Representation theory of the two-boundary Temperley-Lieb algebra - - PowerPoint PPT Presentation

Representation theory of the two-boundary Temperley-Lieb algebra

Zajj Daugherty (Joint work in progress with Arun Ram) September 10, 2014

Temperley-Lieb algebras

The Temperley-Lieb algebra TLk(q) is the algebra of non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relation = q + q−1

Temperley-Lieb algebras

The Temperley-Lieb algebra TLk(q) is the algebra of non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relation = q + q−1 Multiplication:

Temperley-Lieb algebras

The Temperley-Lieb algebra TLk(q) is the algebra of non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relation = q + q−1 Multiplication:

Temperley-Lieb algebras

The Temperley-Lieb algebra TLk(q) is the algebra of non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relation = q + q−1 Multiplication:

Temperley-Lieb algebras

The Temperley-Lieb algebra TLk(q) is the algebra of non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relation = q + q−1 Multiplication: = ∗(q + q−1)2

Temperley-Lieb algebras

The one-boundary Temperley-Lieb algebra TL(1)

k (q, z0) is the

algebra of one-walled non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relations = q + q−1 and

if even # connections below

= 1

• r

if odd # connections below

= z0.

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1, with relations eiei±1ei = ei for i ≥ 1 =

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1, with relations eiei±1ei = ei for i ≥ 1 =

• r

=

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1, with relations eiei±1ei = ei for i ≥ 1 =

• r

= e2

i = aei

= (q + q−1)

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1, with relations eiei±1ei = ei for i ≥ 1 =

• r

= e2

i = aei

= (q + q−1)

• r

= z0

Odd/even relations

The algebra TL(1)

k (q, z0) is generated by

ei =

i i

and e0 =

1 1

for i = 1, . . . , k − 1, with relations eiei±1ei = ei for i ≥ 1 =

• r

= e2

i = aei

= (q + q−1)

• r

= z0 Side loops are resolved with a 1 or a z0 depending on whether there are an even or odd number of connections below their lowest point.

Temperley-Lieb algebras

The one-boundary Temperley-Lieb algebra TL(1)

k (q, z0) is the

algebra of one-walled non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

with multiplication given by stacking diagrams, subject to the relations = q + q−1 and

if even # connections below

= 1

• r

if odd # connections below

= z0.

Our main object: two-boundary Temperley-Lieb algebra

Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL(2)

k (q, z0, zk) = Tk is

the algebra of two-walled non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q−1 and

if even # connections below

= = 1

• r

if odd # connections below

= z0, = zk.

Our main object: two-boundary Temperley-Lieb algebra

Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL(2)

k (q, z0, zk) = Tk is

the algebra of two-walled non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q−1 and

if even # connections below

= = 1

• r

if odd # connections below

= z0, = zk.

Our main object: two-boundary Temperley-Lieb algebra

Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL(2)

k (q, z0, zk) = Tk is

the algebra of two-walled non-crossing pairings on 2k vertices

1 1 2 2 3 3 4 4 k k

ei = e0 = ek =

so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q−1 and

if even # connections below

= = 1

• r

if odd # connections below

= z0, = zk.

Our main object: two-boundary Temperley-Lieb algebra

TLk is finite-dimensional (nth Catalan number)

Our main object: two-boundary Temperley-Lieb algebra

TLk is finite-dimensional (nth Catalan number) TL(1)

k

is finite-dimensional

Our main object: two-boundary Temperley-Lieb algebra

TLk is finite-dimensional (nth Catalan number) TL(1)

k

is finite-dimensional TL(2)

k

= Tk is infinite-dimensional! 2ℓ

Our main object: two-boundary Temperley-Lieb algebra

TLk is finite-dimensional (nth Catalan number) TL(1)

k

is finite-dimensional TL(2)

k

= Tk is infinite-dimensional! 2ℓ de Gier, Nichols (2008): Explored representation theory of Tk.

1 Take quotients giving

= z to get finite-dimensional algebras.

2 Establish connection to the affine Hecke algebras of type A and C

to facilitate calculations.

3 Use diagrammatics and an action on (C2)⊗k to help classify

representations in quotient (most modules are 2k dim’l; some split).

Our main object: two-boundary Temperley-Lieb algebra

TLk is finite-dimensional (nth Catalan number) SWD TL(1)

k

is finite-dimensional SWD TL(2)

k

= Tk is infinite-dimensional! 2ℓ de Gier, Nichols (2008): Explored representation theory of Tk.

1 Take quotients giving

= z to get finite-dimensional algebras.

2 Establish connection to the

SWD

• affine Hecke algebras of type A and C

to facilitate calculations.

3 Use diagrammatics and an action on (C2)⊗k to help classify

representations in quotient (most modules are 2k dim’l; some split).

Quantum groups and braids

Fix q ∈ C∗. Let U = Uqg be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g. Let V, M be U-modules. Then U ⊗ U has invertible R =

R R1 ⊗ R2 that yields a map

ˇ RV M : V ⊗ M − → M ⊗ V v ⊗ m − →

• R

R1m ⊗ R2v M ⊗ V V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of Uqg.

Quantum groups and braids

Fix q ∈ C∗. Let U = Uqg be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g. Let V, M be U-modules. Then U ⊗ U has invertible R =

R R1 ⊗ R2 that yields a map

ˇ RV M : V ⊗ M − → M ⊗ V v ⊗ m − →

• R

R1m ⊗ R2v M ⊗ V V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of Uqg. The braid group shares a commuting action with Uqg on V ⊗k: V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V

Quantum groups and braids

Fix q ∈ C∗. Let U = Uqg be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g. Let V, M be U-modules. Then U ⊗ U has invertible R =

R R1 ⊗ R2 that yields a map

ˇ RV M : V ⊗ M − → M ⊗ V v ⊗ m − →

• R

R1m ⊗ R2v M ⊗ V V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of Uqg. The one-boundary/affine braid group shares a commuting action with Uqg on N ⊗ V ⊗k: V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V N⊗ N⊗

Around the pole:

N⊗V N⊗V

= ˇ RNV ˇ RV N

Quantum groups and braids

Fix q ∈ C∗. Let U = Uqg be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g. Let V, M be U-modules. Then U ⊗ U has invertible R =

R R1 ⊗ R2 that yields a map

ˇ RV M : V ⊗ M − → M ⊗ V v ⊗ m − →

• R

R1m ⊗ R2v M ⊗ V V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of Uqg. The two-boundary braid group shares a commuting action with Uqg on N ⊗ V ⊗k ⊗ M: V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V ⊗ ⊗ V V N⊗ N⊗ ⊗M ⊗M

Around the pole:

N⊗V N⊗V

= ˇ RNV ˇ RV N

Affine type C Hecke algebra and two-boundary braids

k 1 2 3 4 k−2 k−1

· · · Fix constants t0, tk, and t = t1 = · · · = tk−1. The affine Hecke algebra

• f type C, Hk, is generated by T0, T1, . . . , Tk with relations

TiTj . . .

mi,j factors

= TjTi . . .

mi,j factors

where mi,j = 2 if

i j

3 if

i j

4 if

i j

and T 2

i = (t1/2 i

− t−1/2

i

)Ti + 1.

Affine type C Hecke algebra and two-boundary braids

k 1 2 3 4 k−2 k−1

· · · Fix constants t0, tk, and t = t1 = · · · = tk−1. The affine Hecke algebra

• f type C, Hk, is generated by T0, T1, . . . , Tk with relations

TiTj . . .

mi,j factors

= TjTi . . .

mi,j factors

and T 2

i = (t1/2 i

− t−1/2

i

)Ti + 1.

Affine type C Hecke algebra and two-boundary braids

k 1 2 3 4 k−2 k−1

· · · Fix constants t0, tk, and t = t1 = · · · = tk−1. The affine Hecke algebra

• f type C, Hk, is generated by T0, T1, . . . , Tk with relations

TiTj . . .

mi,j factors

= TjTi . . .

mi,j factors

and T 2

i = (t1/2 i

− t−1/2

i

)Ti + 1. The two-boundary (two-pole) braid group Bk is generated by Tk = T0 = and Ti =

i i i+1 i+1

for 1 ≤ i ≤ k − 1.

Affine type C Hecke algebra and two-boundary braids

k 1 2 3 4 k−2 k−1

· · · Fix constants t0, tk, and t = t1 = · · · = tk−1. The affine Hecke algebra

• f type C, Hk, is generated by T0, T1, . . . , Tk with relations

TiTj . . .

mi,j factors

= TjTi . . .

mi,j factors

and T 2

i = (t1/2 i

− t−1/2

i

)Ti + 1. The two-boundary (two-pole) braid group Bk is generated by Tk = T0 = and Ti =

i i i+1 i+1

for 1 ≤ i ≤ k − 1. Relations: TiTi+1Ti = = = Ti+1TiTi+1

Affine type C Hecke algebra and two-boundary braids

k 1 2 3 4 k−2 k−1

· · · Fix constants t0, tk, and t = t1 = · · · = tk−1. The affine Hecke algebra

• f type C, Hk, is generated by T0, T1, . . . , Tk with relations

TiTj . . .

mi,j factors

= TjTi . . .

mi,j factors

and T 2

i = (t1/2 i

− t−1/2

i

)Ti + 1. The two-boundary (two-pole) braid group Bk is generated by Tk = T0 = and Ti =

i i i+1 i+1

for 1 ≤ i ≤ k − 1. Relations: TiTi+1Ti = = = Ti+1TiTi+1 T1T0T1T0 = = = T0T1T0T1

Theorem (D.-Ram, degenerate versions of 1&2 in [D. 10])

(1) Let U = Uqg for any complex reductive Lie algebras g. Let M, N, and V be finite-dimensional modules. The two-boundary braid group Bk acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. (2) If g = gln, then (for good simple choices of M, N, and V ), the affine Hecke algebra of type C, Hk, acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U.

Theorem (D.-Ram, degenerate versions of 1&2 in [D. 10])

(1) Let U = Uqg for any complex reductive Lie algebras g. Let M, N, and V be finite-dimensional modules. The two-boundary braid group Bk acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. (2) If g = gln, then (for good simple choices of M, N, and V ), the affine Hecke algebra of type C, Hk, acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. Now using braid diagrammatics, [GN 08] says that by identifying = t1/2 − , c0 = t1/2 − , and ck = t1/2

k

−

(where ci = t1/2

i

t−1/2 + t−1/2

i

t1/2),

then Tk is a quotient of Hk by. . .

Theorem (D.-Ram, degenerate versions of 1&2 in [D. 10])

(1) Let U = Uqg for any complex reductive Lie algebras g. Let M, N, and V be finite-dimensional modules. The two-boundary braid group Bk acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. (2) If g = gln, then (for good simple choices of M, N, and V ), the affine Hecke algebra of type C, Hk, acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. Now using braid diagrammatics, [GN 08] says that by identifying = t1/2 − , c0 = t1/2 − , and ck = t1/2

k

−

(where ci = t1/2

i

t−1/2 + t−1/2

i

t1/2),

then Tk is a quotient of Hk by eiei±1ei for 1 ≤ i ≤ k−1 : =

• r

=

• and

reverses

Theorem (D.-Ram, degenerate versions of 1&2 in [D. 10])

(1) Let U = Uqg for any complex reductive Lie algebras g. Let M, N, and V be finite-dimensional modules. The two-boundary braid group Bk acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. (2) If g = gln, then (for good simple choices of M, N, and V ), the affine Hecke algebra of type C, Hk, acts on N ⊗ (V )⊗k ⊗ M and this action commutes with the action of U. Now using braid diagrammatics, [GN 08] says that by identifying = t1/2 − , c0 = t1/2 − , and ck = t1/2

k

−

(where ci = t1/2

i

t−1/2 + t−1/2

i

t1/2),

then Tk is a quotient of Hk by eiei±1ei for 1 ≤ i ≤ k−1 : =

• r

=

• and

reverses

• (3) When g = gl2, Tk acts on N ⊗ (V )⊗k ⊗ M (for good choices).

Consider the fin-dim’l simple Uqgln-modules L(λ) indexed by partitions: λ =

Consider the fin-dim’l simple Uqgln-modules L(λ) indexed by partitions: λ =

1 2 3

• 1
• 2

The content of a box is its diagonal number.

Consider the fin-dim’l simple Uqgln-modules L(λ) indexed by partitions: λ =

1 2 3

• 1
• 2

The content of a box is its diagonal number. Fix V = L( ). The generators of Hk acting on N ⊗ V ⊗k ⊗ M look like Tk =

V ⊗M V ⊗M

T0 =

N ⊗V N ⊗V

and Ti =

V ⊗ V V ⊗ V

The eigenvalues of these operators (of which there should be two, since (Tk−t1/2

k

)(Tk+t−1/2

k

) = (T0−t1/2 )(T0+t−1/2 ) = (Ti−t1/2)(Ti+t−1/2) = 0 ) are controlled by contents of addable boxes.

Consider the fin-dim’l simple Uqgln-modules L(λ) indexed by partitions: λ =

1 2 3

• 1
• 2

The content of a box is its diagonal number. Fix V = L( ). The generators of Hk acting on N ⊗ V ⊗k ⊗ M look like Tk =

V ⊗M V ⊗M

T0 =

N ⊗V N ⊗V

and Ti =

V ⊗ V V ⊗ V

The eigenvalues of these operators (of which there should be two, since (Tk−t1/2

k

)(Tk+t−1/2

k

) = (T0−t1/2 )(T0+t−1/2 ) = (Ti−t1/2)(Ti+t−1/2) = 0 ) are controlled by contents of addable boxes. So let M and N be indexed by rectangular partitions, which have two addable boxes: (ac) = c a

a

• c

Consider the fin-dim’l simple Uqgln-modules L(λ) indexed by partitions: λ =

1 2 3

• 1
• 2

The content of a box is its diagonal number. Fix V = L( ). The generators of Hk acting on N ⊗ V ⊗k ⊗ M look like Tk =

V ⊗M V ⊗M

T0 =

N ⊗V N ⊗V

and Ti =

V ⊗ V V ⊗ V

The eigenvalues of these operators (of which there should be two, since (Tk−t1/2

k

)(Tk+t−1/2

k

) = (T0−t1/2 )(T0+t−1/2 ) = (Ti−t1/2)(Ti+t−1/2) = 0 ) are controlled by contents of addable boxes. So let M and N be indexed by rectangular partitions, which have two addable boxes: (ac) = c a

a

• c

Hk has a commuting action with Uqgln on the space L((bd)) ⊗

• L( )

⊗k ⊗ L((ac)) with c, d < n

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Then M ⊗ N = L((ac)) ⊗ L((bd)) =

• λ∈Λ

L(λ),

(multiplicity one!)

where Λ is the following set of partitions:

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Then M ⊗ N = L((ac)) ⊗ L((bd)) =

• λ∈Λ

L(λ),

(multiplicity one!)

where Λ is the following set of partitions: a b c d

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Then M ⊗ N = L((ac)) ⊗ L((bd)) =

• λ∈Λ

L(λ),

(multiplicity one!)

where Λ is the following set of partitions: a b c d

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Then M ⊗ N = L((ac)) ⊗ L((bd)) =

• λ∈Λ

L(λ),

(multiplicity one!)

where Λ is the following set of partitions:

Exploring tensor space structure

Move the right pole to the left:

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

=

N N M M ⊗ ⊗ ⊗ ⊗ V V V V V V V V V V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

New favorite generators: T0 = , Ti =

i i i+1 i+1

and Yj =

j j

Then M ⊗ N = L((ac)) ⊗ L((bd)) =

• λ∈Λ

L(λ),

(multiplicity one!)

where Λ is the following set of partitions. . . (ac) ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕

Exploring tensor space structure

M

c a

Exploring tensor space structure

M k = 0

c a

Exploring tensor space structure

M k = 0

c a

Exploring tensor space structure

M k = 0 k = 1

c a

Central characters

The Hecke algebra Hk features invertible, pairwise commuting elements Y1, . . . , Yk (weight lattice part). Yj =

j j

Central characters

The Hecke algebra Hk features invertible, pairwise commuting elements Y1, . . . , Yk (weight lattice part). The Weyl group W of type C (the group of signed permutations) acts on C[Y ±1

1

, . . . , Y ±1

k

] by permuting the subscripts, with Y−i = Y −1

i

. Then the center of Hk is symmetric Laurent polynomials Z(Hk) = C[Y ±1

1

, . . . , Y ±1

k

]W. Yj =

j j

Central characters

The Hecke algebra Hk features invertible, pairwise commuting elements Y1, . . . , Yk (weight lattice part). The Weyl group W of type C (the group of signed permutations) acts on C[Y ±1

1

, . . . , Y ±1

k

] by permuting the subscripts, with Y−i = Y −1

i

. Then the center of Hk is symmetric Laurent polynomials Z(Hk) = C[Y ±1

1

, . . . , Y ±1

k

]W. We can encode central characters as maps γ : {Y ±1

1

, . . . , Y ±1

k

} → C× with equivalence under W action;

Central characters

The Hecke algebra Hk features invertible, pairwise commuting elements Y1, . . . , Yk (weight lattice part). The Weyl group W of type C (the group of signed permutations) acts on C[Y ±1

1

, . . . , Y ±1

k

] by permuting the subscripts, with Y−i = Y −1

i

. Then the center of Hk is symmetric Laurent polynomials Z(Hk) = C[Y ±1

1

, . . . , Y ±1

k

]W. We can encode central characters as maps γ : {Y ±1

1

, . . . , Y ±1

k

} → C× with equivalence under W action; i.e. representative k-tuples γ = (γ1, . . . , γk) with γ(Y ±1

i

) = (γi)±1

Central characters

The Hecke algebra Hk features invertible, pairwise commuting elements Y1, . . . , Yk (weight lattice part). The Weyl group W of type C (the group of signed permutations) acts on C[Y ±1

1

, . . . , Y ±1

k

] by permuting the subscripts, with Y−i = Y −1

i

. Then the center of Hk is symmetric Laurent polynomials Z(Hk) = C[Y ±1

1

, . . . , Y ±1

k

]W. We can encode central characters as maps γ : {Y ±1

1

, . . . , Y ±1

k

} → C× with equivalence under W action; i.e. representative k-tuples γ = (γ1, . . . , γk) with γ(Y ±1

i

) = (γi)±1 c = (c1, . . . , ck) with γ(Y ±1

i

) = t±ci (when c is real, favorite representatives satisfy 0 ≤ c1 ≤ · · · ≤ ck.)

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

c1 = c2 (c1, c2)

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

(c1, c2)

hα2 hα2+2α1 hα1+α2 hα1

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

hα2 hα2+2α1 hα1+α2 hα1

c2 = c1 + 1

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

hα2 hα2+2α1 hα1+α2 hα1

c2 = c1 + 1 c2 = −c1 + 1 c2 = −c1 + 1 c2 = c1 − 1

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

hα2 hα2+2α1 hα1+α2 hα1

c2 = c1 + 1 c2 = −c1 + 1 c2 = −c1 − 1 c2 = c1 − 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

hα2 hα2+2α1 hα1+α2 hα1

c2 = c1 + 1 c2 = −c1 + 1 c2 = −c1 − 1 c2 = c1 − 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2 c2 = −r1 c2 = −r2 c1 = −r1 c1 = −r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points

Restrict to real points. Fav equivalence class reps: 0 ≤ c1 ≤ · · · ≤ ck. When k = 2:

hα2 hα2+2α1 hα1+α2 hα1

c2 = c1 + 1 c2 = −c1 + 1 c2 = −c1 − 1 c2 = c1 − 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2 c2 = −r1 c2 = −r2 c1 = −r1 c1 = −r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

Central characters as points; Calibrated reps as “skew local regions”

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

The ris depend on Hk’s parameters t0 and tk: r1 = logt(t0/tk), r2 = logt(t0tk)

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

• Thm. (D.-Ram)

(1) Calibrated representations of Hk are indexed by skew local regions at regular (interior red) points.

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

• Thm. (D.-Ram)

(1) Calibrated representations of Hk are indexed by skew local regions at regular (interior red) points. (2) The representations of Hk which factor through the TL quotient are (see above).

hα2 hα1 hα2+2α1

c2 = c1 + 1 c2 = −c1 + 1 c2 = r1 c2 = r2 c1 = r1 c1 = r2

• Thm. (D.-Ram)

(1) Calibrated representations of Hk are indexed by skew local regions at regular (interior red) points. (2) The representations of Hk which factor through the TL quotient are (see above).

a a b b k k

1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 ℓ =

k + a + b − ℓ ℓ