Lectures 12/9

Noncommutative algebras in- Rep Theory- Alg. Geom- Number Theory .

Rep Theory :

G- finite gps

modular repsk=I char ( k ) =p .

P / # G .

k [ G].

not semisimple .

-

infinitegroup

's

g topological ring Ifield .

e. g . Glen ( Op) - Ip-adic groups

GLn ( Eq K¥1) ) .

/

GLn ( E )

G Ln ( LR ).

automorphic repn theory studies reps of

G ( Aa ) = PTI"

G ( Op ) x GCR ).

mm

GEE Glen ( Op ) ] - mod

= ( abstract) E - reps of Gln ( Op ) .Me

⑥Ln ( Op ) has topology .

(Gij ) is close to In

if upffij - Sig ) ⇒ 0.

E - v. s.

care about continuous reps possibly -dim

g :Gln Cop ) → GL ( IL )

↳ discrete.

continuous ⇒ each vector v E V

is fixed by an open subgpc GL

n( Chp ) .

Opensub

gps

Glen (Ap ) D Glen ( Ep ) o K,ok

,o - - -

-

-

compact. [ nbhd basisJg Glen ( Zp ) g

"

open .

of In.

K.am/g=- In mod pi }.

V = U *K

Kc Gln Cop) G = Glen ( Chp) .

open .

compact .

Hak V'←

Hyuk- Lt : G → a }^

Eighths's't

.in?.:E.Kfs*:.EE:i.hY...:iiiiK

c- s

Ren ( G ) → Hqk - mod .Ing

k = Gln ( Zp ) .

H④iscommu-tatiue.GLnczpfh.CN/GLnCzp) → E.

(P"

Pa ?.

.

.

pan) a9n⇐ 2

I ⇐ f g E Gln ( Ep) , gmoop is upper twang}Iwahori

subspnez . ( p** ** ) * t Ep -

HG,I II.fin ( Op )/z

I

4:÷÷:? n.es!2- ( Ha

,z) is large .

-

Ha,I is f. g .

over 2- ( Ha,# )

-

R -zSpec ZCR) .

Mr

R lives over Spee Z ( R) .

as a sheaf of algebra .

¥Rp , THER . RIKI !

Heike alg . appears more generally .

¥① e G .

f : irrep of It

End ( Ind ,t (g ) )GG .

-

Hf

Homme#( g , Res , India, ( g ) ) .

Mr

H.fr/m.--.-sE#FgD.

9=1 . HGH.

e. g .

6=6 Ln ( Fg ) .

H -- HB ( Fg ) uppertriang .

HG,H= ① ( B City*↳ (

"TVB City ) )-

In" deformed

"

E [Sn ).

abstractly . Hg H EEE Q Csn ].

= -

Ind Bain!!)12 ) = Q [ Fln legs ]

.

-

o C V , c. . - C Un = Fg.

"

Gln ( Cl).

G- Ln ( IR).

8 .

connectedcompact Lie gps K e. s . Un

.

Irrep ( K ) → Irrep ( g )

This is a bijection G - (lie K )④ IC.

for K ⇐ Sieted.

sit. SUN

Repfd ( kn ) = Repp, ( ke ) = Repfd ( g ) .

"

Sun Ghul 'd.

E.÷: -⇐÷t¥00000

cg.lk?-mo@q qK

G µSimilar to Hq , - mod .

HA i,⇐

Glance ) - can

Gln ( IR ) > On.

Gln ( Q).

( g , K ) - mod

→ certain stategozofUII-mod.TT( category O : Beil in son

EE

Bernstein

Gelfand ).

g=$lz :( K) . K, Vo

:*::"in

..com .

( simple objects) pe, tf ,

f

-

l : :L- -

-

ek, e

V-z e Kine Vo

←→ . . .# .

I•

ef=o on Vo.

Vi = eigensp of h , eigenvalue 25.

Bettinson - Bernstein

cat 0 . - D - modules on FEIN.

§,

( systems of linear PDEs ).

Kazhdan'

- kuesztig Iconjecture .

.

i-

-

Pes on Fln.

Ulg ) - R →

Weyl algebra① ( Xi

,- -

- n

, Xn ,'21,

- - -

s 2n ).\

--

filtrationRei=

gr R = to RR± ,is commutative

.

( and E polynomial ring ) .

R is a quantization of gr R.

D - mod on X.

-

limit

( ooh .) sheaves on

C-

quantization 1-* X

.

( R,Rei) g⑦

a a

( tf , F) has a supportin Spee (gsk)

Dx-med M → subset of T*X .

( singular Supp .)

Alg .

Geom. X = projective smooth/ E

.

D¥(*) .

"

complexes of v. b . on X"

-

\ controlled by a"

small"

non - comm ring .

e. g .

X =P ?v w

Q -

- ( • Is . ).

F-- - - -

- -

yIb kept = DIKE?:-#0,OKA

Ro,

= Ezde).

Number Theory 1 Arithmetic

central simple alg - Ip .

Br ( Ohp).

→ 0/272.

-

Local class field theory .

abelian extras of F ( e.g .

FEE Chp ) .

⇒ Gal ( EIF)ab

-

mm

Gal ( FTE)ab- F

*

i* d Irae

7£ c- z

Br CF ) = H2 ( Gal ( ETH,

E't )

.-50/2

"

fundamental class"

-

Semi - linear algebrain studying geometry / char p

or p-adic

.

6 & k eg .

char Ck) =p .

or :( a ) = XP -

KI.CH

⇐ VIVCav) = old 44 )

.

Fom~pslp-doin.gsDieudonne

'

theory .I -n.si#.o:ig.X/k.charlk7-p.

-

H'toys ( Xlwck ) ) module over

mfs Wck)(¥?*¥ k¥7

"%fqe' -Manin classification

.

of modules over

WHIFF ) ( x ⇒.

-fi .