Table Of ContentSpringer theory and the geometry of
quiver (cid:29)ag varieties
Julia Anneliese Sauter
Submitted in accordance with the requirements for the degree of Doctor of
Philosophy
The University of Leeds
School of Mathematics
September 2013
The candidate con(cid:28)rms that the work submitted is her own and that appropriate
credit has been given where reference has been made to the work of others. This
copy has been supplied on the understanding that it is copyright material and that
no quotation from the thesis may be published without proper acknowledgement.
(cid:13)c The University of Leeds and Julia Anneliese Sauter
2
Acknowledgements
I thank my supervisor Dr. Andrew Hubery for his constant support through regular
meetings, he helped to answer a great many questions and to solve mathematical prob-
lems. Also, I would like to thank my second supervisor Professor William Crawley-Boevey
for helping me with clearing concepts and going through parts of the thesis pointing out
mistakes. Both very generously lent me their time and sharp minds for my project.
For helpful discussions and explaining some arguments I am thankful to Dr. Greg Steven-
son, Professor Markus Reineke and Dr. Michael Bate. Also, I thank Professor Henning
Krause and the CRC 701 for (cid:28)nancial support during several guest stays in Bielefeld.
For accompanying me through this time I want to thank my family, especially my
parents for their (cid:28)nancial and moral support. My special thanks go to my friends from the
pg-satellite o(cid:30)ce in Leeds and to the algebra group for the nice atmosphere at the algebra
dinners.
3
Abstract
The thesis consists of the following chapters:
1. Springer theory.
For any projective map E → V, Chriss and Ginzburg de(cid:28)ned an algebra structure
on the (Borel-Moore) homology Z := H (E× E), which we call Steinberg algebra.
∗ V
(Graded)ProjectiveandsimpleZ-modulesarecontrolledbytheBBD-decomposition
associated to E → V. We restrict to collapsings of unions of homogeneous vector
bundlesoverhomogeneousspacesbecausewehavethecellular(cid:28)brationtechniqueand
for equivariant Borel-Moore homology we can use localization to torus-(cid:28)xed points.
ExamplesofSteinbergalgebrasincludegroupringsofWeylgroups,Khovanov-Lauda-
Rouquier algebras, nil Hecke algebras.
2. Steinberg algebras.
We choose a class of Steinberg algebras and give generators and relations for them.
This fails if the homogeneous spaces are partial and not complete (cid:29)ag varieties, we
call this the parabolic case.
3. The parabolic case.
In the parabolic cases, we realize the Steinberg algebra ZP as corner algebra in a
Steinberg algebra ZB associated to Borel groups (this means ZP = eZBe for an
idempotent element e ∈ ZB).
4. Monoidal categories.
We explain how to construct monoidal categories from families of collapsings of ho-
mogeneous bundles.
5. Construct collapsings.
We construct collapsing maps over given loci which are resolutions of singularities or
generic Galois coverings. For closures of homogeneous decomposition classes of the
Kronecker quiver these maps are new.
6. Quiver (cid:29)ag varieties.
Quiver(cid:29)agvarietiesarethe(cid:28)bresofcertaincollapsingsofhomogeneousbundles. We
investigate when quiver (cid:29)ag varieties have only (cid:28)nitely many orbits and we describe
the category of (cid:29)ags of quiver representations as a ∆-(cid:28)ltered subcategory for the
quasi-hereditary algebra KQ⊗KA .
n
7. A -equioriented.
n
For the A -equioriented quiver we (cid:28)nd a cell decompositions of the quiver (cid:29)ag vari-
n
eties, which are parametrized by certain multi-tableaux.
4
Contents
1 A survey on Springer theory 9
1.1 De(cid:28)nition of a Springer theory . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Convolution modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 The Steinberg algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 The Steinberg algebra HA(Z) as module over H−∗(pt). . . . . . . . 13
[∗] A
1.3.2 The Steinberg algebra HA(Z) and H∗(E) . . . . . . . . . . . . . . . 15
∗ A
1.4 Indecomposable projective graded modules over HA(Z) and their tops for
[∗]
a di(cid:27)erent grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Indecomposable projectives in the category of graded left HA(Z)-
[∗]
modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.2 Simples in the category of graded (cid:28)nitely generated left HA (Z)-
<∗>
modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3 Springer (cid:28)bre modules in the category of graded HA(Z)-modules . . 22
?
1.5 The Springer functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Orbital varieties, Springer (cid:28)bres and strata in the Steinberg variety . . . . . 29
1.7 What is Springer theory ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8 Classical Springer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8.1 Parametrizing simple modules over Hecke algebras. . . . . . . . . . . 37
1.9 Quiver-graded Springer theory. . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.9.1 Monoidal categori(cid:28)cations of the negative half of the quantum group 42
1.9.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2 Generalized quiver Hecke algebras 52
2.1 Generalized quiver-graded Springer theory . . . . . . . . . . . . . . . . . . . 54
2.2 Relationship between parabolic groups in G and G . . . . . . . . . . . . . . 59
2.3 The equivariant cohomology of (cid:29)ag varieties . . . . . . . . . . . . . . . . . . 61
2.4 Computation of (cid:28)xed points . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.1 Notation for the (cid:28)xed points . . . . . . . . . . . . . . . . . . . . . . 65
2.4.2 The (cid:28)bres over the (cid:28)xpoints . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Relative position strati(cid:28)cation . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5.1 In the (cid:29)ag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5.2 In the Steinberg variety . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6 A short lamentation on the parabolic case . . . . . . . . . . . . . . . . . . . 70
5
2.7 Convolution operation on the equivariant Borel-Moore homology of the
Steinberg variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Computation of some Euler classes . . . . . . . . . . . . . . . . . . . . . . . 74
2.9 Localization to the torus (cid:28)xed points . . . . . . . . . . . . . . . . . . . . . . 77
2.9.1 The W-operation on E : . . . . . . . . . . . . . . . . . . . . . . . . 78
G
2.9.2 Calculations of some equivariant multiplicities . . . . . . . . . . . . . 79
2.9.3 Convolution on the (cid:28)xed points . . . . . . . . . . . . . . . . . . . . . 80
2.10 Generators for Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
G
2.11 Relations for Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
G
3 Parabolic Nil Hecke algebras and parabolic Steinberg algebras 96
3.1 The parabolic (a(cid:30)ne) nil Hecke algebra . . . . . . . . . . . . . . . . . . . . 96
3.2 On parabolic Steinberg algebras . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2.1 Reineke’ s Example (cp. end of [Rei03]) . . . . . . . . . . . . . . . . 104
3.2.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 From Springer theory to monoidal categories 106
4.1 (I-)Graded Springer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Monoidal categori(cid:28)cation of a multiplicative sequence of algebras . . . . . . 111
4.2.1 Alternative description of C as category of projective graded modules 113
4.3 Lusztig’s perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4 Example: Quiver-graded Springer theory . . . . . . . . . . . . . . . . . . . . 117
4.4.1 Quiver-graded Springer theory - Borel case . . . . . . . . . . . . . . 117
4.5 Example: Symplectic quiver-graded Springer theory. . . . . . . . . . . . . . 121
4.5.1 The Steinberg algebra and its horizontal product. . . . . . . . . . . . 126
4.5.2 Lusztig’s Perverse sheaves/Projective modules corresponding to the
vertices of the symmetric quiver. . . . . . . . . . . . . . . . . . . . . 132
4.5.3 Monoidal categori(cid:28)cation . . . . . . . . . . . . . . . . . . . . . . . . 135
4.6 A discussion on the search for Hall algebras for symmetric quiver represen-
tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5 Constructing collapsings of homogeneous bundles over quiver loci 146
5.0.1 Explicit equations for the image of the Springer map . . . . . . . . . 147
5.1 The orbit lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2 Quiver-graded Springer maps . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2.1 The generic composition monoid . . . . . . . . . . . . . . . . . . . . 149
5.2.2 When is the quiver-graded Springer map a resolution of singularities
of an orbit closure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.2.3 Resolution pairs for Dynkin quivers . . . . . . . . . . . . . . . . . . . 155
5.2.4 Resolution pairs for the oriented cycle . . . . . . . . . . . . . . . . . 156
5.2.5 Resolution pairs for extended Dynkin quivers . . . . . . . . . . . . . 158
5.3 Springer maps for homogeneous decomposition classes . . . . . . . . . . . . 160
6
5.3.1 Tube polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3.2 Springer maps for the Jordan quiver . . . . . . . . . . . . . . . . . . 166
5.3.3 Springer maps for homogeneous decomposition classes of the Kro-
necker quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Quiver (cid:29)ag varieties of (cid:28)nite type 176
6.0.4 A locally trivial (cid:28)bre bundle . . . . . . . . . . . . . . . . . . . . . . . 177
6.1 Categories of (cid:29)ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.1.1 What is a (cid:29)ag? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.1.2 On the tensor product KQ⊗KA . . . . . . . . . . . . . . . . . . . 180
n
6.1.3 Categories of monomorphisms . . . . . . . . . . . . . . . . . . . . . . 184
6.1.4 Description as ∆-(cid:28)ltered modules over the quasi-hereditary algebra
Λ = KQ⊗KA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
ν
6.1.5 When is X representation-(cid:28)nite? . . . . . . . . . . . . . . . . . . . . 188
6.2 Tangent methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2.1 An example of a not generically reduced quiver (cid:29)ag variety . . . . . 192
6.2.2 Detecting irreducible components . . . . . . . . . . . . . . . . . . . . 193
6.3 Strati(cid:28)cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.3.1 Strati(cid:28)cation in orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.3.2 Reineke’s strati(cid:28)cation . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.4 A conjecture on generic reducedness of Dynkin quiver (cid:29)ag varieties . . . . 197
6.4.1 Schemes de(cid:28)ned by rank conditions . . . . . . . . . . . . . . . . . . . 197
6.4.2 Quiver-related schemes de(cid:28)ned by rank condition . . . . . . . . . . . 199
6.4.3 Canonical decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.5 AnexampleofaclosureofaReinekestratumwhichisnotaunionofReineke
strata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.5.1 A -Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
2
6.5.2 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7 A -equioriented quiver (cid:29)ag varieties 216
n
7.0.3 Notation and basic properties for A -equioriented representations . 216
n
7.1 Reineke strata and root tableaux . . . . . . . . . . . . . . . . . . . . . . . . 218
7.1.1 Swapping numbered boxes in row root tableaux . . . . . . . . . . . . 221
7.1.2 Dimension of root tableau . . . . . . . . . . . . . . . . . . . . . . . . 221
7.2 rb-strata and row root tableaux . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.2.1 Split Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.3 rb-strati(cid:28)cation as a(cid:30)ne cell decomposition . . . . . . . . . . . . . . . . . . 228
7.3.1 Betti numbers for complete A -equioriented quiver (cid:29)ag varieties . . . 234
n
7.4 Conjectural part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.4.1 Canonical decomposition for A -equioriented quiver (cid:29)ag varieties . . 235
n
7.4.2 Submodules in terms of matrix normal forms . . . . . . . . . . . . . 236
7.4.3 Remarks on partial A -quiver (cid:29)ags . . . . . . . . . . . . . . . . . . . 239
n
7
7.5 Root tableau of hook type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
8 Appendix on equivariant (co)homology 246
8.0.1 A Lemma from Slodowy’s book . . . . . . . . . . . . . . . . . . . . . 246
8.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Equivariant Borel-Moore homology . . . . . . . . . . . . . . . . . . . . . . . 249
8.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2.2 Set theoretic convolution . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.2.3 Convolution in equivariant Borel-Moore homology . . . . . . . . . . 251
8.3 Duality between equivariant cohomology and equivariant Borel-Moore ho-
mology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.3.1 Equivariant derived category of sheaves after Bernstein and Lunts . 252
8.3.2 The functor formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.3.4 Localization for equivariant Borel-Moore homology . . . . . . . . . . 255
8.3.5 Cellular (cid:28)bration for equivariant Borel-Moore homology . . . . . . . 255
8.4 The Serre cohomology spectral sequence with arbitrary coe(cid:30)cients . . . . . 256
8.4.1 A lemma from the survey on Springer theory . . . . . . . . . . . . . 259
8.4.2 The cohomology rings of (cid:29)ag varieties. . . . . . . . . . . . . . . . . . 261
8.4.3 Forgetful maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.5 Equivariant perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.5.1 Perverse sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.5.2 Equivariant perverse sheaves . . . . . . . . . . . . . . . . . . . . . . 265
Bibliography 267
8
Chapter 1
A survey on Springer theory
Summary. A Springer map is for us a union of collapsings of (complex) homogeneous
vector bundles and a Steinberg variety is the cartesian product of a Springer map with
itself. Chriss and Ginzberg constructed on the (equivariant) Borel-Moore homology and
on the (equivariant) K-theory of a Steinberg variety a convolution product making it an
associativealgebra,wecallthisaSteinbergalgebra(cp. [CG97],2.7,5.2forthenonequiv-
ariant case). The decomposition theorem for perverse sheaves gives the indecomposable,
projective graded modules over the Steinberg algebra. Also this convolution yields a mod-
ule structure on the respective homology groups of the (cid:28)bres under the Springer maps,
which we call Springer (cid:28)bre modules. In short, for us a Springer theory is the study
of a Steinberg algebra together with its graded modules.
We give two examples: Classical Springer theory and quiver-graded Springer theory.
(1) De(cid:28)nitions and basic properties.
(2) Examples
(a) Classical Springer theory.
(b) Quiver-graded Springer theory.
(3) We discuss literature on the two examples.
1.1 De(cid:28)nition of a Springer theory
Roughly, following the introduction of Chriss and Ginzburg’s book ([CG97])1, Springer
theory is a uniform geometric construction for a wide class of (non-commutative) algebras
together with families of modules over these algebras. Examples include
(1) Group algebras of Weyl groups together with their irreducible representations,
(2) a(cid:30)ne Hecke algebras together with their standard modules and irreducible represen-
tations,
1Wetakeamoregeneralapproach,whatusuallyisconsideredasSpringertheoryyou(cid:28)ndintheexample
classical Springer theory. Nevertheless, our approach is still only a special case of [CG97], chapter 8.
9
(3) Hecke algebras with unequal parameters,
(4) Khovanov-Lauda-Rouquier-algebras(orshortlyKLR-algebras)andalternativelycalled
quiver Hecke algebras
(5) Quiver Schur algebras
For an algebraic group G and closed subgroup P (over C) we call G → G/P a principal
homogeneous bundle. For a given P-variety F we have the associated bundle de(cid:28)ned
by the quotient
G×P F := G×F/ ∼ , (g,f) ∼ (g(cid:48),f(cid:48)): ⇐⇒ there is p ∈ P: (g,f) = (g(cid:48)p,p−1f(cid:48))
and G×P F → G/P,(g,f) (cid:55)→ gP. Given a representation ρ: P → Gl(F), i.e. a morphism
of algebraic groups, we call associated bundles of the form G×P F → G/P homogeneous
vector bundles (over a homogeneous space).
De(cid:28)nition 1. The uniform geometric construction in all cases is given by the following:
Given (G,P ,V,F ) with I some (cid:28)nite set,
i i i∈I
(∗) G a connetcted reductive group with parabolic subgroups P .
i
We also assume there exists a maximal torus T ⊂ G which is contained in every P .
i
(∗) V a (cid:28)nite dimensional G-representation, F ⊂ V a P -subrepresentation of V, i ∈ I.
i i
We identify V,F with the a(cid:30)ne spaces having the vector spaces as C-valued points. Let
i
Ei := G×Pi Fi,i ∈ I and consider the following morphisms of algebraic varieties2:
(cid:70)
(cid:121)(cid:121)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)π(cid:116)(cid:116)E(cid:116)(cid:116)(cid:116):(cid:116)= i∈I(cid:79)E(cid:79)(cid:79)i(cid:79)(cid:79)(cid:79)µ(cid:79)(cid:79)(cid:79)(cid:79)(cid:79)(cid:39)(cid:39) (cid:123)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)(cid:120)[(cid:120)((cid:56)g,fi(cid:7))(cid:71)](cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:35)
(cid:70)
V i∈IG/Pi gfi gPi
(cid:70)
Then, E → V × G/P ,[(g,f )] (cid:55)→ (gf ,gP ) is a closed embedding (see [Slo80b],
i∈I i i i i
p.25,26), it follows that π is projective. We call the algebraic correspondence3 (E,π,µ)
Springer triple, the map π Springer map, its (cid:28)bres Springer (cid:28)bres. Via restriction
of E → V ×(cid:70) G/P to π−1(s) → {x}×(cid:70) G/P one sees that all Springer (cid:28)bres are
i∈I i i∈I i
(cid:70)
via µ closed subschemes of G/P .
i∈I i
2algebraic variety = separated integral scheme of (cid:28)nite type over a (cid:28)eld
3two scheme morpisms X (cid:111)(cid:111) p Z q (cid:47)(cid:47)Y are called algebraic correspondence, if p is proper and q
is (cid:29)at
10
