Quantization in charpLecture F

1 Hilbert schemes4Procesi bundles1 1 Varieties ofgl e Be Q Cartan diagmatrices

Way V503 AW symplecticn Y VW singular PoissonvarietyT E a V tata v 2 Iti v ti'd descends to 7Subtori Th It t ll teat acts vie Hamiltonianaction

To It tilted has contracting action on V Kony

Y has conical symplectic resolution X Hilton E parametritingcodim n ideals in ClayTAX from Ta OlayHave natl map p X 7 104 Sn Hilbert Chowideal to support counted w multiplicitiesTequiv't projective an isomorphism over thelocusofa pairwisedistinctpts so p is birational

Since Y isnormal p't GLY I Q x

12 Procesi bundle H DEV WG QEv3 Ew

graded fly Cvt algebra actually bigraded w5 in Leg oo b in Leg er and W in deg ee

Let P be a vectorbundle on X s t End P Itt an iso

of 144 algebras WAPbe vectorbundle autom's

Lemma eachfiber of P is AW as a Wmodule

Proof enough toshow I xeX1Px wQW be X is connectedV us p pale V64 pi pj i j verlWe 13

p X Y is an iso ever VYWEnd P to H specialize to xeV9W MaeICV ismax idealEnd R I H m Ecu ma W

PI av moIs

piggingirredmoduleofLim IWI

regular Wmodule

Renew a

Corollary usual and sign invariants PS P's are linebundles

Fact Pic X IR w Ola being ampleGin In

Definition Thm A Preastbundle on X is theunique Tequivariantvector bundle P w Fquirt fly alg Isom'm End P H

gs t

i Ext P P a firein PS t Oy Fequiv Isom'miii Pss I o

Constructions Haiman BetrukarnikerKaledin viaquantizations incharp tobe explained later Ginzburg

Uniqueness I L

Rem Cangeneralize this to Y V15 TcSp v isfinitesubgroup s t Y admits a symplic reselin Teequiv bundlesPW End P H KEV T i Ciii are classifiedby edits ofNamiKawa Weylgrip of Y 72272for 7 5 5 1 5

13 Connection to Combinatorics

Fact Thfixedpoints in X Hilton a are T fixed Emonomial ideals in Glxy Youngdiagrams X Youngdiagramfill it w monomials

pygtakethe ideal spanofremaining

monomials

X X E X fiberPa carries a Taction commuting

Waction i e is Gigraded Wmodule EW

Symmetricpolynomial w coeffis in g't t Frobeniuschar

Thm Macdonaldpositivity Hainan Thispolynomial is themodifiedMacdonaldpolynomial InAlternativeproof was foundby Betrukarnikor Finkelberg usingquantins incharp One advantage generalizes to wreathMacdonald

polynomials assoc te Sak 9 72

Anotherapplication ofconstrin via quantins incharp dooBoixedaAlvarez I L H PHP 01211 103for in

PHP Old RT PHP Old l H bimodulerelatedto t

2 Hamiltonian reduction

2.11 Setting U 0 CGL a all Ri End u UN G3 A g Vert R Jr E3

T R RORY Enda I End U via trpairing End u Uea0 ABi jaMomentmap j T R g f ggu A B i j Jp Ai Bj D E A B Ji jtr A B ij soyuckB i j AB tijeEndul g

2 2 Y VW as Hamiltonian reduction

Thm Gen Ginzburg Almostcommuting

Y Ju o G Poisson 2T equiv't Th RetR'tsimilarly tobefore

Sketch ofproofSteph GGprovedgu lo is the union ofna irreduciblecomponentsof codim n ji lo is acomplete intersection Eachcomponenthas a free Gorbit so ju is a submersion onthisorbit Soju lo is

generically reduced hence it's complete intersection reducedHence

ji lo115 is reduced

Step2 produce Y y 61116 Have V c juicexp Xn ly Yu 4 diag x 41 diagLyeyn 90imagesof Snconjugatepts are conjugate via monomialmatriceshence lie in the same Corbit

V ji lo

In jokeStep3 c is aclosedembedding Result ofWeye says thatv is generatedby Esxty glee Consider

Free GillesG Fe AB i j tr ABe ctfk.esExtySo C't is surjective

Step4 Since c is a closedembedding into areducedscheme so

to prove it's iso it'ssurjective ptsof ji lolls a closedGorbits inpilots CABij CABLE

Fact ric AB so A basiswheretheoperators AB are uppertriangular

Exercise Showthateveryclosed Gorbit injille intersects thelocus

Light al diagly yal he

c is surjective I

2 3 Hilton E as GIT Hamiltonianreduction

for CAji lol wat certain character

Basics on CITquotient G reductivegroup G GAZ affinevarietyorfinitetypescheme G C Qtgradedalgebra Clashing semiinvariants

GITquotient 11G Prying6639nA

gluedfromopen affine charts ofthefollowingform

f e Elzyint n on Zf 262176 03hG affine IfHESpec EEZ11G isgluedfrom21114 21116 Zp119 areglued

alongtheir commonopensubsetEffi119

1106parametrizes closedorbits in A semistablelocusIt z621 I n of fee 239 s t flt to

Example E T R G LetClassical invariant theory shows that

Z algebra g 14239 isgenerated by elements ofthe

Let f IAB i f AB i f AB i f the Qcxy

RJ's AB i j one ofLet's isnonzero U Span fIAB if AB i QcAB is UGaction on this locus isfree So

Ju o 09parameterites all orbits in f los

is smooth

symplectic

Identification w Hilbertscheme It LetExercise AB i j Ej lot's j o AB oRecAB is UIdentification g let G Hill a

AYB.is A fetag IflaB so teaB i oExercise showthis is a bijection ofsets

Rems Have comm've diagram Z

b b quotientmorphisms

119 119

wherep Prey E 6639 Spec ECZI natural

projective morphism

If Z j lol then p is the HilbertChowmorphismHilton ay e Isn

Amplegenerator 040 on X is the linebundleobtainedfromcharr Let G E by equivit descentforguko

slo G