Graded Parabolic Induction for Category Ojens/data/talkparabolicinduction.pdf · Category Ofrom the...

Graded Parabolic Induction for Category O

Albert-Ludwigs-Universität Freiburg

Jens Niklas Eberhardt20. Juni 2016

Motivation for Category O

Fundamental problem: Describe symmetries of some space X.

G#X

semi-simple Lie group,e.g. SLn(R), SO(n,m), SLn(C) . . .acting on some smooth manifold,e.g. some quotient G/H.

Let us turn this into algebra!

20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 2 / 14



G#X






G#X

semi-simple Lie group,e.g. SLn(R), SO(n,m), SLn(C) . . .

acting on some smooth manifold,e.g. some quotient G/H.





G#X






G#X






G# Fun(X ,C)

semi-simple Lie group,e.g. SLn(R), SO(n,m), SLn(C) . . .acting on some space of functions,e.g. C ∞(X ),Lp(X ), . . .




First dilemma:

no canonical choice of Fun(X ,C),

still in the realm of functional/harmonic analysis: full of∫

and∞

∑.



First dilemma:no canonical choice of Fun(X ,C),


and∞

∑.



First dilemma:no canonical choice of Fun(X ,C),


and∞

∑.



Solution by Harish-Chandra (1960’s):

(g,K )#

g = Lie(G), the Lie algebra,K ⊂G, a maximal compact subgroupacting on K -finite vectors in Fun(X ,C),i.e. v ∈ Fun(X ,C) with dimC〈Kv〉< ∞

This is called a Harish-Chandra module.

Theorem (Casselman-Wallach, Kashiwara-Schmid):

You can functorially globalize a Harish-Chandra module to G.




(g,K )#

g = Lie(G), the Lie algebra,K ⊂G, a maximal compact subgroup

acting on K -finite vectors in Fun(X ,C),i.e. v ∈ Fun(X ,C) with dimC〈Kv〉< ∞







(g,K )# Fun(X ,C)K








(g,K )# Fun(X ,C)K








(g,K )# Fun(X ,C)K







Special case: G complex Lie group, e.g. G = SLn(C). Then

H C (g,K )#

the category of Harish-Chandra modulesacts via tensor product onCategory O from the title of this talk!

Category O is purely algebraic and just depends on g.

Insight (Bernstein-Gelfand-Gelfand 1971):

You can understand H C (g,K ) via just using O(g).




H C (g,K )#

the category of Harish-Chandra modulesacts via tensor product on

Category O from the title of this talk!







H C (g,K )#O(g)








H C (g,K )#O(g)








H C (g,K )#O(g)






Definition of Category O

Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def=

{L(λ )|λ ∈ h∗}

ext .

L(λ ) simple highest weight module, i.e. generated by anhighest weight vector v+ 6= 0 with

1 H.v+ = λ (H)v+ for H ∈ h and2 b.v+ ⊆ 〈v+〉C.

Take all extensions of those! (in the category of weightmodules)



Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def=

{L(λ )|λ ∈ h∗}

ext .






Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def=

{L(λ )|λ ∈ h∗}

ext .






Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def=

{L(λ )|λ ∈ h∗}

ext .


1 H.v+ = λ (H)v+ for H ∈ h and

2 b.v+ ⊆ 〈v+〉C.




Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def=

{L(λ )|λ ∈ h∗}

ext .






Cartan Borel

h b g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(g) def= 〈L(λ )|λ ∈ h∗〉ext .





Parabolic Induction

Parabolic induction: inductively construct modules in O(g):

Levi Parabolic

l p g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)


Parabolic Induction

Parabolic induction: inductively construct modules in O(g):Levi Parabolic

l p g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)


Parabolic Induction


l p g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(l)

M

∈

Category O for l (block matrices)


Parabolic Induction


l p g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(l)

M

∈

Category O for l (block matrices)


Parabolic Induction


l p g

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)

O(l)

M M

∈

Category O for l (block matrices)extend trivially to p


Parabolic Induction


l p g n

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)0 0

.. .

∗ 0

O(l)

M M

∈



Parabolic Induction


l p g n

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)0 0

.. .

∗ 0

O(l)

M M C[n]⊗C M

∈


tensor with free n module.20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14

Parabolic Induction


l p g n

∗ 0

...

0 ∗

∗ ∗

. . .

0 ∗

sln(C)0 0

.. .

∗ 0

O(l) O(g)

M M C[n]⊗C M

Indgp

∈ ∈


tensor with free n module.20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14

First Result

Goal: Understand parabolic induction better!

First step: Make the problem finite:

Theorem (E.):

Parabolic induction Indgp :O(l)→O(g) can be understood in

terms of a finite family of functors

Indw : O0(l) → O0(g)

subcategories generatedby finitely many simples:Principal blocks


First Result

Goal: Understand parabolic induction better!First step: Make the problem finite:

Theorem (E.):






First Result


Theorem (E.):






First Result


Theorem (E.):






Geometric Representation Theory

Second step: apply methods from geometric representationtheory.

Philosophy:

Solve problems in representation theory using geometric andcohomological methods!

Kazdhan-Lusztig (1979):

O(g)!Geometry of Schubert varieties.




Philosophy:







Philosophy:







Philosophy:



O(g)!

Geometry of Schubert varieties.




Philosophy:





Flag variety

G∨ = SLn(C)⊃ B = ∗0

[ ]Langlands dual group to g⊃ b.

Flag variety:

X∨ = G∨/B = {0⊆ V1 ⊆ ·· · ⊆ Vn−1 ⊆ Cn|dimVi = i}

=⊎

w∈Sn

BwB/B

For w ∈ Sn, BwB/B is called a Schubert variety.Singularities in Schubert varieties reflect complexity ofsimple modules in O(g).


Flag variety

G∨ = SLn(C)⊃ B = ∗0


Flag variety:

X∨ = G∨/B = {0⊆ V1 ⊆ ·· · ⊆ Vn−1 ⊆ Cn|dimVi = i}

=⊎

w∈Sn

BwB/B



Flag variety

G∨ = SLn(C)⊃ B = ∗0


Flag variety:

X∨ = G∨/B = {0⊆ V1 ⊆ ·· · ⊆ Vn−1 ⊆ Cn|dimVi = i}

=⊎

w∈Sn

BwB/B

For w ∈ Sn, BwB/B is called a Schubert variety.

Singularities in Schubert varieties reflect complexity ofsimple modules in O(g).


Flag variety

G∨ = SLn(C)⊃ B = ∗0


Flag variety:

X∨ = G∨/B = {0⊆ V1 ⊆ ·· · ⊆ Vn−1 ⊆ Cn|dimVi = i}

=⊎

w∈Sn

BwB/B



Graded Geometric Parabolic Induction

Same for l andthe associatedflag variety Y∨.

Soergel-Wendt 2015:Derived category ofcertain motives (up-graded sheaves) onX∨.

Graded version ofderived category O.

DerZ,b(C) (Y∨) DerZ,b(B) (X∨)

Derb(O0(l)) Derb(O0(g))

Is there a geometric and graded version of parabolic induction?

Yes! Use motivic six functor formalism!







Derb(O0(l)) Derb(O0(g))

v









⊕n∈Z

Hom(M,N〈n〉)

Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))

v









⊕n∈Z

Hom(M,N〈n〉)


v v









⊕n∈Z

Hom(M,N〈n〉)

Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))Indw

v v









⊕n∈Z

Hom(M,N〈n〉)


?

Indw

v v









⊕n∈Z

Hom(M,N〈n〉)


?

Indw

v v









⊕n∈Z

Hom(M,N〈n〉)


hw ,∗ pr!

Indw

v v

Theorem (E.):

There are maps Y∨pr←− Y∨×Cl(w) hw−→ X∨ such that ? = hw ,∗pr! .


Preprint

Jens Niklas Eberhardt, Graded parabolic induction andstratified mixed Tate motives, arXiv:1603.00327 (2016)


Further Directions

Joint with Shane Kelly (postdoc at the RTG), construction of amodular analogue:

DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))

We construct category of cer-tain motives with a motivic sixfunctor formalism (using Ayoub2007)

Derived modular category O(defined in Soergel 2001) Representation theory of al-gebraic groups over Fp

Jens Niklas Eberhardt and Shane Kelly, A motivic six functorformalism for the graded modular category O, in preparation


Further Directions


DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))





Further Directions


DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))v





Further Directions


DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))v





Thank you foryour attention!


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