Some special discrete groups of linear transformations
W. Rossmann
– p. 1/1
Character Table
SL(2, R)
SO(2)
+κ(λ)
X
FD(λ)
DS± (λ)
e
−κ(λ)
∞
X
e
κ(λ)
∞
X
ikx
±ikx
eiλx − e−iλx
= − ix/2
e
− e−ix/2
±e±λx
= ix/2
e
− e−ix/2
eikx = δ(x)
PS(λ)
−∞
−κ(λ)
+κ(λ)
– p. 2/1
Character Table
SL(2, R)
SO(2)
+κ(λ)
X
FD(λ)
DS± (λ)
e
−κ(λ)
∞
X
e
ikx
±ikx
κ(λ)
∞
X
eiλx − e−iλx
= − ix/2
e
− e−ix/2
±e±λx
= ix/2
e
− e−ix/2
eikx = δ(x)
PS(λ)
−∞
+κ(λ)
−κ(λ)
PS =DS−
+
FD
+
DS+
– p. 2/1
Orbit Contour Integrals
0.69
0.69
0.69
– p. 3/1
Orbit Contour Integrals
0.69
DS+
0.69
0.69
_
DS
FD
PS = DS ++ DS _ + FD
– p. 3/1
Orbit Contour Integrals
0.69
DS+
0.69
0.69
_
DS
FD
PS = DS ++ DS _ + FD
‘combinatorial’ geometry of representations
– p. 3/1
Orbit Contour Integrals
0.69
DS+
0.69
0.69
_
DS
FD
PS = DS ++ DS _ + FD
‘combinatorial’ geometry of representations
(GL(n): Young diagram algorithms)
– p. 3/1
Orbit Contour Integrals
0.69
DS+
0.69
0.69
_
DS
FD
PS = DS ++ DS _ + FD
‘combinatorial’ geometry of representations
(GL(n): Young diagram algorithms)
Theorem. (Contour Integral Formula for Characters)
Z
1
ehX,ξi+σλ (dξ)
Θ(Γ, λ, exp X) =
D(X) ξ∈Γ
– p. 3/1
Orbit Contour Integrals
0.69
DS+
0.69
0.69
_
DS
FD
PS = DS ++ DS _ + FD
‘combinatorial’ geometry of representations
(GL(n): Young diagram algorithms)
Theorem. (Contour Integral Formula for Characters)
Z
1
ehX,ξi+σλ (dξ)
Θ(Γ, λ, exp X) =
D(X) ξ∈Γ
Γ :=contour on complex coadjoint orbit Ωλ
σλ :=complex symplectic 2-from on Ωλ ,
D(X) = det1/2 (
sinh(adX/2)
).
adX/2
>
– p. 3/1
Wave Front Sets
The wave front set of a representation
– p. 4/1
Wave Front Sets
The wave front set of a representation
The A1 singularity
– p. 4/1
Wave Front Sets
The wave front set of a representation
The A1 singularity
limλ→0 FD(λ)= 0 (vanishing cycle)
– p. 4/1
Wave Front Sets
The wave front set of a representation
The A1 singularity
limλ→0 FD(λ)= 0 (vanishing cycle)
Theorem. (Asymptotics of characters.) As λ → 0
Z
Γ
eX+σλ ∼
X
cO (Γ, λ)
nilp. orb.
Θ(Γ, λ, exp X) ∼
X
Z
eX+σO
O
cO (Γ, λ)ΘO
– p. 4/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
– p. 5/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
Theorem. The representation of W on coherent families of characters decomposes
according to the filtration of wave front sets by nilpotent orbits.
– p. 5/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
Theorem. The representation of W on coherent families of characters decomposes
according to the filtration of wave front sets by nilpotent orbits.
(a) W acts naturally on coherent families of characters
– p. 5/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
Theorem. The representation of W on coherent families of characters decomposes
according to the filtration of wave front sets by nilpotent orbits.
(a) W acts naturally on coherent families of characters
(b) W does not act naturally on characters
– p. 5/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
Theorem. The representation of W on coherent families of characters decomposes
according to the filtration of wave front sets by nilpotent orbits.
(a) W acts naturally on coherent families of characters
(b) W does not act naturally on characters
(c) W acts on characters
– p. 5/1
Weyl Group Representations
W = {1, s | s2 = 1};
sΘ(λ) = Θ(s−1 λ)
sDS± =DS± +FD
sFD= −FD
(s = reflection along FD)
Theorem. The representation of W on coherent families of characters decomposes
according to the filtration of wave front sets by nilpotent orbits.
(a) W
(b) W
(c) W
(d) W
acts naturally on coherent families of characters
does not act naturally on characters
acts on characters
acts on contours
– p. 5/1
Picard–Lefschetz Theory
DS
DS
1.81
s
=
FD
– p. 6/1
Picard–Lefschetz Theory
DS
DS
1.81
s
=
FD
PL deformation of a contour: fixed ends!
– p. 6/1
Picard–Lefschetz Theory
DS
DS
1.81
s
=
FD
PL deformation of a contour: fixed ends!
universal PL Formula: sDS=DS+FD
– p. 6/1
Picard–Lefschetz Theory
DS
DS
1.81
s
=
FD
PL deformation of a contour: fixed ends!
universal PL Formula: sDS=DS+FD
W acts on contours
– p. 6/1
Picard–Lefschetz Theory
DS
DS
1.81
s
=
FD
PL deformation of a contour: fixed ends!
universal PL Formula: sDS=DS+FD
W acts on contours
R
W acts on characters Θ = Γ · · ·
>
– p. 6/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
Toric: a (big!) “toric group” GL(1, C) × · · · ×GL(1, C) acts with an open orbit, eg
componentwise on Pn = {X0 : X1 : · · · : Xn }.
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
Toric: a (big!) “toric group” GL(1, C) × · · · ×GL(1, C) acts with an open orbit, eg
componentwise on Pn = {X0 : X1 : · · · : Xn }.


flag (birationally) if γ = γ0
Deformation: Bγ ≈ Bott-Samelson
if γ = generic


toric
if γ = γ∞
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
Toric: a (big!) “toric group” GL(1, C) × · · · ×GL(1, C) acts with an open orbit, eg
componentwise on Pn = {X0 : X1 : · · · : Xn }.


flag (birationally) if γ = γ0
Deformation: Bγ ≈ Bott-Samelson
if γ = generic


toric
if γ = γ∞
The toric group contains the compact part of the Cartan
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
Toric: a (big!) “toric group” GL(1, C) × · · · ×GL(1, C) acts with an open orbit, eg
componentwise on Pn = {X0 : X1 : · · · : Xn }.


flag (birationally) if γ = γ0
Deformation: Bγ ≈ Bott-Samelson
if γ = generic


toric
if γ = γ∞
The toric group contains the compact part of the Cartan
Moral: look for a toric group acting on a representation space by splitting of
weights.
– p. 7/1
Flag Manifold
Complex group G =SL(2, C), G/B = P1 = C ∪ {∞}, zg =
az+b
.
cz+d
Representation of real forms and the complex flag manifold:
FD: holomorphic sections on P1 = {z}
DS± : L2 -holomorphic sections on {±Im(z) > 0}
PS: L2 sections on {Im(z) = 0}
Fact (Bott). The complex structure on any flag manifold G/B admits (birationally)
a deformation Bγ with a toric limit.
Toric: a (big!) “toric group” GL(1, C) × · · · ×GL(1, C) acts with an open orbit, eg
componentwise on Pn = {X0 : X1 : · · · : Xn }.


flag (birationally) if γ = γ0
Deformation: Bγ ≈ Bott-Samelson
if γ = generic


toric
if γ = γ∞
The toric group contains the compact part of the Cartan
Moral: look for a toric group acting on a representation space by splitting of
weights.
Example: Gelfand-Tsetlin basis for finite dimensional representations
– p. 7/1
Harmonic Oscillator
– p. 8/1
Harmonic Oscillator
“PART TWO”
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
x(t), p(t) = trigonometric functions of t;
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
x(t), p(t) = trigonometric functions of t;
{(x(t), p(t))} = orbit of an action of R/Z
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
x(t), p(t) = trigonometric functions of t;
{(x(t), p(t))} = orbit of an action of R/Z
n
X 1 ∂2
∂
dϕ
= H(x,
)ϕ =
( 2 + αi xi2 )ϕ
quantum:
dt
∂x
2 ∂xi
i=1
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
x(t), p(t) = trigonometric functions of t;
{(x(t), p(t))} = orbit of an action of R/Z
n
X 1 ∂2
∂
dϕ
= H(x,
)ϕ =
( 2 + αi xi2 )ϕ
quantum:
dt
∂x
2 ∂xi
i=1
2
ϕ = functions of the type (polynomial) × e−αx (Hermite functions)
– p. 8/1
Harmonic Oscillator
“PART TWO”
n
X
1 2
Hamiltonian H(x, p) :=
(pi + αi xi2 )
2
i=1
classical
d(xi , pi )
∂H
∂H
=(
,−
) = (pi , −αi xi )
dt
∂pi
∂xi
x(t), p(t) = trigonometric functions of t;
{(x(t), p(t))} = orbit of an action of R/Z
n
X 1 ∂2
∂
dϕ
= H(x,
)ϕ =
( 2 + αi xi2 )ϕ
quantum:
dt
∂x
2 ∂xi
i=1
2
ϕ = functions of the type (polynomial) × e−αx (Hermite functions)
ϕ(t) =orbit of a representation of R/2−1 Z.
– p. 8/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
exp(tH) belongs to G =Sp(V ), V = {xp}
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
exp(tH) belongs to G =Sp(V ), V = {xp}
The Cartan T = Tx1 × · · · × Txn is a ’big’ toric group.
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
exp(tH) belongs to G =Sp(V ), V = {xp}
The Cartan T = Tx1 × · · · × Txn is a ’big’ toric group.
Ωφ = V − {0} is the minimal orbit of Sp(V ).
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
exp(tH) belongs to G =Sp(V ), V = {xp}
The Cartan T = Tx1 × · · · × Txn is a ’big’ toric group.
Ωφ = V − {0} is the minimal orbit of Sp(V ).
Q. What’s the point of the ξ1 , · · · , ξn ?
– p. 9/1
PDE
Theorem. The quantum oscillator
elliptic coordinates (ξ1 , · · · , ξn ).
dϕ
dt
= H(x,
∂
)ϕ
∂x
is completely integrable in
Meaning:
The representation exp(tH) of T = R/2−1 Z on H = {ϕ(x)} splits as a tensor
product H = Hξ1 ⊗ · · · ⊗ Hξn via ϕ(x) = ϕ(ξ1 , · · · , ξn ) and exp(tH) acts
coordinatewise via T → Tξ1 × · · · × Tξn (“toric group”)
Much simpler:
= H(x,
The quantum oscillator dϕ
dt
coordinates x1 , · · · , xn .
∂
)ϕ
∂x
is completely integrable in Euclidean
Moreover:
exp(tH) belongs to G =Sp(V ), V = {xp}
The Cartan T = Tx1 × · · · × Txn is a ’big’ toric group.
Ωφ = V − {0} is the minimal orbit of Sp(V ).
Q. What’s the point of the ξ1 , · · · , ξn ?
A. Ωφ is rigid in sp{xp}∗ but becomes mobile along with x = x(ξ, λ)
– p. 9/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
Q
f (z) := (z − λi ) encodes the parameters, z ∈ P1
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
Q
f (z) := (z − λi ) encodes the parameters, z ∈ P1
’Consider’ the ODE
d2
ϕi (z) = Ai (z)ϕi (z),
du2
p
[du := dz/ f , ]
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
Q
f (z) := (z − λi ) encodes the parameters, z ∈ P1
’Consider’ the ODE
Ai (z) :=
Pn−2
k=0
d2
ϕi (z) = Ai (z)ϕi (z),
du2
ak z k + (−αi λi + βi )z n−1 + (
p
[du := dz/ f , ]
P
−αi )z n
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
Q
f (z) := (z − λi ) encodes the parameters, z ∈ P1
’Consider’ the ODE
Ai (z) :=
Pn−2
k=0
d2
ϕi (z) = Ai (z)ϕi (z),
du2
ak z k + (−αi λi + βi )z n−1 + (
p
[du := dz/ f , ]
P
−αi )z n
Theorem (main point). Let ϕ1 (z), · · · , ϕn (z) be n solutions of the ODE with
Q
arbitrary polynomials of the form Ai (z). Then ϕ(ξ) = ϕi (ξi ) is a solution of the
P
oscillator PDE Hϕ = Eϕ with E = (−αi λi + βi ).
– p. 10/1
Separation of Variables
coordinate variables x = (x1 , · · · , xn ), ξ = (ξ1 , · · · , ξn )
parameters λ = (λ1 , · · · , λn )
equation (partial fractions)
Q
X x2
(z − ξi )
i
Q
=1−
z − λi
(z − λi )
(i = 1, · · · , n)
the elliptic coordordinate map x = x(ξ, λ) depends on λ
Q
f (z) := (z − λi ) encodes the parameters, z ∈ P1
’Consider’ the ODE
Ai (z) :=
Pn−2
k=0
d2
ϕi (z) = Ai (z)ϕi (z),
du2
ak z k + (−αi λi + βi )z n−1 + (
p
[du := dz/ f , ]
P
−αi )z n
Theorem (main point). Let ϕ1 (z), · · · , ϕn (z) be n solutions of the ODE with
Q
arbitrary polynomials of the form Ai (z). Then ϕ(ξ) = ϕi (ξi ) is a solution of the
P
oscillator PDE Hϕ = Eϕ with E = (−αi λi + βi ).
– p. 10/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
Q
X x2
(z − ξi )
i
Q
=1−
(z − λi )
z − λi
(i = 1, · · · , 3)
solution (“generic point” ξx):
– p. 11/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
Q
X x2
(z − ξi )
i
Q
=1−
(z − λi )
z − λi
(i = 1, · · · , 3)
solution (“generic point” ξx):
– p. 11/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
Q
X x2
(z − ξi )
i
Q
=1−
(z − λi )
z − λi
solution (“generic point” ξx):
1
ξ1 = ℘(u1 ) + (λ1 + λ2 + λ3 ),
3
(℘(u1 ) − λ3 )(℘(u2 ) − λ3 )(℘(u3 ) − λ3 )
x1 =
,
(λ1 − λ3 )(λ2 − λ3 )
(i = 1, · · · , 3)
ξ2 = · · ·
ξ3 = · · ·
x2 = · · ·
x3 = · · ·
– p. 11/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
Q
X x2
(z − ξi )
i
Q
=1−
(z − λi )
z − λi
solution (“generic point” ξx):
1
ξ1 = ℘(u1 ) + (λ1 + λ2 + λ3 ),
3
(℘(u1 ) − λ3 )(℘(u2 ) − λ3 )(℘(u3 ) − λ3 )
x1 =
,
(λ1 − λ3 )(λ2 − λ3 )
Rz
℘–function z = ℘(u) ⇔ u = ∞ $ [du = √dz
f (z)
(i = 1, · · · , 3)
ξ2 = · · ·
ξ3 = · · ·
x2 = · · ·
x3 = · · ·
]
– p. 11/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
1
ξ2 = · · ·
ξ1 = ℘(u1 ) + (λ1 + λ2 + λ3 ),
3
(℘(u1 ) − λ3 )(℘(u2 ) − λ3 )(℘(u3 ) − λ3 )
x1 =
,
x2 = · · ·
(λ1 − λ3 )(λ2 − λ3 )
Rz
℘–function z = ℘(u) ⇔ u = ∞ $ [du = √dz ]
f (z)
R
dz
I(γ, λ) = γ √
ξ3 = · · ·
x3 = · · ·
(z−λ1 )(z−λ2 )(z−λ3 )
γ1
z
γ2
λ1
λ2
λ3
– p. 11/1
Elliptic Contour Integrals
n = 3; f (z) := (z − λ1 )(z − λ2 )(z − λ3 )
1
ξ2 = · · ·
ξ1 = ℘(u1 ) + (λ1 + λ2 + λ3 ),
3
(℘(u1 ) − λ3 )(℘(u2 ) − λ3 )(℘(u3 ) − λ3 )
x1 =
,
x2 = · · ·
(λ1 − λ3 )(λ2 − λ3 )
Rz
℘–function z = ℘(u) ⇔ u = ∞ $ [du = √dz ]
f (z)
R
dz
I(γ, λ) = γ √
ξ3 = · · ·
x3 = · · ·
(z−λ1 )(z−λ2 )(z−λ3 )
γ1
z
γ2
λ1
λ2
λ3
elliptic curve over {z} : Eλ = {w 2 = f (z)}; branch cuts [λ1 , λ2 ], [λ3 , ∞],
1-homology H1 (Eλ ) = {γ = m1 γ1 + m2 γ2 };
H
H
period lattice Λλ = {µ = m1 ω1 + m2 ω2 }, ω1 := γ $, ω2 := γ $
1
2
>
– p. 11/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
γ1
z
γ2
λ1
λ2
λ3
– p. 12/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
γ1
z
γ2
λ1
λ2
λ3
– p. 12/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
γ1
z
γ2
λ2
λ1
λ3
φ=π
φ=0
γ1
λ1
γ2
λ2
λ2
λ1
λ2
λ1
– p. 12/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
φ=π
φ=0
γ1
λ1
γ2
λ2
λ2
λ1
λ2
λ1
parameter motion forces contour deformation: T : γ 1 7→ γ1 + γ2 , γ2 7→ γ2 ,
– p. 12/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
φ=π
φ=0
γ1
λ1
γ2
λ2
λ2
λ1
λ2
λ1
parameter motion forces contour deformation: T : γ 1 7→ γ1 + γ2 , γ2 7→ γ2 ,
generators for SL(2, Z). T (ω1 , ω2 ) := (ω1 + ω2 , ω2 ), S(ω1 , ω2 ) := (−ω2 , ω1 )
– p. 12/1
Elliptic Picard-Lefschetz Theory
I(γ, λ) =
R
γ
dz
(z−λ1 )(z−λ2 )(z−λ3 )
√
φ=π
φ=0
γ1
λ1
γ2
λ2
λ2
λ1
λ2
λ1
parameter motion forces contour deformation: T : γ 1 7→ γ1 + γ2 , γ2 7→ γ2 ,
generators for SL(2, Z). T (ω1 , ω2 ) := (ω1 + ω2 , ω2 ), S(ω1 , ω2 ) := (−ω2 , ω1 )
Transformation Formulas (very old). On the elliptic curve E λ = {w 2 = f (z)} the
functions xi (λ, ξ) split into numerator and denominator in the form
Xi (λ, ξ)
xi (λ, ξ) =
X0 (λ, ξ)
1
ξi = ℘(ui ) + (λ1 + λ2 + λ3 ) ,
3
(Primeform factorization). The xi (λ, ξ) transform under SL(2, Z) = hS, T i according to
the oscillator representation of SL(2, Z/2Z)×SL(2, Z/2Z) on C 4 = {X0 , · · · , X4 }.
>(to be explained) >
– p. 12/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
Finite versions V2 := Λ02 /Λ2 ≈ (Z/2Z)2 ,
Hb(V2 ),
H2 (Λ) := H(V )Λ̃2
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
Finite versions V2 := Λ02 /Λ2 ≈ (Z/2Z)2 ,
Hb(V2 ),
H2 (Λ) := H(V )Λ̃2
Klein-Hurwitz Representation.
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
Finite versions V2 := Λ02 /Λ2 ≈ (Z/2Z)2 ,
Hb(V2 ),
H2 (Λ) := H(V )Λ̃2
Klein-Hurwitz Representation.
(a) The functions X0 , X1 , X2 , X4 form a basis for H2 (Λ).
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
Finite versions V2 := Λ02 /Λ2 ≈ (Z/2Z)2 ,
Hb(V2 ),
H2 (Λ) := H(V )Λ̃2
Klein-Hurwitz Representation.
(a) The functions X0 , X1 , X2 , X4 form a basis for H2 (Λ).
(b) The representation of Hb(V2 ) in H2 (Λ) is the canonical representation of
Hb(V2 )×Hb(V2 ).
– p. 13/1
Lattice Model
period lattice Λ = {µ = m1 ω1 + m2 ω2 | m1 , m2 ∈ Z}
real symplectic space V = {u = x1 ω1 + x2 ω2 }, (u | u0 ) := x1 x02 − x2 x01
real Heisenberg group Hb(V ) := {eiθ eu | u ∈ V }, eu ev = e2πi(u|v) ev eu
maximal abelian subgroup Λ̃ := {eiθ eµ | µ ∈ Λ}
canonical representation of Hb(V ) (lattice model) H(V ) :=Ind
Hb(V ) iθ
e
Λ̃
Sp(V ) acts naturally on Hb(V ) and on PH(V ) (oscillator rep, lattice model)
dual lattices Λ2 ⊂ Λ02 in V : h2ω1 , ω2 i ⊂ hω1 , 2−1 ω2 i
Finite versions V2 := Λ02 /Λ2 ≈ (Z/2Z)2 ,
Hb(V2 ),
H2 (Λ) := H(V )Λ̃2
Klein-Hurwitz Representation.
(a) The functions X0 , X1 , X2 , X4 form a basis for H2 (Λ).
(b) The representation of Hb(V2 ) in H2 (Λ) is the canonical representation of
Hb(V2 )×Hb(V2 ).
(c) The representation of SL(2, Z) = hS, T i in H 2 (Λ) is the oscillator
representation of Sp(V2 )×Sp(V2 ).
<
– p. 13/1
Analogies
Weyl group W
complex orbit Ωλ
contour Γ on Ωλ
orbit integrals for Θ(λ)
W rep on 0 H(Ωλ )
Cartan TR
rep of GR
?!
R
contours Γ · · · <
deformations Γ = Γ(λ) <
modular group SL(2, Z)
phase space {(x(λ, ξ), p(λ, ξ))}
contour γ on Eλ ≈ V /Λλ
elliptic integrals for xi (λ)
SL(2, Z) rep on H(V2 (λ))
osc T = {exp (tH)})
osc rep of Sp(V ) on {ϕ(x)}
toric group on {ϕ(ξ)}
R
γ ··· <
γ = γ(λ) <
– p. 14/1
Residue Lemma
Notation.
Y
M (z) :=
(z − ξi )
X Ri (ξ)
1
,
=
M (z)
z − ξi
(i = 1, · · · , n)
[Ri (ξ) = Res (
z=ξi
1
1
)=
]
M (z)
M 0 (ξi )
∞
X
1 (k)
U (z) =
U (0)z k holomorphic on {z 6= ∞}
k!
k=0
– p. 15/1
Residue Lemma
Notation.
Y
M (z) :=
(z − ξi )
X Ri (ξ)
1
,
=
M (z)
z − ξi
(i = 1, · · · , n)
[Ri (ξ) = Res (
z=ξi
1
1
)=
]
M (z)
M 0 (ξi )
∞
X
1 (k)
U (z) =
U (0)z k holomorphic on {z 6= ∞}
k!
k=0


if l < n − 1
0
Sl (ξ) := 1
if l = n − 1

P

k1
kn
if l > n − 1
k1 +···+kn +n=l+1 ξ1 · · · ξn
– p. 15/1
Residue Lemma
Notation.
Y
M (z) :=
(z − ξi )
(i = 1, · · · , n)
X Ri (ξ)
1
,
=
M (z)
z − ξi
[Ri (ξ) = Res (
z=ξi
1
1
)=
]
M (z)
M 0 (ξi )
∞
X
1 (k)
U (z) =
U (0)z k holomorphic on {z 6= ∞}
k!
k=0


if l < n − 1
0
Sl (ξ) := 1
if l = n − 1

P

k1
kn
if l > n − 1
k1 +···+kn +n=l+1 ξ1 · · · ξn
Residue Lemma. With this notation, the residue relation
X
i
1
2πi
I
(ξi )
1
U (z)dz
=−
Q
2πi
j (z − ξj )
I
(∞)
U (z)dz
Q
j (z − ξj )
reads
X
X
U (ξi )Ri (ξ) =
U (l) (ξi )Sl (ξ).
i
l
– p. 15/1
Residue Lemma
Notation.
Y
M (z) :=
(z − ξi )
(i = 1, · · · , n)
X Ri (ξ)
1
,
=
M (z)
z − ξi
[Ri (ξ) = Res (
z=ξi
1
1
)=
]
M (z)
M 0 (ξi )
∞
X
1 (k)
U (z) =
U (0)z k holomorphic on {z 6= ∞}
k!
k=0


if l < n − 1
0
Sl (ξ) := 1
if l = n − 1

P

k1
kn
if l > n − 1
k1 +···+kn +n=l+1 ξ1 · · · ξn
Residue Lemma. With this notation, the residue relation
X
i
1
2πi
I
(ξi )
1
U (z)dz
=−
Q
2πi
j (z − ξj )
I
(∞)
U (z)dz
Q
j (z − ξj )
reads
X
X
U (ξi )Ri (ξ) =
U (l) (ξi )Sl (ξ).
i
l
Proof. Residue theorem and geometric series. – p. 15/1
Theorem
Theorem. In elliptic coordinates, the oscillator PDE
n
X
1 ∂2
( 2 + αi xi2 )ϕ(x) = Eϕ(x)
2 ∂xi
i=1
becomes
X
Ri (ξ) (
p
∂ 2
f (ξi )
) − Ai (ξi ) ϕ(ξ) = 0
∂ξi
for any functions Ai (z) of the form
Ai (z) :=
n−2
X
k
ak z + (−αi λi + βi )z
n−1
k=0
with
P
+(
X
−αi )z n
(−αi λi + βi ) = E.
Proof. By calculation based on the Residue Lemma.
– p. 16/1
Theorem
Theorem. In elliptic coordinates, the oscillator PDE
n
X
1 ∂2
( 2 + αi xi2 )ϕ(x) = Eϕ(x)
2 ∂xi
i=1
becomes
X
Ri (ξ) (
p
∂ 2
f (ξi )
) − Ai (ξi ) ϕ(ξ) = 0
∂ξi
for any functions Ai (z) of the form
Ai (z) :=
n−2
X
k=0
with
P
k
ak z + (−αi λi + βi )z
n−1
+(
X
−αi )z n
(−αi λi + βi ) = E.
Proof. By calculation based on the Residue Lemma.
Corollary. Let ϕ1 (z), · · · , ϕn (z) be n solutions of the ODE
√
Q
d2
ϕ
(z)
=
A
(z)ϕ
(z),
[du
:=
dz/
f
].
Then
ϕ(ξ)
=
ϕi (ξi ) is a solution of the
i
i
i
du2
P
oscillator PDE with E = (−αi λi + βi ).
– p. 16/1
Current Groups
Residue Splitting.
X
K(x(z), p(z))
K(x(z), p(z))
] ⇒ H(x, p) = −
Res [ Q
]
H(x, p) = Res [ Q
z=∞
z=ξ
(z − ξi )
(z − ξi )
i
– p. 17/1
Current Groups
Residue Splitting.
X
K(x(z), p(z))
K(x(z), p(z))
] ⇒ H(x, p) = −
Res [ Q
]
H(x, p) = Res [ Q
z=∞
z=ξ
(z − ξi )
(z − ξi )
i
Elliptic coordinates
Qn
j=1 (z − ξj )
Qn
i=1 (z − λi )
=1−
n
X
i=1
x2i
z − λi
– p. 17/1
Current Groups
Residue Splitting.
X
K(x(z), p(z))
K(x(z), p(z))
] ⇒ H(x, p) = −
Res [ Q
]
H(x, p) = Res [ Q
z=∞
z=ξ
(z − ξi )
(z − ξi )
i
Elliptic coordinates
Qn
j=1 (z − ξj )
Qn
i=1 (z − λi )
=1−
n
X
i=1
Symbol calculus (normal ordering). H(x, p) 7→ H(x,
x2i
z − λi
∂
),
∂x
P
ck pk 7→
P
∂ k
ck (x)( ∂x
) .
– p. 17/1
Current Groups
Residue Splitting.
X
K(x(z), p(z))
K(x(z), p(z))
] ⇒ H(x, p) = −
Res [ Q
]
H(x, p) = Res [ Q
z=∞
z=ξ
(z − ξi )
(z − ξi )
i
Elliptic coordinates
Qn
j=1 (z − ξj )
Qn
i=1 (z − λi )
=1−
n
X
i=1
Symbol calculus (normal ordering). H(x, p) 7→ H(x,
Current groups (loop groups, affine groups)
x2i
z − λi
∂
),
∂x
P
ck pk 7→
P
∂ k
ck (x)( ∂x
) .
Spz = SpVz , Vz = {xz pz }
Πz Spz = {g(z) = arbitrary function}
Πz,mero Spz = {g(z) = meromorphic}
Πz,{ξi } Sp z = {g(z) has poles only at z = ξi }
(Compare: local, global, adelic.)
– p. 17/1
Picture
– p. 18/1
Picture
Fz :
x2
1
z−λ1
+
x2
2
z−λ2
= 1,
∅
z
z
−∞ −→ λ1 −→ λ2 −→ +∞
– p. 18/1
Picture
Fz :
x2
1
z−λ1
+
x2
2
z−λ2
= 1,
∅
z
z
−∞ −→ λ1 −→ λ2 −→ +∞
– p. 18/1
Picture
Fz :
x2
1
z−λ1
+
x2
2
z−λ2
= 1,
∅
z
z
−∞ −→ λ1 −→ λ2 −→ +∞
Compare
x22
x21
(z − ξ1 )(z − ξ2 )
−
=1−
(z − λ1 )(z − λ2 )
z − λ1
z − λ2
with z = ξi (u, λ), x = xi (u, λ).
– p. 18/1