Unipotent Cells in Kac-Moody Groups

Unipotent cells in Kac-Moody groups

Bernard Leclerc

joint with Christof Geiss and Jan Schr¨oer

ICTP Trieste, February 2010 In this talk : Lie-theoretic preliminaries for lectures of Schr¨oer and Geiss.

(3) Understand better this class of cluster algebras by means of semicanonical bases.

(2) Categorify these coordinate rings using preprojective algebras.

(1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups.

Aims of our joint project : In this talk : Lie-theoretic preliminaries for lectures of Schr¨oer and Geiss.

(3) Understand better this class of cluster algebras by means of semicanonical bases.

(2) Categorify these coordinate rings using preprojective algebras.

Aims of our joint project :

(1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups. In this talk : Lie-theoretic preliminaries for lectures of Schr¨oer and Geiss.

(3) Understand better this class of cluster algebras by means of semicanonical bases.

Aims of our joint project :

(1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups.

(2) Categorify these coordinate rings using preprojective algebras. In this talk : Lie-theoretic preliminaries for lectures of Schr¨oer and Geiss.

Aims of our joint project :

(1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups.

(2) Categorify these coordinate rings using preprojective algebras.

(3) Understand better this class of cluster algebras by means of semicanonical bases. Aims of our joint project :

(1) Describe (following Berenstein-Fomin-Zelevinsky) cluster algebra structures on coordinate rings of unipotent cells of Kac-Moody groups.

(2) Categorify these coordinate rings using preprojective algebras.

(3) Understand better this class of cluster algebras by means of semicanonical bases.

In this talk : Lie-theoretic preliminaries for lectures of Schr¨oer and Geiss. Preprojective algebras and semicanonical bases

Unipotent cells and their coordinate rings

Unipotent subgroups and Kac-Moody groups

Kac-Moody algebras

Plan Preprojective algebras and semicanonical bases

Unipotent cells and their coordinate rings

Unipotent subgroups and Kac-Moody groups

Plan

Kac-Moody algebras Preprojective algebras and semicanonical bases

Unipotent cells and their coordinate rings

Plan

Kac-Moody algebras

Unipotent subgroups and Kac-Moody groups Preprojective algebras and semicanonical bases

Plan

Kac-Moody algebras

Unipotent subgroups and Kac-Moody groups

Unipotent cells and their coordinate rings Plan

Kac-Moody algebras

Unipotent subgroups and Kac-Moody groups

Unipotent cells and their coordinate rings

Preprojective algebras and semicanonical bases Kac-Moody algebras Example: Q = 1 / 2 / 3

Γ = 1 2 3

 2 −2 0  C = −2 2 −1 0 −1 2

C = [cij ] where cij = 2δij − ]{edges between i and j in Γ}.

Γ : underlying unoriented graph.

Q : acyclic quiver with vertex set I = {1, 2,..., n}.

Cartan matrix Example: Q = 1 / 2 / 3

Γ = 1 2 3

 2 −2 0  C = −2 2 −1 0 −1 2

C = [cij ] where cij = 2δij − ]{edges between i and j in Γ}.

Γ : underlying unoriented graph.

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2,..., n}. Example: Q = 1 / 2 / 3

Γ = 1 2 3

 2 −2 0  C = −2 2 −1 0 −1 2

C = [cij ] where cij = 2δij − ]{edges between i and j in Γ}.

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2,..., n}. Γ : underlying unoriented graph. Example: Q = 1 // 2 / 3

Γ = 1 2 3

 2 −2 0  C = −2 2 −1 0 −1 2

Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2,..., n}. Γ : underlying unoriented graph.

C = [cij ] where cij = 2δij − ]{edges between i and j in Γ}. Cartan matrix

Q : acyclic quiver with vertex set I = {1, 2,..., n}. Γ : underlying unoriented graph.

C = [cij ] where cij = 2δij − ]{edges between i and j in Γ}.

Example: Q = 1 / 2 / 3

Γ = 1 2 3

 2 −2 0  C = −2 2 −1 0 −1 2 n+ = n := hei | i ∈ I i, n− := hfi | i ∈ I i.

g : Lie algebra over C with generators ei , fi (i ∈ I ), h ∈ h, and relations :

0 [h, h ] = 0, [h, ei ] = αi (h)ei , [h,fi ] = −αi (h)fi ,

[ei , fj ] = δij hi ,

1−c 1−c ad(ei ) ij (ej ) = ad(fi ) ij (fj ) = 0, (i 6= j).

∗ {h1,..., hn} ⊂ h, {α1,..., αn} ⊂ h linearly independent s.t.

αi (hj ) = cij

h : C-vector space of dimension 2n − rk C

Kac-Moody algebra n+ = n := hei | i ∈ I i, n− := hfi | i ∈ I i.

g : Lie algebra over C with generators ei , fi (i ∈ I ), h ∈ h, and relations :

0 [h, h ] = 0, [h, ei ] = αi (h)ei , [h,fi ] = −αi (h)fi ,

[ei , fj ] = δij hi ,

1−c 1−c ad(ei ) ij (ej ) = ad(fi ) ij (fj ) = 0, (i 6= j).

∗ {h1,..., hn} ⊂ h, {α1,..., αn} ⊂ h linearly independent s.t.

αi (hj ) = cij

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C n+ = n := hei | i ∈ I i, n− := hfi | i ∈ I i.

g : Lie algebra over C with generators ei , fi (i ∈ I ), h ∈ h, and relations :

0 [h, h ] = 0, [h, ei ] = αi (h)ei , [h,fi ] = −αi (h)fi ,

[ei , fj ] = δij hi ,

1−c 1−c ad(ei ) ij (ej ) = ad(fi ) ij (fj ) = 0, (i 6= j).

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C ∗ {h1,..., hn} ⊂ h, {α1,..., αn} ⊂ h linearly independent s.t.

αi (hj ) = cij n+ = n := hei | i ∈ I i, n− := hfi | i ∈ I i.

Kac-Moody algebra

h : C-vector space of dimension 2n − rk C ∗ {h1,..., hn} ⊂ h, {α1,..., αn} ⊂ h linearly independent s.t.

αi (hj ) = cij

g : Lie algebra over C with generators ei , fi (i ∈ I ), h ∈ h, and relations :

0 [h, h ] = 0, [h, ei ] = αi (h)ei , [h,fi ] = −αi (h)fi ,

[ei , fj ] = δij hi ,

1−c 1−c ad(ei ) ij (ej ) = ad(fi ) ij (fj ) = 0, (i 6= j). Kac-Moody algebra

h : C-vector space of dimension 2n − rk C ∗ {h1,..., hn} ⊂ h, {α1,..., αn} ⊂ h linearly independent s.t.

αi (hj ) = cij

g : Lie algebra over C with generators ei , fi (i ∈ I ), h ∈ h, and relations :

0 [h, h ] = 0, [h, ei ] = αi (h)ei , [h,fi ] = −αi (h)fi ,

[ei , fj ] = δij hi ,

1−c 1−c ad(ei ) ij (ej ) = ad(fi ) ij (fj ) = 0, (i 6= j).

n+ = n := hei | i ∈ I i, n− := hfi | i ∈ I i. We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

+ L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+).

∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g.

∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}.

W is a Coxeter group with length function w 7→ `(w).

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

Weyl group and Roots We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

+ L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+).

∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g.

∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}.

W is a Coxeter group with length function w 7→ `(w).

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi . We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

+ L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+).

∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g.

∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}.

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W is a Coxeter group with length function w 7→ `(w). We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

+ L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+).

∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g.

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W is a Coxeter group with length function w 7→ `(w). ∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

+ L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+).

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W is a Coxeter group with length function w 7→ `(w). ∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g. We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}.

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W is a Coxeter group with length function w 7→ `(w). ∗ For α ∈ h , let gα := {x ∈ g | [h, x] = α(h)x, h ∈ h}. ∗ ∆ := {α ∈ h | α 6= 0 and gα 6= 0} : root system of g. + L + + Set R = i∈I Nαi , and ∆ := ∆ ∩ R . We have ∆ = ∆+ t (−∆+). We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

+ − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re .

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}. We have ]∆w = dim n(w) = `(w).

Set M n(w) := gα ⊂ n.

α∈∆w

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}. + − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re . We have ]∆w = dim n(w) = `(w).

Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}. + − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re . Set M n(w) := gα ⊂ n.

α∈∆w Weyl group and Roots

∗ W : subgroup of GL(h ) generated by the reﬂexions si (i ∈ I ):

si (α) = α − α(hi )αi .

W acts on ∆. Deﬁne ∆re := W {α1, . . . , αn}. + − For w ∈ W , let ∆w := {α ∈ ∆ | w(α) ∈ ∆ } ⊂ ∆re . Set M n(w) := gα ⊂ n.

α∈∆w

We have ]∆w = dim n(w) = `(w). n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

w = s2s1s2s1.

2 2 W = hs1, s2 | s1 = s2 = 1i.

Q = 1 / 2

An example n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

w = s2s1s2s1.

2 2 W = hs1, s2 | s1 = s2 = 1i.

An example

Q = 1 / 2 n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

w = s2s1s2s1.

An example

Q = 1 / 2

2 2 W = hs1, s2 | s1 = s2 = 1i. n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

An example

Q = 1 / 2

2 2 W = hs1, s2 | s1 = s2 = 1i.

w = s2s1s2s1. n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i.

An example

Q = 1 / 2

2 2 W = hs1, s2 | s1 = s2 = 1i.

w = s2s1s2s1.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}. An example

Q = 1 / 2

2 2 W = hs1, s2 | s1 = s2 = 1i.

w = s2s1s2s1.

∆w := {α1, 2α1 + α2, 3α1 + 2α2, 4α1 + 3α2}.

n(w) := Spanhe1, [e1, [e2, e1]], [e1, [e2, [e1, [e2, e1]]]], [e1, [e2, [e1, [e2, [e1, [e2, e1]]]]]]i. Unipotent groups and Kac-Moody groups ∗ By construction, U(n)gr = C[N].

∗ ∗ Let N := maxSpec(U(n)gr ) = Hom alg (U(n)gr , C). This is the pro-unipotent pro-group with Lie algebra Y bn = gα. α∈∆+

∗ L ∗ Let U(n)gr := d∈R+ U(n)d be the graded dual. This is a commutative Hopf algebra.

U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei ) := αi .

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0

The group N ∗ By construction, U(n)gr = C[N].

∗ ∗ Let N := maxSpec(U(n)gr ) = Hom alg (U(n)gr , C). This is the pro-unipotent pro-group with Lie algebra Y bn = gα. α∈∆+

∗ L ∗ Let U(n)gr := d∈R+ U(n)d be the graded dual. This is a commutative Hopf algebra.

U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei ) := αi .

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0 ∗ By construction, U(n)gr = C[N].

∗ ∗ Let N := maxSpec(U(n)gr ) = Hom alg (U(n)gr , C). This is the pro-unipotent pro-group with Lie algebra Y bn = gα. α∈∆+

∗ L ∗ Let U(n)gr := d∈R+ U(n)d be the graded dual. This is a commutative Hopf algebra.

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0

U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei ) := αi . ∗ By construction, U(n)gr = C[N].

∗ ∗ Let N := maxSpec(U(n)gr ) = Hom alg (U(n)gr , C). This is the pro-unipotent pro-group with Lie algebra Y bn = gα. α∈∆+

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0

U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei ) := αi . ∗ L ∗ Let U(n)gr := d∈R+ U(n)d be the graded dual. This is a commutative Hopf algebra. ∗ By construction, U(n)gr = C[N].

The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0

U(n) is a cocommutative Hopf algebra, with an R+-grading given by deg(ei ) := αi . ∗ L ∗ Let U(n)gr := d∈R+ U(n)d be the graded dual. This is a commutative Hopf algebra. ∗ ∗ Let N := maxSpec(U(n)gr ) = Hom alg (U(n)gr , C). This is the pro-unipotent pro-group with Lie algebra Y bn = gα. α∈∆+ The group N

U(n) : universal enveloping algebra of n. It is generated by ei (i ∈ I ), with relations

1−cij X k (k) (1−cij −k) (−1) ei ej ei = 0, (i 6= j). k=0

∗ By construction, U(n)gr = C[N]. Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring

N0(w) 0 0 C[N] = f ∈ C[N] | f (nn ) = f (n), n ∈ N, n ∈ N(w)

Multiplication yields a bijection N(w) × N0(w) →∼ N.

Let N0(w) be the subgroup of N with Lie algebra

0 Y n (w) := gα ⊂ bn. α6∈∆w

Let N(w) be the subgroup of N with Lie algebra n(w).

The unipotent subgroup N(w) Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring

N0(w) 0 0 C[N] = f ∈ C[N] | f (nn ) = f (n), n ∈ N, n ∈ N(w)

Multiplication yields a bijection N(w) × N0(w) →∼ N.

Let N0(w) be the subgroup of N with Lie algebra

0 Y n (w) := gα ⊂ bn. α6∈∆w

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring

N0(w) 0 0 C[N] = f ∈ C[N] | f (nn ) = f (n), n ∈ N, n ∈ N(w)

Multiplication yields a bijection N(w) × N0(w) →∼ N.

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N0(w) be the subgroup of N with Lie algebra

0 Y n (w) := gα ⊂ bn. α6∈∆w Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring

N0(w) 0 0 C[N] = f ∈ C[N] | f (nn ) = f (n), n ∈ N, n ∈ N(w)

The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N0(w) be the subgroup of N with Lie algebra

0 Y n (w) := gα ⊂ bn. α6∈∆w

Multiplication yields a bijection N(w) × N0(w) →∼ N. The unipotent subgroup N(w)

Let N(w) be the subgroup of N with Lie algebra n(w). Let N0(w) be the subgroup of N with Lie algebra

0 Y n (w) := gα ⊂ bn. α6∈∆w

Multiplication yields a bijection N(w) × N0(w) →∼ N.

Proposition The coordinate ring C[N(w)] is isomorphic to the invariant subring

N0(w) 0 0 C[N] = f ∈ C[N] | f (nn ) = f (n), n ∈ N, n ∈ N(w) For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

∼ We have NormG (H)/H = W .

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

Let G be the group attached to g by Kac-Peterson.

The Kac-Moody group G For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

∼ We have NormG (H)/H = W .

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

∼ We have NormG (H)/H = W .

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−. For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

∼ We have NormG (H)/H = W .

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G. For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

∼ We have NormG (H)/H = W . For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H).

The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

∼ We have NormG (H)/H = W . For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H). The Kac-Moody group G

Let G be the group attached to g by Kac-Peterson. This is an aﬃne ind-variety. It has a reﬁned Tits system

(G, NormG (H), N+, N−, H),

where Lie(H) = h, Lie(N+) = n+, and Lie(N−) = n−.

Note : In general N 6⊂ G. They can both be regarded as subgroups of a bigger group G max constructed by Tits. Then N+ = N ∩ G.

∼ We have NormG (H)/H = W . For i ∈ I , put

si := exp(fi ) exp(−ei ) exp(fi ) ∈ NormG (H).

For w = sir ··· si1 with `(w) = r, put w = sir ··· si1 , a representative of w in NormG (H). If Q is wild, no “concrete” realization of G is known.

Moreover,

N+ '{g ∈ G Xn (C[z]) | g|z=0 ∈ NXn (C)}.

If Q is of aﬃne Dynkin type Xen, then G is a central extension ∗ −1 by C of G Xn (C[z, z ]).

If Q is of Dynkin type Xn, then G = G Xn (C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).)

Examples If Q is wild, no “concrete” realization of G is known.

Moreover,

N+ '{g ∈ G Xn (C[z]) | g|z=0 ∈ NXn (C)}.

If Q is of aﬃne Dynkin type Xen, then G is a central extension ∗ −1 by C of G Xn (C[z, z ]).

Examples

If Q is of Dynkin type Xn, then G = G Xn (C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).) If Q is wild, no “concrete” realization of G is known.

Moreover,

N+ '{g ∈ G Xn (C[z]) | g|z=0 ∈ NXn (C)}.

Examples

If Q is of Dynkin type Xn, then G = G Xn (C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).)

If Q is of aﬃne Dynkin type Xen, then G is a central extension ∗ −1 by C of G Xn (C[z, z ]). If Q is wild, no “concrete” realization of G is known.

Examples

If Q is of Dynkin type Xn, then G = G Xn (C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).)

If Q is of aﬃne Dynkin type Xen, then G is a central extension ∗ −1 by C of G Xn (C[z, z ]). Moreover,

N+ '{g ∈ G Xn (C[z]) | g|z=0 ∈ NXn (C)}. Examples

If Q is of Dynkin type Xn, then G = G Xn (C) is a connected simply-connected algebraic group of type Xn over C. (Ex: If Xn = An, G = SL(n + 1, C).)

If Q is of aﬃne Dynkin type Xen, then G is a central extension ∗ −1 by C of G Xn (C[z, z ]). Moreover,

N+ '{g ∈ G Xn (C[z]) | g|z=0 ∈ NXn (C)}.

If Q is wild, no “concrete” realization of G is known. Unipotent cells G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

Let x 7→ x$i denote the corresponding character of H.

∗ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij .

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

Let G0 = N−HN+.

Generalized minors G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

Let x 7→ x$i denote the corresponding character of H.

∗ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij .

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

Generalized minors

Let G0 = N−HN+. G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

Let x 7→ x$i denote the corresponding character of H.

∗ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij .

Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+. G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

Let x 7→ x$i denote the corresponding character of H.

Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

∗ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij . G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

∗ $ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij . Let x 7→ x i denote the corresponding character of H. G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

∗ $ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij . Let x 7→ x i denote the corresponding character of H.

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0). G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }.

Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

∗ $ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij . Let x 7→ x i denote the corresponding character of H.

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw). Generalized minors

Let G0 = N−HN+.

Proposition

G0 is open dense in G. Every g ∈ G0 has a unique factorization g = [g]−[g]0[g]+ with [g]− ∈ N−, [g]0 ∈ H, [g]+ ∈ N+.

∗ $ For i ∈ I , let $i ∈ h s. t. $i (hj ) = δij . Let x 7→ x i denote the corresponding character of H.

There is a unique regular function ∆$i ,$i on G such that

$i ∆$i ,$i (g) = [g]0 (g ∈ G0).

For w ∈ W , set ∆$i ,w($i )(g) := ∆$i ,$i (gw).

G0 = {g ∈ G | ∆$i ,$i (g) 6= 0, i ∈ I }. Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }.

w T For w ∈ W , deﬁne N := N+ B−wB−.

The group G has a Bruhat decomposition G G = B−wB−. w∈W

Let B− = N−H.

The unipotent cell Nw Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }.

w T For w ∈ W , deﬁne N := N+ B−wB−.

The group G has a Bruhat decomposition G G = B−wB−. w∈W

The unipotent cell Nw

Let B− = N−H. Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }.

w T For w ∈ W , deﬁne N := N+ B−wB−.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G G = B−wB−. w∈W Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }.

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G G = B−wB−. w∈W

w T For w ∈ W , deﬁne N := N+ B−wB−. Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I

The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G G = B−wB−. w∈W

w T For w ∈ W , deﬁne N := N+ B−wB−.

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }. The unipotent cell Nw

Let B− = N−H. The group G has a Bruhat decomposition G G = B−wB−. w∈W

w T For w ∈ W , deﬁne N := N+ B−wB−.

−1 Let Ow := {n ∈ N(w) | ∆$i ,w ($i )(n) 6= 0, i ∈ I }.

Proposition ∼ w w We have an isomorphism Ow → N . It follows that C[N ] is the N0(w) localization of C[N(w)] ' C[N] at Y −1 ∆w := ∆$i ,w ($i ). i∈I Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ?

w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f .

∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw .

Let w = si1 ··· sir be a reduced decomposition.

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ).

Concrete calculations Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ?

w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f .

∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw .

Let w = si1 ··· sir be a reduced decomposition.

Concrete calculations

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ). Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ?

w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f .

∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw .

Concrete calculations

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ).

Let w = si1 ··· sir be a reduced decomposition. Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ?

w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f .

Concrete calculations

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ).

Let w = si1 ··· sir be a reduced decomposition. ∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw . Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ?

Concrete calculations

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ).

Let w = si1 ··· sir be a reduced decomposition. ∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw . w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f . Concrete calculations

Set xi (t) := exp(tei )(t ∈ C, i ∈ I ).

Let w = si1 ··· sir be a reduced decomposition. ∗ r The image of the map (C ) → N given by

(t1,..., tr ) 7→ xi1 (t1) ··· xir (tr )

is a dense subset of Nw . w If f ∈ C[N ], then (t1,..., tr ) 7→ f (xi1 (t1) ··· xir (tr )) is a polynomial function, which completely determines f .

Problem

How to calculate f (xi1 (t1) ··· xir (tr )), for example when f is a generalized minor ? Preprojective algebras and semicanonical bases Theorem (Lusztig, Geiss-L-Schr¨oer)

There exits a unique ϕM ∈ C[N] such that for all j = (j1,..., jk ) a a X t1 1 ··· tk k ϕ (x (t ) ··· x (t )) = χ a M j1 1 jk k M,j a ! ··· a ! k 1 k a∈N a where j = (j1,..., j1,..., jk ,..., jk ) | {z } | {z } a1 ak

χM,i := χ(FM,i) ∈ Z (Euler characteristic).

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M.

nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules.

Λ : the preprojective algebra attached to Q.

The preprojective algebra Theorem (Lusztig, Geiss-L-Schr¨oer)

χM,i := χ(FM,i) ∈ Z (Euler characteristic).

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M.

nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules.

The preprojective algebra

Λ : the preprojective algebra attached to Q. Theorem (Lusztig, Geiss-L-Schr¨oer)

χM,i := χ(FM,i) ∈ Z (Euler characteristic).

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M.

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules. Theorem (Lusztig, Geiss-L-Schr¨oer)

χM,i := χ(FM,i) ∈ Z (Euler characteristic).

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules.

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M. Theorem (Lusztig, Geiss-L-Schr¨oer)

The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules.

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M.

χM,i := χ(FM,i) ∈ Z (Euler characteristic). The preprojective algebra

Λ : the preprojective algebra attached to Q. nil(Λ) : category of ﬁnite-dimensional nilpotent Λ-modules.

For M ∈ nil(Λ) and i := (i1,..., ik ), let FM,i be the variety of type i composition series of M.

χM,i := χ(FM,i) ∈ Z (Euler characteristic).

Theorem (Lusztig, Geiss-L-Schr¨oer)

There exits a unique ϕM ∈ C[N] such that for all j = (j1,..., jk ) a a X t1 1 ··· tk k ϕ (x (t ) ··· x (t )) = χ a M j1 1 jk k M,j a ! ··· a ! k 1 k a∈N a where j = (j1,..., j1,..., jk ,..., jk ) | {z } | {z } a1 ak 2 3 ϕM2 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 t3

2 2 2 ϕM1 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 + 2t1t2t4 + t1t4 + t3t4

M = 1 1 M = 2 2 2 1 < 2 < < < ÐÒÒ < ÐÒÒ < ÐÒÒ 2 1 1 < < ÐÒÒ 2

Q = 1 // 2

An example 2 3 ϕM2 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 t3

2 2 2 ϕM1 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 + 2t1t2t4 + t1t4 + t3t4

M = 1 1 M = 2 2 2 1 < 2 < < < ÐÒÒ < ÐÒÒ < ÐÒÒ 2 1 1 < < ÐÒÒ 2

An example

Q = 1 // 2 2 3 ϕM2 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 t3

2 2 2 ϕM1 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 + 2t1t2t4 + t1t4 + t3t4

An example

Q = 1 // 2

M = 1 1 M = 2 2 2 1 < 2 < < < ÐÒÒ < ÐÒÒ < ÐÒÒ 2 1 1 < < ÐÒÒ 2 2 3 ϕM2 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 t3

An example

Q = 1 // 2

M = 1 1 M = 2 2 2 1 < 2 < < < ÐÒÒ < ÐÒÒ < ÐÒÒ 2 1 1 < < ÐÒÒ 2

2 2 2 ϕM1 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 + 2t1t2t4 + t1t4 + t3t4 An example

Q = 1 // 2

M = 1 1 M = 2 2 2 1 < 2 < < < ÐÒÒ < ÐÒÒ < ÐÒÒ 2 1 1 < < ÐÒÒ 2

2 2 2 ϕM1 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 + 2t1t2t4 + t1t4 + t3t4

2 3 ϕM2 (x2(t1)x1(t2)x2(t3)x1(t4)) = t1t2 t3 Proposition (Geiss-L-Schr¨oer) Let i be a reduced word for w −1 ∈ W . Then

∆ = ϕ . $k ,w($k ) soc i(bIk )

bIk : injective envelope of Sk (inﬁnite-dimensional).

For i = (i1,..., im) there is a unique sequence

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M ∼ such that soc ik (M/Mk−1) = Mk /Mk−1 for 1 ≤ k ≤ m. Deﬁne soc i(M) = Mm.

For M ∈ Mod Λ and i ∈ I , let soc i (M) be the Si -isotypic component of soc M.

Generalized minors Proposition (Geiss-L-Schr¨oer) Let i be a reduced word for w −1 ∈ W . Then

∆ = ϕ . $k ,w($k ) soc i(bIk )

bIk : injective envelope of Sk (inﬁnite-dimensional).

For i = (i1,..., im) there is a unique sequence

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M ∼ such that soc ik (M/Mk−1) = Mk /Mk−1 for 1 ≤ k ≤ m. Deﬁne soc i(M) = Mm.

Generalized minors

For M ∈ Mod Λ and i ∈ I , let soc i (M) be the Si -isotypic component of soc M. Proposition (Geiss-L-Schr¨oer) Let i be a reduced word for w −1 ∈ W . Then

∆ = ϕ . $k ,w($k ) soc i(bIk )

bIk : injective envelope of Sk (inﬁnite-dimensional).

Generalized minors

For M ∈ Mod Λ and i ∈ I , let soc i (M) be the Si -isotypic component of soc M.

For i = (i1,..., im) there is a unique sequence

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M ∼ such that soc ik (M/Mk−1) = Mk /Mk−1 for 1 ≤ k ≤ m. Deﬁne soc i(M) = Mm. Proposition (Geiss-L-Schr¨oer) Let i be a reduced word for w −1 ∈ W . Then

∆ = ϕ . $k ,w($k ) soc i(bIk )

Generalized minors

For M ∈ Mod Λ and i ∈ I , let soc i (M) be the Si -isotypic component of soc M.

For i = (i1,..., im) there is a unique sequence

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M ∼ such that soc ik (M/Mk−1) = Mk /Mk−1 for 1 ≤ k ≤ m. Deﬁne soc i(M) = Mm.

bIk : injective envelope of Sk (inﬁnite-dimensional). Generalized minors

For M ∈ Mod Λ and i ∈ I , let soc i (M) be the Si -isotypic component of soc M.

For i = (i1,..., im) there is a unique sequence

0 = M0 ⊆ M1 ⊆ · · · ⊆ Mm ⊆ M ∼ such that soc ik (M/Mk−1) = Mk /Mk−1 for 1 ≤ k ≤ m. Deﬁne soc i(M) = Mm.

bIk : injective envelope of Sk (inﬁnite-dimensional).

Proposition (Geiss-L-Schr¨oer) Let i be a reduced word for w −1 ∈ W . Then

∆ = ϕ . $k ,w($k ) soc i(bIk ) ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

F Irr(Λ) := I Irr(Λ ). d∈N d

Irr(Λd) : set of irreducible components of Λd.

Λd : variety of nilpotent Λ-modules of dimension vector d.

I d ∈ N : dimension vector for Λ.

The semicanonical basis ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

F Irr(Λ) := I Irr(Λ ). d∈N d

Irr(Λd) : set of irreducible components of Λd.

Λd : variety of nilpotent Λ-modules of dimension vector d.

The semicanonical basis

I d ∈ N : dimension vector for Λ. ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

F Irr(Λ) := I Irr(Λ ). d∈N d

Irr(Λd) : set of irreducible components of Λd.

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d. ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

F Irr(Λ) := I Irr(Λ ). d∈N d

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. F Irr(Λ) := I Irr(Λ ). d∈N d ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. F Irr(Λ) := I Irr(Λ ). d∈N d For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U. ∗ The generalized minors ∆$k ,w($k ) belong to S .

S∗ is dual to Lusztig’s semicanonical basis of U(n).

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. F Irr(Λ) := I Irr(Λ ). d∈N d For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N]. ∗ The generalized minors ∆$k ,w($k ) belong to S .

The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. F Irr(Λ) := I Irr(Λ ). d∈N d For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

S∗ is dual to Lusztig’s semicanonical basis of U(n). The semicanonical basis

I d ∈ N : dimension vector for Λ. Λd : variety of nilpotent Λ-modules of dimension vector d.

Irr(Λd) : set of irreducible components of Λd. F Irr(Λ) := I Irr(Λ ). d∈N d For Z ∈ Irr(Λ), there is a dense U ⊂ Z such that M 7→ ϕM is constant on U. Set ϕZ := ϕM for M ∈ U.

Theorem (Lusztig, Geiss-L-Schr¨oer) ∗ S := {ϕZ | Z ∈ Irr(Λ)} is a basis of C[N].

S∗ is dual to Lusztig’s semicanonical basis of U(n). ∗ The generalized minors ∆$k ,w($k ) belong to S . → talks of Jan Schr¨oer and Christof Geiss.

Study these cluster algebras by means of preprojective algebras and semicanonical bases.

w Categorify C[N(w)] and C[N ], and obtain cluster algebra structures.

Aims → talks of Jan Schr¨oer and Christof Geiss.

Study these cluster algebras by means of preprojective algebras and semicanonical bases.

Aims

w Categorify C[N(w)] and C[N ], and obtain cluster algebra structures. → talks of Jan Schr¨oer and Christof Geiss.

Aims

w Categorify C[N(w)] and C[N ], and obtain cluster algebra structures.

Study these cluster algebras by means of preprojective algebras and semicanonical bases. Aims

w Categorify C[N(w)] and C[N ], and obtain cluster algebra structures.

Study these cluster algebras by means of preprojective algebras and semicanonical bases.

→ talks of Jan Schr¨oer and Christof Geiss.