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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration Koszul Duality and Geometric Satake for SL 2 ( R ) Oliver Straser Annual conference of the DFG priority programme in representation theory, SPP 1388 March 27, 2013

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Koszul Duality and Geometric Satake for SL2(R)

Oliver Straser Annual conference of the DFG priority programme in representation theory, SPP 1388 March 27, 2013

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Motivation

G = PSL2(C), ˇ g := sl2(R) ⊗ C and K = SO2(R)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Motivation

G = PSL2(C), ˇ g := sl2(R) ⊗ C and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Motivation

G = PSL2(C), ˇ g := sl2(R) ⊗ C and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character X Adams-Barbasch-Vogan parameter space of G. (will be defined later)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Motivation

G = PSL2(C), ˇ g := sl2(R) ⊗ C and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character X Adams-Barbasch-Vogan parameter space of G. (will be defined later) Koszul Duality roughly identifies: H C Db

G(X)

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

G = PSL2(C), ˇ g := Lie SL2(C) and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character X Adams-Barbasch-Vogan parameter space of G. (will be defined later) Koszul Duality roughly identifies: H C

F⊗

  • Db

G(X)

  • F finite dimensional SL2(C) representation.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

G = PSL2(C), ˇ g := Lie SL2(C) and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character X Adams-Barbasch-Vogan parameter space of G. (will be defined later) Koszul Duality roughly identifies: H C

F⊗

  • Db

G(X)

  • ?
  • F finite dimensional SL2(C) representation.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

G = PSL2(C), ˇ g := Lie SL2(C) and K = SO2(R) H C := {(ˇ g, K)-modules of finite length} with integral infinitisimal character X Adams-Barbasch-Vogan parameter space of G. (will be defined later) Koszul Duality roughly identifies: H C

F⊗

  • Db

G(X)

  • ?
  • F finite dimensional SL2(C) representation.

Question Is there a nice (geometric) desciption of “?”

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Outline

1

Koszul Duality

2

Geometric Satake

3

Connecting both pictures

4

Geometric Tensoration

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Notation

G = PSL2(C), fix a Borel B and a maximal torus T ⊂ B determining a system of pos. roots.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Notation

G = PSL2(C), fix a Borel B and a maximal torus T ⊂ B determining a system of pos. roots. ˇ g ⊃ ˇ h cartanian. Via the Harish-Chandra isomorphism we indentify the set

  • f integral infinitisimal characters with N0

{Inf. Chars. of ˇ g} ∼ = ˇ h∗/W ⊃ N0ρ ∼ = N0

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Notation

G = PSL2(C), fix a Borel B and a maximal torus T ⊂ B determining a system of pos. roots. ˇ g ⊃ ˇ h cartanian. Via the Harish-Chandra isomorphism we indentify the set

  • f integral infinitisimal characters with N0

{Inf. Chars. of ˇ g} ∼ = ˇ h∗/W ⊃ N0ρ ∼ = N0 For any infinitisimal character n ∈ N0 H C n = {M ∈ H C | (n(z) − z)kM = 0 for k ≫ 0 and ∀z ∈ Z(U(ˇ g))} and the category of Harish-Chandra Modules with integral infinitisimal character H C = ⊕n∈N0H C n is stable under tensoring with finite dimensional SL2(C) representations.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some More Notation

Let X be a complex variety (equipped with the analytic topology) acted upon by some linear algebraic group G, then Db

G(X)

denotes the Bernstein-Lunts equivariant derived category.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some More Notation

Let X be a complex variety (equipped with the analytic topology) acted upon by some linear algebraic group G, then Db

G(X)

denotes the Bernstein-Lunts equivariant derived category. If the G-orbits on X are of finite number then these orbits define a stratification of X.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some More Notation

Let X be a complex variety (equipped with the analytic topology) acted upon by some linear algebraic group G, then Db

G(X)

denotes the Bernstein-Lunts equivariant derived category. If the G-orbits on X are of finite number then these orbits define a stratification of X. The category of equivariant perverse sheaves PervG(X) is artinian. (i.e. abelian and every object has finite length)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some More Notation

Let X be a complex variety (equipped with the analytic topology) acted upon by some linear algebraic group G, then Db

G(X)

denotes the Bernstein-Lunts equivariant derived category. If the G-orbits on X are of finite number then these orbits define a stratification of X. The category of equivariant perverse sheaves PervG(X) is artinian. (i.e. abelian and every object has finite length) Let Dss

G (X) full additive subcategory of semisimple objects of Db G(X). (An

  • bject in Db

G(X) is called semisimple if it is isomorphic to finte direct sum

  • f shifted simple objects of PervG(X))

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Geometric Parameter Spaces

Definition (Adams-Barbasch-Vogan) Let Y := {g ∈ G|g 2 = 1},

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Geometric Parameter Spaces

Definition (Adams-Barbasch-Vogan) Let Y := {g ∈ G|g 2 = 1}, X(n) :=

  • G ×B Y

if n > 0 Y if n = 0 is called the Geometric Parameter Space for n ∈ N0.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Geometric Parameter Spaces

Definition (Adams-Barbasch-Vogan) Let Y := {g ∈ G|g 2 = 1}, X(n) :=

  • G ×B Y

if n > 0 Y if n = 0 is called the Geometric Parameter Space for n ∈ N0. Theorem (Soergel, 02) For any n ∈ N0 there exists an graded Version H C Z

n of H C n

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Geometric Parameter Spaces

Definition (Adams-Barbasch-Vogan) Let Y := {g ∈ G|g 2 = 1}, X(n) :=

  • G ×B Y

if n > 0 Y if n = 0 is called the Geometric Parameter Space for n ∈ N0. Theorem (Soergel, 02) For any n ∈ N0 there exists an graded Version H C Z

n of H C n such that

PH C Z

n ∼

= Dss

G (X(n))

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ]

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O) tn := (t → t−n 1

  • ) ∈ hom(C×, T) ⊂ G(K)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O) tn := (t → t−n 1

  • ) ∈ hom(C×, T) ⊂ G(K)

G(K)n := G(O)tnG(O)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O) tn := (t → t−n 1

  • ) ∈ hom(C×, T) ⊂ G(K)

G(K)n := G(O)tnG(O) G(K) =

  • n∈N0

G(K)n “Bruhat decomposition”

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O) tn := (t → t−n 1

  • ) ∈ hom(C×, T) ⊂ G(K)

G(K)n := G(O)tnG(O) G(K) =

  • n∈N0

G(K)n “Bruhat decomposition” Grn :=

  • 0≤k≤n

G(K)k/G(O) is a projective variety

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Affine Grassmannians

K = C( (t) ) O := C[ [t] ] G(K) and G(O) tn := (t → t−n 1

  • ) ∈ hom(C×, T) ⊂ G(K)

G(K)n := G(O)tnG(O) G(K) =

  • n∈N0

G(K)n “Bruhat decomposition” Grn :=

  • 0≤k≤n

G(K)k/G(O) is a projective variety Grn ֒ → Grn+1 is a closed embedding

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian”

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian” Gr carries a natural (left)-G(O)-action.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian” Gr carries a natural (left)-G(O)-action. Db

G(O)(Gr) := lim

− → Db

G(O)(Gri) and PervG(O)(Gr) := lim

− → PervG(O)(Gri)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian” Gr carries a natural (left)-G(O)-action. Db

G(O)(Gr) := lim

− → Db

G(O)(Gri) and PervG(O)(Gr) := lim

− → PervG(O)(Gri) Gr carries also a “continious” G(K)-action.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian” Gr carries a natural (left)-G(O)-action. Db

G(O)(Gr) := lim

− → Db

G(O)(Gri) and PervG(O)(Gr) := lim

− → PervG(O)(Gri) Gr carries also a “continious” G(K)-action. This gives a-bifunctor ∗ : Db

G(O)(Gr) × Db G(O)(Gr) → Db G(O)(Gr)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Gr := G(K)/G(O) ∼ = lim − → Grn “Affine Grassmannian” Gr carries a natural (left)-G(O)-action. Db

G(O)(Gr) := lim

− → Db

G(O)(Gri) and PervG(O)(Gr) := lim

− → PervG(O)(Gri) Gr carries also a “continious” G(K)-action. This gives a-bifunctor ∗ : Db

G(O)(Gr) × Db G(O)(Gr) → Db G(O)(Gr)

Theorem (Geometric Satake Isomorphism, Mirkovic-Vilonen) There exists an equivalence of tensor categories S : (PervG(O)(Gr), ∗) → (RepC(SL2(C)), ⊗)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far:

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr H C ← → G (

i∈N0 X(n))

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr H C ← → G (

i∈N0 X(n))

We may define X :=

  • i∈N0

X(n)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr H C ← → G (

i∈N0 X(n))

We may define X :=

  • i∈N0

X(n) and Db

G(X) :=

  • n∈N0

Db

G(X(n))

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr H C ← → G (

i∈N0 X(n))

We may define X :=

  • i∈N0

X(n) and Db

G(X) :=

  • n∈N0

Db

G(X(n))

We would like to have a convolution product PervG(O)(Gr) × Db

G(X) → Db G(X)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

So far: Rep(SL2(C)) ← → G(O) Gr H C ← → G (

i∈N0 X(n))

We may define X :=

  • i∈N0

X(n) and Db

G(X) :=

  • n∈N0

Db

G(X(n))

We would like to have a convolution product PervG(O)(Gr) × Db

G(X) → Db G(X)

Problem Unfortunately G(K) has no obvious action on X

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Lemma Let X n := G(O)tnG(O) ×G(O) Y . There exists an equivalence of categories Db

G(O)(X n) ∼

= Db

G(X(n))

preserving the perverse t-structure.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Lemma Let X n := G(O)tnG(O) ×G(O) Y . There exists an equivalence of categories Db

G(O)(X n) ∼

= Db

G(X(n))

preserving the perverse t-structure. So we may replace X by

  • X :=
  • n∈N0

X n (each X n is open and closed in X )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Lemma Let X n := G(O)tnG(O) ×G(O) Y . There exists an equivalence of categories Db

G(O)(X n) ∼

= Db

G(X(n))

preserving the perverse t-structure. So we may replace X by

  • X :=
  • n∈N0

X n (each X n is open and closed in X ) We have an EQUALITY OF SETS

  • X = G(K) ×G(O) Y

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Lemma Let X n := G(O)tnG(O) ×G(O) Y . There exists an equivalence of categories Db

G(O)(X n) ∼

= Db

G(X(n))

preserving the perverse t-structure. So we may replace X by

  • X :=
  • n∈N0

X n (each X n is open and closed in X ) We have an EQUALITY OF SETS

  • X = G(K) ×G(O) Y

but still G(K) does not operate continously on X

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Now we will change the topology of X a little, such that G(K) acts continiously as follows: Xn :=

0≤k≤n X k is a quasi-projective variety

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Now we will change the topology of X a little, such that G(K) acts continiously as follows: Xn :=

0≤k≤n X k is a quasi-projective variety

Xn ֒ → Xn+1 is a closed embedding

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Now we will change the topology of X a little, such that G(K) acts continiously as follows: Xn :=

0≤k≤n X k is a quasi-projective variety

Xn ֒ → Xn+1 is a closed embedding X := G(K) ×G(O) Y = lim − → Xn “Affine Parameter Space”

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Now we will change the topology of X a little, such that G(K) acts continiously as follows: Xn :=

0≤k≤n X k is a quasi-projective variety

Xn ֒ → Xn+1 is a closed embedding X := G(K) ×G(O) Y = lim − → Xn “Affine Parameter Space” Db

G(O)(X ) := lim

− → Db

G(O)(Xn)

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Now we will change the topology of X a little, such that G(K) acts continiously as follows: Xn :=

0≤k≤n X k is a quasi-projective variety

Xn ֒ → Xn+1 is a closed embedding X := G(K) ×G(O) Y = lim − → Xn “Affine Parameter Space” Db

G(O)(X ) := lim

− → Db

G(O)(Xn)

Proposition There exists a bi-functor ∗ : Db

G(O)(Gr) × Db G(O)(X ) → Db G(O)(X )

generalizing the convolution bi-functor on the Affine Grassmannian.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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SLIDE 52

Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows:

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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SLIDE 54

Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows: Db

G(O)(

X )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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SLIDE 55

Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows: Db

G(O)(

X )

Rj!

Db

G(O)(X )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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SLIDE 56

Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows: Db

G(O)(

X )

Rj!

Db

G(O)(X ) F∗

  • Db

G(O)(X )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows: Db

G(O)(

X )

Rj!

Db

G(O)(X ) F∗

  • Db

G(O)(

X ) Db

G(O)(X ) j∗

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Remark Note that

  • X = X

since X ⊃ X n =

  • 0≤k∈{n,n−2,n−4,...} X k
  • X ⊃ X n

= X n but we have a canonical inclusion j : X ֒ → X Definition Let F ∈ PervG(O)(Gr). Geometric Tensorfunctor F˜ ∗ is defined as follows: Db

G(O)(

X )

Rj!

  • F ˜

∗

  • Db

G(O)(X ) F∗

  • Db

G(O)(

X ) Db

G(O)(X ) j∗

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Main Theorem

Recall: By Koszul Duality Dss

G(O)(

X ) is a graded version of the “projective

  • bjects” of H C .

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Main Theorem

Recall: By Koszul Duality Dss

G(O)(

X ) is a graded version of the “projective

  • bjects” of H C .

Under this equivalence we have a functor for : Dss

G(O)(

X ) → PH C “forget the grading”

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Main Theorem

Recall: By Koszul Duality Dss

G(O)(

X ) is a graded version of the “projective

  • bjects” of H C .

Under this equivalence we have a functor for : Dss

G(O)(

X ) → PH C “forget the grading” Theorem Geometric Tensoration is the Koszul-Dual of Tensoration with finite dimensional representations.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Main Theorem

Recall: By Koszul Duality Dss

G(O)(

X ) is a graded version of the “projective

  • bjects” of H C .

Under this equivalence we have a functor for : Dss

G(O)(

X ) → PH C “forget the grading” Theorem Geometric Tensoration is the Koszul-Dual of Tensoration with finite dimensional representations. If F is a finite dimensional rational SL2(C)-representation and S(F) it’s Satake equivalent, then the following diagram commutes. PH C

F⊗

  • Dss

G (

X )

S(F)˜ ∗

  • for
  • PH C

Dss

G (

X )

for

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Corollary For each finite dimensional representation F of SL2(C) there exists a graded lift F ⊗ : H C Z → H C Z of the functor F ⊗ : H C → H C ,

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Corollary For each finite dimensional representation F of SL2(C) there exists a graded lift F ⊗ : H C Z → H C Z of the functor F ⊗ : H C → H C , such that For the trivial representation C we have an isomorphism of functors C ⊗ ∼ = id

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Corollary For each finite dimensional representation F of SL2(C) there exists a graded lift F ⊗ : H C Z → H C Z of the functor F ⊗ : H C → H C , such that For the trivial representation C we have an isomorphism of functors C ⊗ ∼ = id For F1, F2 ∈ Rep(SL2(C)) we have an isomorphism of functors (F1 ⊗ F2) ⊗ ∼ = F1 ⊗(F2 ⊗ )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Properties of the Geometric Tensor Functor

Definition (Geometric Translation Functors) For k, m, n ∈ N0 let L(|n − m|) be the unique simple finite dimensional representation with highest weight |n − m| and jk : X k → X the inclusion.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Properties of the Geometric Tensor Functor

Definition (Geometric Translation Functors) For k, m, n ∈ N0 let L(|n − m|) be the unique simple finite dimensional representation with highest weight |n − m| and jk : X k → X the inclusion. Db

G(O)(X m) Rjm! Tm,n

  • Db

G(O)(X ) S(L(|n−m|))∗

  • Db

G(O)(X n)

Db

G(O)(X ) j∗

n

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Some Properties of the Geometric Tensor Functor

Definition (Geometric Translation Functors) For k, m, n ∈ N0 let L(|n − m|) be the unique simple finite dimensional representation with highest weight |n − m| and jk : X k → X the inclusion. Db

G(O)(X m) Rjm! Tm,n

  • Db

G(O)(X ) S(L(|n−m|))∗

  • Db

G(O)(X n)

Db

G(O)(X ) j∗

n

  • Theorem

For any F ∈ PervG(O)(Gr), there exists an isomorphism of functors F˜ ∗ ∼ = some finite sum of compositions

  • f geometric translation functors
  • non canonically.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Theorem Let π : G ×B Y → Y the quotient map. Then Tm,n ∼ =      π∗[1] if m = 0, n = 0 π∗[1] if n = 0, m = 0 id else

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Theorem Let π : G ×B Y → Y the quotient map. Then Tm,n ∼ =      π∗[1] if m = 0, n = 0 π∗[1] if n = 0, m = 0 id else Corollary i) Geometric Tensoration preserves semi-simplicity.

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Theorem Let π : G ×B Y → Y the quotient map. Then Tm,n ∼ =      π∗[1] if m = 0, n = 0 π∗[1] if n = 0, m = 0 id else Corollary i) Geometric Tensoration preserves semi-simplicity. ii) Geometric Tensoration is (weakly) associative, this means there exists isomorphism of functors (F1 ∗ F2)˜ ∗ ∼ = F1˜ ∗(F2˜ ∗ )

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Thank You!

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Convolution

Assume G(K) acts continously on X, where X ∈ {Gr, X }. Philisophically the convolution bi-functor ∗ : Db

G(O)(Gr) × Db G(O)(X)

→ Db

G(O)(X)

is defined as follows:

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Convolution

Assume G(K) acts continously on X, where X ∈ {Gr, X }. Philisophically the convolution bi-functor ∗ : Db

G(O)(Gr) × Db G(O)(X)

→ Db

G(O)(X)

is defined as follows: For F ∈ Db

G(O)(Gr) define F ∗

by Db

G(O)(X) F∗

  • F⊠ Db

G(O)×G(O)(Gr × X) p∗

1 ×id

  • Db

G(O)3(G(K) × X) res∆

  • Db

G(O)2(G(K) × X) (q∗)−1

  • Db

G(O)(X)

Db

G(O)2(G(K) ×G(O) X) m!

  • Oliver Straser

Koszul Duality and Geometric Satake for SL2(R)

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Koszul Duality Geometric Satake Connecting both pictures Geometric Tensoration

Summerschool on Category O

Freiburg: 6.8-9.8.2013 http://home.mathematik.uni-freiburg.de/ostraser/cato.html

Oliver Straser Koszul Duality and Geometric Satake for SL2(R)