Graded Parabolic Induction for Category Ojens/data/talkparabolicinduction.pdf · Category Ofrom the...
Transcript of 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
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
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
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
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
Motivation for Category O
Fundamental problem: Describe symmetries of some space 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 ), . . .
Let us turn this into algebra!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 2 / 14
Motivation for Category O
First dilemma:
no canonical choice of Fun(X ,C),
still in the realm of functional/harmonic analysis: full of∫
and∞
∑.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 3 / 14
Motivation for Category O
First dilemma:no canonical choice of Fun(X ,C),
still in the realm of functional/harmonic analysis: full of∫
and∞
∑.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 3 / 14
Motivation for Category O
First dilemma:no canonical choice of Fun(X ,C),
still in the realm of functional/harmonic analysis: full of∫
and∞
∑.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 3 / 14
Motivation for Category O
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 4 / 14
Motivation for Category O
Solution by Harish-Chandra (1960’s):
(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〉< ∞
This is called a Harish-Chandra module.
Theorem (Casselman-Wallach, Kashiwara-Schmid):
You can functorially globalize a Harish-Chandra module to G.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 4 / 14
Motivation for Category O
Solution by Harish-Chandra (1960’s):
(g,K )# Fun(X ,C)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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 4 / 14
Motivation for Category O
Solution by Harish-Chandra (1960’s):
(g,K )# Fun(X ,C)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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 4 / 14
Motivation for Category O
Solution by Harish-Chandra (1960’s):
(g,K )# Fun(X ,C)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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 4 / 14
Motivation for Category O
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 5 / 14
Motivation for Category O
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 on
Category 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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 5 / 14
Motivation for Category O
Special case: G complex Lie group, e.g. G = SLn(C). Then
H C (g,K )#O(g)
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 5 / 14
Motivation for Category O
Special case: G complex Lie group, e.g. G = SLn(C). Then
H C (g,K )#O(g)
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 5 / 14
Motivation for Category O
Special case: G complex Lie group, e.g. G = SLn(C). Then
H C (g,K )#O(g)
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 5 / 14
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)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
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)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
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)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
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 and
2 b.v+ ⊆ 〈v+〉C.
Take all extensions of those! (in the category of weightmodules)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
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)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
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)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 6 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):
Levi Parabolic
l p g
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)
O(l)
M
∈
Category O for l (block matrices)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)
O(l)
M
∈
Category O for l (block matrices)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)
O(l)
M M
∈
Category O for l (block matrices)extend trivially to p
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g n
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)0 0
.. .
∗ 0
O(l)
M M
∈
Category O for l (block matrices)extend trivially to p
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g n
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)0 0
.. .
∗ 0
O(l)
M M C[n]⊗C M
∈
Category O for l (block matrices)extend trivially to p
tensor with free n module.20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 7 / 14
Parabolic Induction
Parabolic induction: inductively construct modules in O(g):Levi Parabolic
l p g n
∗ 0
...
0 ∗
∗ ∗
. . .
0 ∗
sln(C)0 0
.. .
∗ 0
O(l) O(g)
M M C[n]⊗C M
Indgp
∈ ∈
Category O for l (block matrices)extend trivially to p
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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 8 / 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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 8 / 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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 8 / 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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 8 / 14
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 9 / 14
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 9 / 14
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 9 / 14
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 9 / 14
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.
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 9 / 14
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 10 / 14
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 10 / 14
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 10 / 14
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).
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 10 / 14
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!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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))
v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))
v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))
v v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))Indw
v v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))
?
Indw
v v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))
?
Indw
v v
Is there a geometric and graded version of parabolic induction?
Yes! Use motivic six functor formalism!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
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∨)
⊕n∈Z
Hom(M,N〈n〉)
Derb(O0(l)) Derb(O0(g)) Hom(v (M),v (N))
hw ,∗ pr!
Indw
v v
Theorem (E.):
There are maps Y∨pr←− Y∨×Cl(w) hw−→ X∨ such that ? = hw ,∗pr! .
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 11 / 14
Preprint
Jens Niklas Eberhardt, Graded parabolic induction andstratified mixed Tate motives, arXiv:1603.00327 (2016)
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 12 / 14
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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 13 / 14
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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 13 / 14
Further Directions
Joint with Shane Kelly (postdoc at the RTG), construction of amodular analogue:
DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))v
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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 13 / 14
Further Directions
Joint with Shane Kelly (postdoc at the RTG), construction of amodular analogue:
DerZ,b(B) (X∨/Fp,Fp) Derb(O(G/Fp))v
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
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 13 / 14
Thank you foryour attention!
20. Juni 2016 Jens Niklas Eberhardt – Graded Parabolic Induction for Category O 14 / 14