LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL ALGEBRAS

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LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL ALGEBRAS OLIVIER SCHIFFMANN Contents Introduction 2 Lecture 1. 6 1.1. Recollections on quivers. 6 1.2. Moduli spaces of representations of quivers. 7 1.3. The induction and restriction functors. 9 1.4. The Lusztig sheaves and the Hall category. 16 1.5. The geometric pairing on the Hall category. 19 Lecture 2. 22 2.1. The simplest of all quivers. 22 2.2. The fundamental relations. 23 2.3. Finite type quivers. 27 2.4. The Jordan quiver and the cyclic quivers. 30 2.5. Affine quivers. 33 Lecture 3. 40 3.1. The graded Grothendieck group of the Hall category. 40 3.2. Relation to quantum groups. 42 3.3. Proof of Lusztig's theorem (finite type). 45 3.4. Fourier-Deligne transform. 46 3.5. Proof of Lusztig's theorem. 51 3.6. The Lusztig graph. 55 3.7. The trace map and purity. 57 Lecture 4. 62 4.1. Kashiwara crystals. 62 4.2. Lusztig's Lagrangian. 67 4.3. Hecke correspondences. 70 4.4. Geometric construction of the crystal. 72 4.5. Relationship to the Hall category. 75 4.6. Relationship to the Lusztig graph. 76 Lecture 5. 79 5.1. Moduli stacks of coherent sheaves on curves. 79 5.2. Convolution functors and the Hall category. 82 5.3.

• canonical bases has


• lusztig's nilpotent


• called canonical


• sheaves


• moduli spaces


• kashiwara's crystal graph


• perverse sheaves generating


• lusztig's theorem

Sujets

• Grenoble
• Schwäbisch Hall
• Sheaves
• Moduli space
• multiplication

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL
ALGEBRAS
OLIVIER SCHIFFMANN
Contents
Introduction 2
Lecture 1. 6
1.1. Recollections on quivers. 6
1.2. Moduli spaces of representations of quivers. 7
1.3. The induction and restriction functors. 9
1.4. The Lusztig sheaves and the Hall category. 16
1.5. The geometric pairing on the Hall. 19
Lecture 2. 22
2.1. The simplest of all quivers. 22
2.2. The fundamental relations. 23
2.3. Finite type quivers. 27
2.4. The Jordan quiver and the cyclic quivers. 30
2.5. A ne quivers. 33
Lecture 3. 40
3.1. The graded Grothendieck group of the Hall category. 40
3.2. Relation to quantum groups. 42
3.3. Proof of Lusztig’s theorem ( nite type). 45
3.4. Fourier-Deligne transform. 46
3.5. Proof of Lusztig’s theorem. 51
3.6. The Lusztig graph. 55
3.7. The trace map and purity. 57
Lecture 4. 62
4.1. Kashiwara crystals. 62
4.2. Lusztig’s Lagrangian. 67
4.3. Hecke correspondences. 70
4.4. Geometric construction of the crystal. 72
4.5. Relationship to the Hall category. 75
4.6. to the Lusztig graph. 76
Lecture 5. 79
5.1. Moduli stacks of coherent sheaves on curves. 79
5.2. Convolution functors and the Hall category. 82
5.3. Examples, curves of low genera. 83
5.4. Higgs bundles and the global nilpotent cone. 85
5.5. Conjectures and relation to the geometric Langlands program. 87
Windows 94
References 98
12 OLIVIER SCHIFFMANN
Introduction
These notes form the support of a series of lectures given for the summer school \Geo-
metric methods in representation theory" at the Institut Fourier in Grenoble in June 2008.
They represent the second half of the lecture series. The rst half of the series was dedi-
cated to the notion of the Hall algebra H of an abelian (or derived) categoryA, and theA
notes for that part are written in [S2]. The present text is a companion to [S2]; we will
use the same notation as in [S2] and sometimes refer to [S2] for de nitions. Nevertheless,
we have tried to make this text as self-contained as (reasonably) possible.
In this part of the lecture series we explain how to translate the purely algebraic
construction of Hall algebras given in [S2] into a geometric one. This geometric lift
amounts to replacing the \naive" space of functions on the setM of objects of a categoryA
A by a suitable category Q of constructible sheaves on the moduli space (or moreA
precisely, moduli stack)M parametrizing the objects ofA. The operations in the HallA
algebra (multiplication and comultiplication) then ought to give rise to functors
m :Q Q !Q ;A A A
:Q !Q Q :A A A
The Faisceaux-Fonctions correspondence of Grothendieck, which associates to a con-
structible sheaf P2Q its trace{a (constructible) function on the \naive" moduli spaceA
M { draws a bridge between the \geometric" Hall algebra (or rather, Hall category)A
Q and the \algebraic" Hall algebra H . Such a geometric lift from H toQ mayA A A A
be thought of as \categori cation" of the Hall algebra (and is, in fact, one of the early
examples of \categori cation").
Of course, for the above scheme to start making any sense, a certain amount of tech-
nology is required : for one thing, the moduli stackM has to be rigorously de ned andA
the accompanying formalism of constructible orl-adic sheaves has to be developped. The
relevant language for a general theory is likely to be [TV1]. Rather than embarking on
the (probably risky) project of de ning the Hall category Q for an arbitrary abelianA
categoryA using that language we believe it will be more useful to focus in these lectures
on several examples. Another reason for this is that, as explained in [S2, Section 5], the
correct setting for the theory of Hall algebras (especially for categories of global dimension
more than one) seems to be that of derived or triangulated categories. The necessary tech-
nology to deal with moduli stacks parametrizing objects in triangulated (or dg) categories
is, as far as we know, still in the process of being fully worked out, see [TV1], [TV2], (this,
in any case, far exceeds the competence of the author).
~The main body of the existing theory is the work of Lusztig whenA =Rep Q is thek
1~category of representations of a quiverQ over a nite eld k (see [Lu5], [Lu6]) , which we
now succintly describe. In that case (see [S2]), there is an embedding
+ : U (g),! H~v Q
of the positive half of the quantum enveloping algebra of the Kac-Moody algebra g associ-
~ated toQ into the Hall algebra. The image of this map is called the composition subalgebra
C of H and is generated by the constant functions 1 for running among the classes~ ~ Q Q
~of simple objects in Rep Q. These classes bijectively correspond to the positive simplek
roots of g and we will call them in this way. The moduli stackM parametrizing objects~Q
1Actually, the theory really originates from Lusztig’s theory of character sheaves in the representation
theory of nite groups of Lie type, see [Lu2]. This, however, has little to do with Hall algebras.LECTURES ON CANONICAL AND CRYSTAL BASES OF HALL ALGEBRAS 3
~of RepQ splits into a disjoint unionk
G

M = M~ ~Q Q
~2K (Q)0
b according to the class in the Grothendieck group. LetD (M ) stand for the triangulated~Q
category of constructible complexes onM (see Lecture 1 for precise de nitions). For~Q
; ~;2 K (Q), letE be the stack parametrizing inclusions M N, where M and N0
~are objects in Rep Q of class + and respectively. There are natural maps p and 1k
p :2
; (0.1) E
Et Et Ep t p1 E 2t Et Et Ett Et
+
M M M~ ~ ~Q Q Q
de ned by p : (M N)7! (M=N;N), p : (M N)7! (M): The map p is proper1 2 2
(the ber of p over M is the Grassmanian of subobjects N of M of class , a projective2
scheme) while p can be shown to be smooth. One then considers the functors1
+