that T = i Tλ`i , where Tλ`i ∈ I ` (λ`i ). Letting λ denote the transpose of λ` ,
⊕λ
`
⊕λ
`
to S0 1 ,
Proposition
T has socle isomorphic to S1 1 , cosocle isomorphic
5.6 implies
`
`
`
`
L
⊕λ
⊕λ
⊕λ
⊕λ
r
that ker T /S1 1 → S0 1 ∈ T r,∞ /T∞
, and that ker T /S1 1 → S0 1 = i Tλ`i −1 ,
where Tλ`i −1 ∈ I r (λ`i − 1). This is precisely the content of (I).
5.4. Characterization of symmetric affine MV polytopes. Along with
[BKT, §1.6 and §7.6], Theorem 5.9 allows one to characterization of MV polytopes
in all symmetric affine types. In this section we state this precisely. This section is
essentially a rewording of results in [BKT].
Definition 5.10. Let g be a symmetric affine Kac-Moody algebra of rank r + 1.
A decorated pseudo-Weyl polytope for g is a convex polytope whose edges are all
translates of integer multiples of roots, along with a partition λF associated to each
(possibly degenerate) r-face F parallel to δ, such that, for each edge e which is a
translate of kδ, the sum of |λF | over all faces F incident to e is k.
68
Remark 5.11. In [BKT] the decorating partitions are indexed by chamber
weights of an underlying finite-type root system; it is straightforward to see that
these in turn index the possible r-faces parallel to δ, so the above wording is equivalent.
Theorem 5.12. Let g be a symmetric affine Kac-Moody algebra of rank r + 1.
The type g MV polytopes are exactly those decorated pseudo-Weyl polytopes P such
that each 2-face S satisfies either:
(i) The roots parallel to S form a rank-2 root system (of type sl2 × sl2 or sl3 ),
and S is an MV polytope of that type.
b 2 . S is a Minkowski sum
(ii) The roots parallel to S form a root system of type sl
P
of a smaller polytope S 0 with the line segment S⊂F λF . For each edge e of
S 0 parallel to δ, let λe = λF for the unique r-face F of P which contains e
b 2 MV
but does not contain S. Then S 0 along with this decoration is a type sl
polytope.
Remark 5.13. An extension of Theorem 5.12 to all (not necessarily symmetric)
affine cases is given in [TW, Theorem B].
69
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