Two-block Springer fibers and Springer representations in types C & D

Two-block Springer fibers and Springer representations in types C & D

Arik Wilbert

Hausdorff Research Institute for Mathematics

November 30, 2017

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 1 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleS_m-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ

m Springer fiber H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleS_m-representations

(up to isomorphism)

1:1

1:1 deeper/direct

connection?

Oλ B_GL^λ

m Springer fiber H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ

m Springer fiber H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ

m Springer fiber H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ

B_GL^λ

m Springer fiber H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ _m Springer fiber

H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ _m Springer fiber

H^∗(B^λ_GL

m,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ _m Springer fiber H^∗(B^λ_GLm,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B_GL^λ

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

GL_m(C)-conjugacy classes of nilpotent endomorphisms ofC^m

partitions λofm

complex, finite-dimensional, irreducibleSm-representations

(up to isomorphism)

1:1 1:1

deeper/direct connection?

Oλ B_GL^λ _m Springer fiber H^∗(B^λ_GLm,C)

Definition (Springer fiber of type A)

x:C^m→C^m nilpotent endomorphism of Jordan typeλ

B_GL^λ _m ={{0}(F1(F2(. . .(Fm=C^m|xFi⊆Fi−1}

Theorem (Springer, 1978)

There exists a gradedSm-action onH^∗(B^λ_GL

m,C)such that H^top(B^λ_GL

m,C)is the irreducibleSm-representation labeled byλ. This yields a correspondence

Irr^f.d._C (S_m)−−→ {^1:1 nilpotent endomorphisms ofC^m}.

GL_m(C).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 2 / 17

Gconnected, reductive, complex, algebraic group

, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even multiplicity.

The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2),(3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent

WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even multiplicity.

The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2),(3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even multiplicity.

The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2),(3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even multiplicity.

The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2),(3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2),(3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2), (3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2), (3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2), (3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2), (3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group, x∈g=Lie(G)nilpotent WG Weyl group

Theorem (Gerstenhaber, 1961)

TheSp2m-conjugacy classes of nilpotent elements insp_2m are in bijective correspondence with partitions of2min which odd parts occur with even

multiplicity. The parts of the partition encode the sizes of the Jordan blocks of an element in the conjugacy class.

partitions of 4:

(1,1,1,1), (2,1,1), (2,2), (3,1), (4)

Theorem (Folklore)

The isomorphism classes of complex, finite-dimensional, irreducible representations of the Weyl groupWSp2m are in bijective correspondence with bipartitions ofm.

bipartitions of m=2:

,∅

, , ∅

, ,

, ∅,

,

∅,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 3 / 17

Gconnected, reductive, complex, algebraic group,

x∈

g=Lie(G)

nilpotent

WG Weyl group

A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible WG-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G)

nilpotent

WG Weyl group

A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible WG-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group

A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible WG-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible WG-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible WG-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible W_G-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible W_G-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

Gconnected, reductive, complex, algebraic group,x∈g=Lie(G) nilpotent WG Weyl group A_x=C_G(x)/C_G⁰(x) component group

Definition (Springer fiber)

B_G^x ={Borel subgroupsB ⊆G|x∈Lie(B)}, x∈gnilpotent

Theorem (Springer, 1978)

I There exist grading-preserving, commuting actions ofWG andAxon H^∗(B^x_G,C).

I The decomposition

H^top(B_G^x,C) =M

λ

H_λ^top(B^x_G,C)

into non-zeroA_x-isotypic subspaces is a decomposition into irreducible W_G-representations.

I This yields the Springer correspondence

Irr^f.d._C (WG),→ {nilpotent elements ing}/G×Irr^f.d._C (A_x).

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 4 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW:

G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW:

G=SO2m WG=WD_m

(type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m

WG=WD_m

(type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆

WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆

WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

I TheW_G-action on cohomology is not induced from an action on the space (restrict action on Springer sheaf to its stalks).

I The topology and geometry of the Springer fiber is poorly understood (in general singular, many irreducible components).

NOW: G=SO2m WG=WD_m (type D)

Sm⊆ WD_m

subgroup generated by s1, . . . , s_m−1

generators: s0, s1, . . . , sm−1

relations: (sisj)^mij =e mij =







1 if i=j, 3 if i j in

2 else. ₀

1

2 3 . . .m−1

Theorem (Lusztig, 2004)

There exists an isomorphism ofC[WD_m]-modules H^∗(B^m,m_SO

2m,C)

| {z }

Springer representation

∼=C⊗_C[Sm]C[WD_m]

| {z }

induced trivial module

.

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 5 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m)

b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras. O^p₀(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra. Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m)

b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras. O^p₀(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra. Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m) b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras. O^p₀(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra. Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m) b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras. O^p₀(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra. Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m) b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras.

O₀^p(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra. Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m) b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras.

O₀^p(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra.

Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

{standard basisbλ} φ

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even

{cupdiagramsonmvertices,}=CKL(m) b_µ=P

λαλ,µbλ

Question

Where else do theαλ,µ appear? Why are they interesting?

I Infinite-dimensional representation theory of Lie algebras.

O₀^p(so_2m(C)) [M(λ) :L(µ)] =α_λ,µ (Kazhdan–Lusztig, Beilinson–Bernstein, Brylinski–Kashiwara, Elias–Williamson)

I Non-semisimple representation theory of the Brauer algebra.

Brm(δ)-mod (δ∈Z) [∆(λ) :L(µ)] =αλ,µ

(Martin, Cox–DeVisscher, Ehrig–Stroppel)

Question

Can we explicitly compute theαλ,µ?

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 6 / 17

φ

∼=

ψ

∼=

{standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨,∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17

φ

∼=

ψ

∼=

{standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨,∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17

φ

∼=

ψ

∼=

{standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨, ∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17

φ

∼=

ψ

∼=

{standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨, ∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17

φ

∼=

ψ

∼= {standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨, ∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17

φ

∼=

ψ

∼= {standard basisbλ}

{Kazhdan-Lusztig basisb_µ}

{∧,∨}-sequences, lengthm,#(∧)even







cup diagrams onmvertices,

#

+ #

even







=CKL(m)

b_µ=P

λαλ,µbλ

Example: (m=4)

∨ ∨ ∨∨, ∨ ∨ ∧∧,∨ ∧ ∨∧,∨ ∧ ∧∨,∧ ∨ ∨∧,∧ ∨ ∧∨,∧ ∧ ∨∨,∧ ∧ ∧∧

Theorem (Lejczyk–Stroppel, 2013)

α_λ,µ=







1, if φ(bλ)

ψ(b_µ) oriented, 0, else.

∧ (∨

∨

∧)

∨ ∨ (∧ ∧)

∨ ∧

, , ,

A. Wilbert (HIM) Two-block Springer fibers of types C & D November 30, 2017 7 / 17