K-Motives and Koszul Duality in
Geometric Representation Theory
I . Back to the 80's : How to come up with Koszul duality?
Tf . Status quo .tf . A K- theoretic perspective .
Jens Niklas Eberhardt , Uni Bonn
Koszul Duality: First Example
A- kenya A ! -- hasher ]
End#CAI -- A Ext;!k ) -- H' Honihtfi, ok )
=kEx4×2 = keisha
at iii.⇒ ⇐ iiiiiiii:# 3,2¥.ae#otaDbfA-mod*t=DbfA.mod2t1omalityofEnd'HEEMAsi ) ⇒ Kiki]
Koszul Duality: Second Example
Oocslzktt-SMlol.MG.DE?eg-nodh.ss.
DbyCPI.ie) P ' f- {0 ,.}
MCs.O) ((s. O)
motmot -- FYI's.o,PG.fi/:neoi--/:yY!o,Iee=E.Ies--ItDC=EndCPCs.0l)--Clarks ( = Ext
'
( Ies ) - H' ( Pic) -- ClerksA- = End (Plot ④Pcso) ) A=ExtTIee④Ies )= at ¥ = at ¥
OF = Atoned# DICK .
u Very mysterious !(PH.PG.AZ ,>,
E SIG, Ies > ers ,
Now what ? ?=
- generalize? - Rep .tk . meaning of 0ft ?- grading on D:(Pic) ? - Nat
. functor ?✓
x r X
Koszul Duality: Generalize
hokey complex reductive Lie algebra TKBVCGV Langlands dualrdalggp.PCw.OIEO.ly ) ,weW W -- Weyl group F- gyp Xj=BuB%jS={KilweW)
C-schyschgyiew-IECXT.de/JgCF.E)C=EndlPCwo.0)) GEXTCIevot-HCH.ci)
C- Mod CjMod#(Pcu .o) ?¥ ( Dow ) ← soergelmodules-fDD.io.ms#SIewlweW7+o.eDA--EndcCtODu)OoZ--A-modZDYCHE )
uu
( pcv.co) >④ is ,>€ SIG Zo
, try
Koszul Duality: Mixed Geometry
How to obtain graded version D§Mf , of Dgbcertellc ) ?(a) Dbc MHM(XI)) mixed Hodge modules
" ( FYI(b) Dmix -
p , Ee ) mixed e-adic sheaves
typo cc) DMCXT#p .Q ) mixed motives
D.(Q-mod't
)=4QCnD, CDM ( spec #p , Q ) (n)#t ! x d Tast¥ shift of grading
DBCQ-mod )= SQ ), C D ( pt.IQ )
DICK! 1=5 www.CDMCXY#p.Q)rs t t to
DIY! 1=50*7, a Duke.IQ)
Koszul Duality: Mixed Geometry (cont.)
Amazing properties of DEMKO ) :• Six functors : ft, f-* if , .fi , ①,How• Chow weight structure (= co - t- structure )
motives of spmoothproj.oar.ltD§MfiQ)w=o = Chow (ftp.QINDYMKQ )
or = f Ie w Cul E2nD④,E
decomp . thm .
E f Dw (Ln)>⑦ c C -Mod#
D}Mf!Q ) Kb CD}Mf!Q )w=o) I Kbc s Dusan>↳ IDondarko's weight complex functor
Koszul Duality: Status Quo
Pcw . o) Icw
Ll ) ( 17523
Dbcoftcg ) )# DYCE ,
Te de
kblprojloftgDIKKIR.sn#IKKDbjmCE,C)w=o)
Alternatively : ch (1) 523
Dime.at#D;mcei.Q)
Koszul Duality: Tabula Rasa
Why not get rid of the grading? ?{ 1) c- ( 1752]
4) D- ( 07cg) )#Djm ( tic ) ( 11h23
it onto:* ? ! it
K-Motives: Introduction ( t ) ( 11523'
DMCXA-p.IQ ) '
✓ /Iommukµ \L L withsixfct . vyid Ducal
, DKCH#p ,lid
P'
K-Motivessua
%÷⇒⇒.
"h!:p!:L.IT#oIooTHomDkcsgCQiQCp7tg3l=Kap-qCS)n0Qvsfp-thtdamseigenspaceltompncgCQ.QCpltqD-HRFCS.de)
K-Motives: On the flag variety
Amazing properties of DKGCHQ ) =L xD, c DK CF,Q)• Six functors : ft, f-* if , .fi , ①,Hom• Chow weight structure (= co - t- structure )
K- motives of spmoothproj.oar.ltDugCfiQ )w=o=kChow (ftp.QINDKglf.Q )
= f Ie w Zo,e ←
" intersection K -theory"
E T Dw>⑦ c C -Mod
D KoffiQ ) Kb ( DKy CfiQ )w=o) I Kbc s DwH )Dondarko's weight complex functor
Koszul Duality: Tabula Rasa (cont.)
"
K - motives are Kosal - dual to constr.sheaves
"
Ll ) c- ( 1752]Ll ) Db ( Oft g) )c- Djm ( tic ) cited]
I t .- t lid D
'
( 0dg) )I DKgH! id
AlternativelyDgb CX , 9)a- Dkgcei , Q )
Koszul Duality: Equivariant/Unipotenly Monodromic
→
m
v r v nm ⑧ Be Bo
H'TBTY → HCDTT KCDT) ← HH*I=KCRepT¥Rti=R11 11 11
Q11
Sciuto) TTSicxtto.IT/SiCxetTo) ①EXIT]
HIBTt-KCBTTo.TK#k.I-Atiyah-Segalcompletionthm .
+ QI.ctD-QLXCTD-kifx.to, duality forton-
HCBTT -_ RI -- IQEITCTDI
Koszul Duality: Equivariant/Monodromic
Why not get rid of grading + completion ?Conjecture : free monoatomic tilting equivariant intersection K-theory complex
①BxB -monk g) I DKBW.sc gu )
x xKb(R -SBImod )
K- theoretic Foergel bimodules
ANTI = R -- Ktlpttq
Koszul Duality: Equivariant/Monodromic (Example)
F- B=g T=Bu=gV
DEEM.NET ) L Dktxtct ) Q- is
-
" "
D ( T ) Dktecpt )T-mon
DIET ) / L "O%cDKHpt) /b" H
Daet , Cpt ) v
u ✓DVR -mod ) R
DTR -mod) R
K-Motives: Further Directions
→
→ Sooyd conjecture
↳ - . -
