The affine Yangian of gl₁, and the infinitesimal Cherednik algebras

186 Pages·2014·6.33 MB·English

The Affine Yangian of of, and the Infinitesimal
Cherednik Algebras
by
Oleksandr Tsymbaliuk
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy MA&SACHUSETTS liNSfl1JTE
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
0B- RA R IES
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
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Author ......
/
Department of Mathematics
April 30, 2014
Signature redacted
Certified by...
Pavel Etingof
Professor of Mathematics
Thesis Supervisor
Signature redacted
Accepted by ..
Alexei Borodin
Chairman, Department Committee on Graduate Theses
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The Affine Yangian of grl and the Infinitesimal Cherednik
Algebras
by
Oleksandr Tsymbaliuk
Submitted to the Department of Mathematics
on April 30, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In the first part of this thesis, we obtain some new results about infinitesimal Chered-
nik algebras. They have been introduced by Etingof-Gan-Ginzburg in [EGG] as
appropriate analogues of the classical Cherednik algebras, corresponding to the re-
ductive groups, rather than the finite ones. Our main result is the realization of those
algebras as particular finite W-algebras of associated semisimple Lie algebras with
nilpotent 1-block elements. To achieve this, we prove its Poisson counterpart first,
which identifies the Poisson infinitesimal Cherednik algebras introduced in [DT] with
the Poisson algebras of regular functions on the corresponding Slodowy slices. As a
consequence, we obtain some new results about those algebras. We also generalize
the classification results of [EGG] from the cases GL, and SP2n to SOl.
In the second part of the thesis, we discuss the loop realization of the affine Yangian
of g~l. Similar objects were recently considered in the work of Maulik-Okounkov on
the quantum cohomology theory, see [MO]. We present a purely algebraic realization
of these algebras by generators and relations. We discuss some families of their
representations. A similarity with the representation theory of the quantum toroidal
algebra of gli is explained by adapting a recent result of Gautam-Toledano Laredo,
see [GTL], to the local setting. We also discuss some aspects of those two algebras such
as the degeneration isomorphism, a shuffle presentation, and a geometric construction
of the Whittaker vectors.
Thesis Supervisor: Pavel Etingof
Title: Professor of Mathematics
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Acknowledgments
First of all, I wish to express my deepest gratitude to Professors Pavel Etingof, Boris
L'vovich Feigin, Michael Finkelberg and Ivan Losev for their guidance and constant
patience for many years. I am sincerely grateful to Pavel Etingof for his beautiful
ideas, generosity, good sense of humor and also for his endless enthusiasm; to Boris
Feigin for the guidance and enlightening discussions which always inspired me, espe-
cially in the most desperate moments; to Michael Finkelberg for getting me addicted
to the representation theory since 2004 and for his endless support and encourage-
ment over the years; and to Ivan Losev for his friendly advice during these years and
for his interesting math talks spiced up with the Belorussian firemen jokes.
I would also like to thank Professors Roman Bezrukavnikov and Dennis Gaitsgory
for teaching me modern aspects of the geometric representation theory, even though
I was never able to follow those things on my own. I wish to thank Professor Valerio
Toledano Laredo for being a part of the Dissertation Committee.
For their help with different non-academic issues I wish to thank the staff of
our department: Cesar Duarte, Michele Gallarelli, Galina Lastovkina, Linda Okun,
Anthony Pelletier, and Barbara Peskin.
I also owe special thanks to my best friends Giorgia Fortuna, Andrei Negut, Sam
Raskin, and Daniele Valeri. Without them my years at MIT would not have been so
colorful. In particular, I would like to thank Giorgia for being such a great and lively
person, always ready to help; Andrei for being such an extraordinary and inspiring
guy; Sam for always knowing how to "categorify" something and being helpful and
patient; and Daniele for his fantastic sense of humor, which I especially appreciate. I
want to thank them for their friendship and support.
Among the people who I want to thank are more of my friends and colleagues:
Alex, Alisa, Andrea, Bhairav, Francesco, Galya, Kostya, Roberto, Roma, Salvatore,
Tsao-Hsien, Yaping.
Finally, I would like to thank my family and my beloved for endless love.
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Contents
1
Introduction 11
1.1 Continuous Hecke algebras . . . . . . . . . . . . 11
1.1.1 Algebraic distributions . . . . . . . . . . 11
1.1.2 Continuous Hecke algebras . . . . . . 12
1.2 Infinitesimal Cherednik algebras . . . . . . . 13
1.2.1 Infinitesimal Cherednik algebras . . . 13
1.2.2 Classifications for gL, and -SP2. . -. 14
1.3 The quantum toroidal and the affine Yangian of oil . . 15
1.4 Organization of the thesis . . . . . . . . . . 15
2 Infinitesimal Cherednik algebras 17
2.1 Representations of HC(g[,) . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 The Shapovalov Form . . . . . ... . . . . . . . . . . . 18
2.1.3 The Casimir Element of Hc(g[,) . . . . . . . . . . . . . 22
2.1.4 The center . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.5 Action of the Casimir Element on the Verma Mo dule . . . . . 24
2.1.6 Finite Dimensional Representations . . . . . . . . . . . . . . . 27
2.1.7 Rectangular Nature of Irreducible Representations . . . . . . . 27
2.1.8 Existence of L(A) with a given shape . . . . . . . . . . . . . . 31
2.2 Poisson Infinitesimal Cherednik Algebras . . . . . . . . . . . . . . . . 34
2.2.1 Poisson Infinitesimal Cherednik Algebras of gl. . . . . . . . . 34
2.2.2 Passing from Commutative to Noncommutative Algebras 37
7
2.2.3 Poisson infinitesimal Cherednik algebras of SP2n. . . . . . . . . 41
2.2.4 Proof of Lemma 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . 45
3 Infinitesimal Cherednik algebras as W-algebras 47
3.1 B asics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Length of the deformation . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Proof of Lemmas 3.1.1, 3.1.2 . . . . . . . . . . . . . . . . . . . 48
3.1.3 Universal algebras Hm(g(n) and Hm(SPn) . . . . . . . . . . 50
2
3.1.4 Poisson counterparts of Hm(g) . . . . . . . . . . . . . . . . . . 51
3.1.5 W -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.6 Additional properties of W-algebras . . . . . . . . . . . . . . . 53
3.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Poisson analogue of Theorem 3.2.2 . . . . . . . . . . . . . . . . . . . 58
3.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1 Centers of Hm(g[n) and Hm(s3 n) . . . . . . . .... . . . . 63
2
3.4.2 Symplectic leaves of Poisson infinitesimal Cherednik algebras . 64
3.4.3 The analogue of Kostant's theorem . . . . . . . . . . . . . . . 65
3.4.4 The category 0 and finite dimensional representations . . . . . 65
3.4.5 Finite dimensional representations of Hm(g[n). . . . . . . . . 67
3.4.6 Explicit isomorphism in the case g = gl . . . . . . . . . . ..6.8
3.4.7 Higher central elements . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Com pletions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.1 Completions of graded deformations of Poisson algebras . . . . 74
3.5.2 Decompositions (3.10) and (3.11) for m = -1, 0 . . . . . . . . 77
4 Generalization to the SON case 81
4.1 Classification results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 The Poisson center of algebras H'(soN) . . . . . ........... 89
4.5 The key isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8
4.5.1 Algebras Hm(soN, VN) . . . . .. ......... . . . . . . 93
4.5.2 Isomorphisms e and ec . . . . . . . . . . . . . . . . . . . ... 94
4.6 The Casimir element . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 The affine Yangian of grl 99
5.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.1 The toroidal algebra of g~l . . . . . . . . . . . . . . . . . . . . 99
5.1.2 Elements ti E Uqiq2,q3(g1) . . . . . . . . . . . . ... .. 100
5.1.3 The affine Yangian of g9l . . . . . . . . . . . . . . . . . . . . . 103
5.1.4 Generating series forkh,h ,h (grl) . . . . . . . . . . . . . . . . 104
1 2 3
5.2 Representation theory via the Hilbert scheme . . . . . . . . . . . . . 106
5.2.1 Correspondences and fixed points for (A2) [n] . . . . . . . . . . 106
5.2.2 Geometric Uq1,q2,q3 (g(1)-action I . . . . . . . . . . . . . . . . . 106
5.2.3 Geometric kh1,h2,h3 (g1)-action I . . . . . . . . . . . . . . . . . 108
5.3 Representation theory via the Gieseker space . . . . . . . . . . . . . . 112
5.3.1 Correspondences and fixed points for M(r, n) . . . . . . . . . 112
5.3.2 Geometric Uq,q ,q (g[)-action II . . . . . . . . . . . . . . . . . 114
2
5.3.3 Geometric Yhl,h ,h(g,)-action II . . . . . . . . . . . . . . . . . 115
2
5.3.4 Sketch of the proof of Theorem 5.3.2 . . . . . . . . . . . . . . 117
5.3.5 Sketch of the proof of Theorem 5.3.4 . . . . . . . . . . . . . . 120
5.4 Some representations of Uqj,q2,q3(gr1) and khjh2,h3(9r1) . . . . . . . . . 123
5.4.1 Vector representations . . . . . . . . . . . . . . . . . . . . . . 123
5.4.2 Fock representations . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.3 The tensor product of Fock modules F(u) . . . . . . . . . . . 126
5.4.4 The tensor product of Fock modules aF(u) . . . . . . . . . . . 128
5.4.5 Other series of representations . . . . . . . . . . . . . . . . . . 130
5.4.6 The categories 0 . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5 Limit algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.5.1 Algebras Dh and 6h . . . . . . . . . . . . . . . . . . . . . . . . 134
5.5.2 Algebras 'Dh and lih . . . . . . . . . . . . . . . . . . . . . . . 134
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5.5.3 The isomorphism To ............... 135
5.5.4 The renormalized algebra U(gL) .................. 135
1
5.5.5 The renormalized algebra h(g ) . . . . . . . . . . . . . . . . 138
1
5.5.6 Proof of Theorem 5.5.5 . . . . . . . . . . . . . . . . . . . . . . 139
5.5.7 Proof of Theorem 5.5.9 . . . . . . . . . . . . . . . . . . . . . . 145
5.6 The homomorphism T . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.6.1 Construction of T . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.6.2 The limit h3 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6.3 The elliptic Hall algebra . . . . . . . . . . . . . . . . . . . . . 152
5.6.4 Flatness of the deformations . . . . . . . . . . . . . . . . . . . 154
5.6.5 The linear map chr . . . . . . . . . . . . . . . . . . . . . . . .. 156
5.7 Small shuffle algebras Sm and Sa . . . . . . . . . . . . . . . . . . . . 157
5.7.1 The shuffle algebra Sm . . . . . . . . . . . . . . . . . . . . . . 157
5.7.2 Commutative subalgebra A' C S" . . . . . . . . . . . . . . . 158
5.7.3 The shuffle algebra Sa . . . . . . . . . . . . . . . . . . . . . . 160
5.7.4 Commutative subalgebra A" C Sa . . . . . . . . . . . . . . . . 161
5.8 The horizontal realization of U',q ,q (gi) . . . . . . . . . . . . . . . . 163
2 3
5.8.1 The horizontal realization via . . . . . . . . . . . . . . . . . 163
5.8.2 Modules V(u), F(u) in the horizontal realization . . . . . . . . 164
5.8.3 The matrix coefficient realization of Am . . . . . . . . . . . . 165
5.8.4 The Whittaker vector in the K-theory case . . . . . . . . . . . 167
5.8.5 The Whittaker vector in the cohomology case . . . . . . . . . 167
5.8.6 Sketch of the proof of Theorem 5.8.5 . . . . . . . . . . . . . . 168
5.8.7 Sketch of the proof of Theorem 5.8.6 . . . . . . . . . . . . . . 170
A Future work 173
A.1 Quantum toroidal q,(s ) and the big shuffle algebras . . . . . . . . 173
A.2 Subalgebras A(s ,..., s) . . . . . . . . . . . . . . . . . . . . . . . . 178
1
A.3 Degeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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