A GNS construction for unitary representations of Lie ... · A GNS construction for unitary...
Transcript of A GNS construction for unitary representations of Lie ... · A GNS construction for unitary...
A GNS construction for unitary representations
of Lie supergroups
Hadi SalmasianDepartment of Mathematics and Statistics
University of Ottawa
March 16, 2012
1 / 80
Supergeometry
Physical motivation: works of Ferrara, Salam, Strathdee, Wess,Zumino,...
Mathematical foundations: works of Berezin, Kac, Kostant,Leites, Manin,...
Basic idea: Functions with both commuting and anticommutingvariables.
xixj = xjxi and ξiξj = −ξjξi for every i, j
Cω(Rm) f(x) =∑
k1,...,km≥0
ck1,··· ,kmxk1
1 · · ·xkm
m
Cω(Rm|n) f(x, ξ) =∑
k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}
ck1,...,km,l1,...,lnxk1
1 · · ·xkm
m · · · ξl11 · · · ξlnn
2 / 80
Supergeometry
Physical motivation: works of Ferrara, Salam, Strathdee, Wess,Zumino,...
Mathematical foundations: works of Berezin, Kac, Kostant,Leites, Manin,...
Basic idea: Functions with both commuting and anticommutingvariables.
xixj = xjxi and ξiξj = −ξjξi for every i, j
Cω(Rm) f(x) =∑
k1,...,km≥0
ck1,··· ,kmxk1
1 · · ·xkm
m
Cω(Rm|n) f(x, ξ) =∑
k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}
ck1,...,km,l1,...,lnxk1
1 · · ·xkm
m · · · ξl11 · · · ξlnn
3 / 80
Supergeometry
Physical motivation: works of Ferrara, Salam, Strathdee, Wess,Zumino,...
Mathematical foundations: works of Berezin, Kac, Kostant,Leites, Manin,...
Basic idea: Functions with both commuting and anticommutingvariables.
xixj = xjxi and ξiξj = −ξjξi for every i, j
Cω(Rm) f(x) =∑
k1,...,km≥0
ck1,··· ,kmxk1
1 · · ·xkm
m
Cω(Rm|n) f(x, ξ) =∑
k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}
ck1,...,km,l1,...,lnxk1
1 · · ·xkm
m · · · ξl11 · · · ξlnn
4 / 80
Supergeometry
Physical motivation: works of Ferrara, Salam, Strathdee, Wess,Zumino,...
Mathematical foundations: works of Berezin, Kac, Kostant,Leites, Manin,...
Basic idea: Functions with both commuting and anticommutingvariables.
xixj = xjxi and ξiξj = −ξjξi for every i, j
Cω(Rm) f(x) =∑
k1,...,km≥0
ck1,··· ,kmxk1
1 · · ·xkm
m
Cω(Rm|n) f(x, ξ) =∑
k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}
ck1,...,km,l1,...,lnxk1
1 · · ·xkm
m · · · ξl11 · · · ξlnn
5 / 80
Supergeometry
Physical motivation: works of Ferrara, Salam, Strathdee, Wess,Zumino,...
Mathematical foundations: works of Berezin, Kac, Kostant,Leites, Manin,...
Basic idea: Functions with both commuting and anticommutingvariables.
xixj = xjxi and ξiξj = −ξjξi for every i, j
Cω(Rm) f(x) =∑
k1,...,km≥0
ck1,··· ,kmxk1
1 · · ·xkm
m
Cω(Rm|n) f(x, ξ) =∑
k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}
ck1,...,km,l1,...,lnxk1
1 · · ·xkm
m · · · ξl11 · · · ξlnn
6 / 80
Supergeometry
Berezin–Kostant–Leites supermanifolds
Λn := Λ(Rn) = 〈ξ1, . . . , xn | ξiξj + ξjξi = 0〉 Λn = (Λn)0 ⊕ (Λn)1.
Rm|n = (Rm,ORm|n) where:
ORm|n(U) = C∞(U)⊗ Λn for every open U ⊆ Rm.
An (m|n)-dimensional supermanifold is a locally ringed space
M = (M,OM)
with an open covering M =⋃
α∈I Uα such that
(Uα,OM
∣
∣
Uα
) ≃ Rm|n for every α ∈ I.
7 / 80
Supergeometry
Berezin–Kostant–Leites supermanifolds
Λn := Λ(Rn) = 〈ξ1, . . . , xn | ξiξj + ξjξi = 0〉 Λn = (Λn)0 ⊕ (Λn)1.
Rm|n = (Rm,ORm|n) where:
ORm|n(U) = C∞(U)⊗ Λn for every open U ⊆ Rm.
An (m|n)-dimensional supermanifold is a locally ringed space
M = (M,OM)
with an open covering M =⋃
α∈I Uα such that
(Uα,OM
∣
∣
Uα
) ≃ Rm|n for every α ∈ I.
8 / 80
Supergeometry
Berezin–Kostant–Leites supermanifolds
Λn := Λ(Rn) = 〈ξ1, . . . , xn | ξiξj + ξjξi = 0〉 Λn = (Λn)0 ⊕ (Λn)1.
Rm|n = (Rm,ORm|n) where:
ORm|n(U) = C∞(U)⊗ Λn for every open U ⊆ Rm.
An (m|n)-dimensional supermanifold is a locally ringed space
M = (M,OM)
with an open covering M =⋃
α∈I Uα such that
(Uα,OM
∣
∣
Uα
) ≃ Rm|n for every α ∈ I.
9 / 80
Lie supergroups
Lie Supergroups
A supermanifold G = (G,OG) is called a Lie supergroup iff it is agroup object in the category of supermanifolds.
µ : G× G → G ε : {∗} → G ι : G → G
µ ◦ (µ× idG) = (µ× idG) ◦ µ µ ◦ (idG × ε) = π1 µ ◦ (ε× idG) = π2
µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε
G = (G,OG) Lie supergroup Lie superalgebra g = g0 ⊕ g1.
[X, Y ] = −(−1)p(X)p(Y )[Y,X]
(−1)p(X)p(Z)[X, [Y,Z]] + (−1)p(Y )p(X)[Y, [Z,X]] + (−1)p(Z)p(Y )[Z, [X,Y ]] = 0
10 / 80
Lie supergroups
Lie Supergroups
A supermanifold G = (G,OG) is called a Lie supergroup iff it is agroup object in the category of supermanifolds.
µ : G× G → G ε : {∗} → G ι : G → G
µ ◦ (µ× idG) = (µ× idG) ◦ µ µ ◦ (idG × ε) = π1 µ ◦ (ε× idG) = π2
µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε
G = (G,OG) Lie supergroup Lie superalgebra g = g0 ⊕ g1.
[X, Y ] = −(−1)p(X)p(Y )[Y,X]
(−1)p(X)p(Z)[X, [Y,Z]] + (−1)p(Y )p(X)[Y, [Z,X]] + (−1)p(Z)p(Y )[Z, [X,Y ]] = 0
11 / 80
Lie supergroups
Lie Supergroups
A supermanifold G = (G,OG) is called a Lie supergroup iff it is agroup object in the category of supermanifolds.
µ : G× G → G ε : {∗} → G ι : G → G
µ ◦ (µ× idG) = (µ× idG) ◦ µ µ ◦ (idG × ε) = π1 µ ◦ (ε× idG) = π2
µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε
G = (G,OG) Lie supergroup Lie superalgebra g = g0 ⊕ g1.
[X, Y ] = −(−1)p(X)p(Y )[Y,X]
(−1)p(X)p(Z)[X, [Y,Z]] + (−1)p(Y )p(X)[Y, [Z,X]] + (−1)p(Z)p(Y )[Z, [X,Y ]] = 0
12 / 80
Lie supergroups
Lie Supergroups
A supermanifold G = (G,OG) is called a Lie supergroup iff it is agroup object in the category of supermanifolds.
µ : G× G → G ε : {∗} → G ι : G → G
µ ◦ (µ× idG) = (µ× idG) ◦ µ µ ◦ (idG × ε) = π1 µ ◦ (ε× idG) = π2
µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε
G = (G,OG) Lie supergroup Lie superalgebra g = g0 ⊕ g1.
[X, Y ] = −(−1)p(X)p(Y )[Y,X]
(−1)p(X)p(Z)[X, [Y,Z]] + (−1)p(Y )p(X)[Y, [Z,X]] + (−1)p(Z)p(Y )[Z, [X,Y ]] = 0
13 / 80
Examples of Lie superalgebras
Example
General Linear Lie superalgebra gl(m|n):
V = V0 ⊕ V1 = Cm ⊕ C
n
We can write End(V ) = End(V )0 ⊕ End(V )1 where
End(V )i ={
T ∈ End(V ) : TVj ⊆ Vi+j for every j ∈ Z2
}
.
Set gl(m|n)i = End(V )i.
Superbracket: [X,Y ] = XY − (−1)p(X)p(Y )Y X .
Special linear Lie superalgebra sl(m|n):
sl(m|n) ={
A ∈ gl(m|n) | str(A) = 0}
A=
m{ n{[L M
N P
] }m
} n⇒ str(A) = trL− trP .
14 / 80
Examples of Lie superalgebras
Example
General Linear Lie superalgebra gl(m|n):
V = V0 ⊕ V1 = Cm ⊕ C
n
We can write End(V ) = End(V )0 ⊕ End(V )1 where
End(V )i ={
T ∈ End(V ) : TVj ⊆ Vi+j for every j ∈ Z2
}
.
Set gl(m|n)i = End(V )i.
Superbracket: [X,Y ] = XY − (−1)p(X)p(Y )Y X .
Special linear Lie superalgebra sl(m|n):
sl(m|n) ={
A ∈ gl(m|n) | str(A) = 0}
A=
m{ n{[L M
N P
] }m
} n⇒ str(A) = trL− trP .
15 / 80
Examples of Lie superalgebras
Example
General Linear Lie superalgebra gl(m|n):
V = V0 ⊕ V1 = Cm ⊕ C
n
We can write End(V ) = End(V )0 ⊕ End(V )1 where
End(V )i ={
T ∈ End(V ) : TVj ⊆ Vi+j for every j ∈ Z2
}
.
Set gl(m|n)i = End(V )i.
Superbracket: [X,Y ] = XY − (−1)p(X)p(Y )Y X .
Special linear Lie superalgebra sl(m|n):
sl(m|n) ={
A ∈ gl(m|n) | str(A) = 0}
A=
m{ n{[L M
N P
] }m
} n⇒ str(A) = trL− trP .
16 / 80
Examples of Lie superalgebras
Example
General Linear Lie superalgebra gl(m|n):
V = V0 ⊕ V1 = Cm ⊕ C
n
We can write End(V ) = End(V )0 ⊕ End(V )1 where
End(V )i ={
T ∈ End(V ) : TVj ⊆ Vi+j for every j ∈ Z2
}
.
Set gl(m|n)i = End(V )i.
Superbracket: [X,Y ] = XY − (−1)p(X)p(Y )Y X .
Special linear Lie superalgebra sl(m|n):
sl(m|n) ={
A ∈ gl(m|n) | str(A) = 0}
A=
m{ n{[L M
N P
] }m
} n⇒ str(A) = trL− trP .
17 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
18 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
19 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
20 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
21 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
22 / 80
Examples of Lie superalgebras
Example
Orthosymplectic Lie superalgebras osp(m|2n):
osp(m|2n) ={
A ∈ sl(m|2n) | 〈Av,w〉+(−1)p(v)p(A)〈v,Aw〉 = 0}
Exceptional Lie superalgebas, Strange series, Cartanseries.
V. Kac ’77, Serganova ’82: Classification of finitedimensional simple Lie superalgebras over R and C.
Further examples
g: Lie superalgebra g⊗A: Lie superalgebraA: associative Z2-graded algebra
[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′
• A = k[t±11 , . . . , t±1
n ] untwisted (multi)loop superalgebras
• g compact Lie algebra superconformal current algebrasA = k[t1, t
−11 , θ] (θ2 = 0) (Kac–Todorov ’85)
23 / 80
Representation theory
Harish–Chandra pairs
G = (G,OG) (G, g)g = g0 ⊕ g1.
Lie(G) = g0.
G acts on g via automorphisms: Ψ : G → Aut(g).
deΨ(x) = adx for every x ∈ g0.
The pair (G, g) is called a Harish–Chandra pair.
Theorem (Kostant ’75, Koszul ’82)
The functor G 7→ (G, g) is an equivalence of the categories SuperGrp and HCPair.
Representations of Lie superalgebras
• A representation (ρ, V ) of a Lie superalgebra g = g0 ⊕ g1 is a morphism
ρ : g → End(V )
in the category of Lie superalgebras (where V = V0 ⊕ V1).
• When V is an inner product space, (ρ, V ) is called unitary if g acts on V
by (super) skew-adjoint operators.24 / 80
Representation theory
Harish–Chandra pairs
G = (G,OG) (G, g)g = g0 ⊕ g1.
Lie(G) = g0.
G acts on g via automorphisms: Ψ : G → Aut(g).
deΨ(x) = adx for every x ∈ g0.
The pair (G, g) is called a Harish–Chandra pair.
Theorem (Kostant ’75, Koszul ’82)
The functor G 7→ (G, g) is an equivalence of the categories SuperGrp and HCPair.
Representations of Lie superalgebras
• A representation (ρ, V ) of a Lie superalgebra g = g0 ⊕ g1 is a morphism
ρ : g → End(V )
in the category of Lie superalgebras (where V = V0 ⊕ V1).
• When V is an inner product space, (ρ, V ) is called unitary if g acts on V
by (super) skew-adjoint operators.25 / 80
Representation theory
Harish–Chandra pairs
G = (G,OG) (G, g)g = g0 ⊕ g1.
Lie(G) = g0.
G acts on g via automorphisms: Ψ : G → Aut(g).
deΨ(x) = adx for every x ∈ g0.
The pair (G, g) is called a Harish–Chandra pair.
Theorem (Kostant ’75, Koszul ’82)
The functor G 7→ (G, g) is an equivalence of the categories SuperGrp and HCPair.
Representations of Lie superalgebras
• A representation (ρ, V ) of a Lie superalgebra g = g0 ⊕ g1 is a morphism
ρ : g → End(V )
in the category of Lie superalgebras (where V = V0 ⊕ V1).
• When V is an inner product space, (ρ, V ) is called unitary if g acts on V
by (super) skew-adjoint operators.26 / 80
Representation theory
Harish–Chandra pairs
G = (G,OG) (G, g)g = g0 ⊕ g1.
Lie(G) = g0.
G acts on g via automorphisms: Ψ : G → Aut(g).
deΨ(x) = adx for every x ∈ g0.
The pair (G, g) is called a Harish–Chandra pair.
Theorem (Kostant ’75, Koszul ’82)
The functor G 7→ (G, g) is an equivalence of the categories SuperGrp and HCPair.
Representations of Lie superalgebras
• A representation (ρ, V ) of a Lie superalgebra g = g0 ⊕ g1 is a morphism
ρ : g → End(V )
in the category of Lie superalgebras (where V = V0 ⊕ V1).
• When V is an inner product space, (ρ, V ) is called unitary if g acts on V
by (super) skew-adjoint operators.27 / 80
Unitary representations
“Drawing from experience in ordinary Lie theory,graded Lie groups are likely to be a useful objectonly insofar as one can develop a corresponding
theory of harmonic analysis.”
– B. Kostant, in a paper published in 1979.
Unitary representations of Lie groups
A unitary representation (π,H ) of a Lie group G is a group homomorphism
π : G → U(H )
such that for every v ∈ H , the orbit map πv : G → H , πv(g) = π(g)v iscontinuous.
U(H ) : group of linear isometries of a Hilbert space H .
Basic idea: A unitary representation of a Harish–Chandra pair (G, g)should be a compound of:
a unitary representation of G,
a unitary representation of g.
28 / 80
Unitary representations
“Drawing from experience in ordinary Lie theory,graded Lie groups are likely to be a useful objectonly insofar as one can develop a corresponding
theory of harmonic analysis.”
– B. Kostant, in a paper published in 1979.
Unitary representations of Lie groups
A unitary representation (π,H ) of a Lie group G is a group homomorphism
π : G → U(H )
such that for every v ∈ H , the orbit map πv : G → H , πv(g) = π(g)v iscontinuous.
U(H ) : group of linear isometries of a Hilbert space H .
Basic idea: A unitary representation of a Harish–Chandra pair (G, g)should be a compound of:
a unitary representation of G,
a unitary representation of g.
29 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
30 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
31 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
32 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
33 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
34 / 80
Smooth and analytic vectors
G : Lie group.
(π,H ) : Unitary representation of G.
Differentiation: dπ(x)v = limt→0
1
t
(π(etx)v − v
)for every x ∈ g = Lie(G).
Problem: Usually dπ(x) is unbounded, i.e., D(dπ(x)) 6= H .
Example
DefineH
∞ ={v ∈ H : g 7→ π(g)v is C
∞}
andH
ω ={v ∈ H : g 7→ π(g)v is Cω
}.
The actions
dπ : g → End(H ∞) and dπ : g → End(H ω)
are well-defined.
Theorem (Nelson ’59)
Hω is a dense subspace of H .
35 / 80
Unitary representations of Harish–Chandra pairs
Definition (Carmeli, Casssinelli, Toigo, Varadarajan ’06)
A smooth unitary representation of a Harish–Chandra pair (G, g) is atriple (π, ρπ,H ) where
(i) H = H0 ⊕ H1 is a Z2-graded Hilbert space.
(ii) (π,H ) is a unitary rep. of G by even operators.
(iii) ρπ : g → EndC(H∞) is a unitary representation of g.
(iv) ρπ(x) = dπ(x)∣
∣
H ∞ for every x ∈ g0.
Notation. Rep∞(G, g) denotes the category of smooth unitary rep.of (G, g).
Theorem (Carmeli, Cassinelli, Toigo, Varadarajan ’06)
Rep∞(G, g) ≃ Rep
ω(G, g)
36 / 80
Unitary representations of Harish–Chandra pairs
Definition (Carmeli, Casssinelli, Toigo, Varadarajan ’06)
An analytic unitary representation of a Harish–Chandra pair (G, g) isa triple (π, ρπ ,H ) where
(i) H = H0 ⊕ H1 is a Z2-graded Hilbert space.
(ii) (π,H ) is a unitary rep. of G by even operators.
(iii) ρπ : g → EndC(Hω) is a unitary representation of g.
(iv) ρπ(x) = dπ(x)∣
∣
H ωfor every x ∈ g0.
Notation. Repω(G, g) denotes the category of analytic unitary rep.of (G, g).
Theorem (Carmeli, Cassinelli, Toigo, Varadarajan ’06)
Rep∞(G, g) ≃ Rep
ω(G, g)
37 / 80
Unitary representations of Harish–Chandra pairs
Definition (Carmeli, Casssinelli, Toigo, Varadarajan ’06)
An analytic unitary representation of a Harish–Chandra pair (G, g) isa triple (π, ρπ ,H ) where
(i) H = H0 ⊕ H1 is a Z2-graded Hilbert space.
(ii) (π,H ) is a unitary rep. of G by even operators.
(iii) ρπ : g → EndC(Hω) is a unitary representation of g.
(iv) ρπ(x) = dπ(x)∣
∣
H ωfor every x ∈ g0.
Notation. Repω(G, g) denotes the category of analytic unitary rep.of (G, g).
Theorem (Carmeli, Cassinelli, Toigo, Varadarajan ’06)
Rep∞(G, g) ≃ Rep
ω(G, g)
38 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).39 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).40 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).41 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).42 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).43 / 80
Quantization for nilpotent Lie supergroups
G = (G, g): Harish–Chandra pair such that:
G is simply connected
g is a nilpotent Lie superalgebra.
For every λ ∈ g∗0 one can define a symmetric bilinear form
Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])
SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }
C (G) is a G-invariant cone in g∗0.
Theorem (S. ’10)
There exists a bijective correspondence:
irr. unitary rep. G ! G–orbits in C (G)
This extends Kirillov’s classical result (1961).44 / 80
Simple Lie supergroups
G = (G, g) Harish-Chandra pair, g real simple Lie superalgebra.
Theorem (Neeb–S. ’11)
G has nontrivial unitary representations if an only if g is not in thefollowing list:
(a) sl(m|n, R) where m > 2 or n > 2.
(b) su(p, q|r, s) where p, q, r, s > 0.
(c) su∗(2p, 2q) where p, q > 0 and p + q > 2.
(d) pq(m) where m > 1.
(e) usp(m) where m > 1.
(f) osp∗(m|p, q) where p, q,m > 0.
(g) osp(p, q|2n) where p, q, n > 0.
(h) Real forms of P(n), n > 1.
(i) psq(n, R) where n > 2, psq∗(n) where n > 2, and psq(p, q), where p, q > 0.
(j) Real forms of W(n), S(n), and S(n).
(k) H(p, q) where p + q > 4.
(l) Complex simple Lie superalgebras.
This unifies the observations of Hirai, Jakobsen, Nishiyama,Wakimoto,...
45 / 80
Simple Lie supergroups
G = (G, g) Harish-Chandra pair, g real simple Lie superalgebra.
Theorem (Neeb–S. ’11)
G has nontrivial unitary representations if an only if g is not in thefollowing list:
(a) sl(m|n, R) where m > 2 or n > 2.
(b) su(p, q|r, s) where p, q, r, s > 0.
(c) su∗(2p, 2q) where p, q > 0 and p + q > 2.
(d) pq(m) where m > 1.
(e) usp(m) where m > 1.
(f) osp∗(m|p, q) where p, q,m > 0.
(g) osp(p, q|2n) where p, q, n > 0.
(h) Real forms of P(n), n > 1.
(i) psq(n, R) where n > 2, psq∗(n) where n > 2, and psq(p, q), where p, q > 0.
(j) Real forms of W(n), S(n), and S(n).
(k) H(p, q) where p + q > 4.
(l) Complex simple Lie superalgebras.
This unifies the observations of Hirai, Jakobsen, Nishiyama,Wakimoto,...
46 / 80
Simple Lie supergroups
G = (G, g) Harish-Chandra pair, g real simple Lie superalgebra.
Theorem (Neeb–S. ’11)
G has nontrivial unitary representations if an only if g is not in thefollowing list:
(a) sl(m|n, R) where m > 2 or n > 2.
(b) su(p, q|r, s) where p, q, r, s > 0.
(c) su∗(2p, 2q) where p, q > 0 and p + q > 2.
(d) pq(m) where m > 1.
(e) usp(m) where m > 1.
(f) osp∗(m|p, q) where p, q,m > 0.
(g) osp(p, q|2n) where p, q, n > 0.
(h) Real forms of P(n), n > 1.
(i) psq(n, R) where n > 2, psq∗(n) where n > 2, and psq(p, q), where p, q > 0.
(j) Real forms of W(n), S(n), and S(n).
(k) H(p, q) where p + q > 4.
(l) Complex simple Lie superalgebras.
This unifies the observations of Hirai, Jakobsen, Nishiyama,Wakimoto,...
47 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.48 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.49 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.50 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.51 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.52 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.53 / 80
Infinite dimensional supermanifolds
Motivation
(Kac, Todorov ’85) Superconformal current algebras.
(Jakobsen ’94) Highest weight unitary rep. of affine Lie superalgebras.
(Iohara ’10) Super Virasoro algebras.
Rm|n
R∞|∞
Functor of points approach(A. Schwarz, Molotkov ’84, Sachse ’07, Alldridge–Laubinger ’10...)
Basic idea: M is uniquely identified by its Λn-points (n ≥ 0).
Λn = (Λn)0 ⊕ (Λn)1
Superpoints: Set ptn = ({∗},Λn).
M(Λn) = Hom(ptn,M) (ordinary manifold)
Λm → Λn ptn → ptm M(Λm) → M(Λn)
A supermanifold is a covariant functor Gr → Man.54 / 80
Banach supermanifolds
Category Gr
ObjGr = {Λn : n = 0, 1, 2, . . .}HomGr(Λp,Λq) = Homs−algebra(Λp,Λq)
DeWitt topology
If F : Gr → Man then G : Gr → Man is called an open subfunctor of F ifG(Λn) ⊆ F(Λn) is open for every n ≥ 0.
Superdomains
Given a Z2-graded Banach space E = E0 ⊕ E1, set
E : Gr → Man , E(Λn) = (E ⊗ Λn)0.
Note: Λm → Λn E(Λm) → E(Λn)
Superdomains: An open subfunctor of E is called a superdomain.
Morphism of superdomains: A morphism from a superdomain U to asuperdomain V is a natural transformation Φ : U → V.
55 / 80
Banach supermanifolds
Category Gr
ObjGr = {Λn : n = 0, 1, 2, . . .}HomGr(Λp,Λq) = Homs−algebra(Λp,Λq)
DeWitt topology
If F : Gr → Man then G : Gr → Man is called an open subfunctor of F ifG(Λn) ⊆ F(Λn) is open for every n ≥ 0.
Superdomains
Given a Z2-graded Banach space E = E0 ⊕ E1, set
E : Gr → Man , E(Λn) = (E ⊗ Λn)0.
Note: Λm → Λn E(Λm) → E(Λn)
Superdomains: An open subfunctor of E is called a superdomain.
Morphism of superdomains: A morphism from a superdomain U to asuperdomain V is a natural transformation Φ : U → V.
56 / 80
Banach supermanifolds
Category Gr
ObjGr = {Λn : n = 0, 1, 2, . . .}HomGr(Λp,Λq) = Homs−algebra(Λp,Λq)
DeWitt topology
If F : Gr → Man then G : Gr → Man is called an open subfunctor of F ifG(Λn) ⊆ F(Λn) is open for every n ≥ 0.
Superdomains
Given a Z2-graded Banach space E = E0 ⊕ E1, set
E : Gr → Man , E(Λn) = (E ⊗ Λn)0.
Note: Λm → Λn E(Λm) → E(Λn)
Superdomains: An open subfunctor of E is called a superdomain.
Morphism of superdomains: A morphism from a superdomain U to asuperdomain V is a natural transformation Φ : U → V.
57 / 80
Banach supermanifolds
Category Gr
ObjGr = {Λn : n = 0, 1, 2, . . .}HomGr(Λp,Λq) = Homs−algebra(Λp,Λq)
DeWitt topology
If F : Gr → Man then G : Gr → Man is called an open subfunctor of F ifG(Λn) ⊆ F(Λn) is open for every n ≥ 0.
Superdomains
Given a Z2-graded Banach space E = E0 ⊕ E1, set
E : Gr → Man , E(Λn) = (E ⊗ Λn)0.
Note: Λm → Λn E(Λm) → E(Λn)
Superdomains: An open subfunctor of E is called a superdomain.
Morphism of superdomains: A morphism from a superdomain U to asuperdomain V is a natural transformation Φ : U → V.
58 / 80
Banach supermanifolds
Category Gr
ObjGr = {Λn : n = 0, 1, 2, . . .}HomGr(Λp,Λq) = Homs−algebra(Λp,Λq)
DeWitt topology
If F : Gr → Man then G : Gr → Man is called an open subfunctor of F ifG(Λn) ⊆ F(Λn) is open for every n ≥ 0.
Superdomains
Given a Z2-graded Banach space E = E0 ⊕ E1, set
E : Gr → Man , E(Λn) = (E ⊗ Λn)0.
Note: Λm → Λn E(Λm) → E(Λn)
Superdomains: An open subfunctor of E is called a superdomain.
Morphism of superdomains: A morphism from a superdomain U to asuperdomain V is a natural transformation Φ : U → V.
59 / 80
Banach supermanifold and Banach–Lie supergroups
A Banach supermanifold is a functor M : Gr → Man with an open covering
M =⋃
α∈I
Mα
where
each Mα is a superdomain.
the fiber product Mα ×M Mβ has a superdomain structure whichmakes the projections
Mα ×M Mβ → Mα and Mα ×M Mβ → Mβ
morphisms.
• The category of Banach supermanifolds will be denoted by SMan.
A Banach–Lie supergroup is a group object in SMan.
Example
Loop supergroups Map(S1, G) have realizations as Hilbert–Lie supergroups.
60 / 80
Banach supermanifold and Banach–Lie supergroups
A Banach supermanifold is a functor M : Gr → Man with an open covering
M =⋃
α∈I
Mα
where
each Mα is a superdomain.
the fiber product Mα ×M Mβ has a superdomain structure whichmakes the projections
Mα ×M Mβ → Mα and Mα ×M Mβ → Mβ
morphisms.
• The category of Banach supermanifolds will be denoted by SMan.
A Banach–Lie supergroup is a group object in SMan.
Example
Loop supergroups Map(S1, G) have realizations as Hilbert–Lie supergroups.
61 / 80
Banach supermanifold and Banach–Lie supergroups
A Banach supermanifold is a functor M : Gr → Man with an open covering
M =⋃
α∈I
Mα
where
each Mα is a superdomain.
the fiber product Mα ×M Mβ has a superdomain structure whichmakes the projections
Mα ×M Mβ → Mα and Mα ×M Mβ → Mβ
morphisms.
• The category of Banach supermanifolds will be denoted by SMan.
A Banach–Lie supergroup is a group object in SMan.
Example
Loop supergroups Map(S1, G) have realizations as Hilbert–Lie supergroups.
62 / 80
Banach supermanifold and Banach–Lie supergroups
A Banach supermanifold is a functor M : Gr → Man with an open covering
M =⋃
α∈I
Mα
where
each Mα is a superdomain.
the fiber product Mα ×M Mβ has a superdomain structure whichmakes the projections
Mα ×M Mβ → Mα and Mα ×M Mβ → Mβ
morphisms.
• The category of Banach supermanifolds will be denoted by SMan.
A Banach–Lie supergroup is a group object in SMan.
Example
Loop supergroups Map(S1, G) have realizations as Hilbert–Lie supergroups.
63 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
64 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
65 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
66 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
67 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
68 / 80
Unitary representations
Passage from a Banach-Lie supergroup G to a Harish–Chandra pair(G, g) is possible.
The categories Rep∞(G, g) and Repω(G, g) are well-defined, but notidentical.
Question 1. Is there an injection Repω(G, g) → Rep∞(G, g)?
(π, ρπ,H ) unitary rep. of (G, g).
x ∈ g ⇒ ρπ(x) : Hω→ H
ω ρπ(x) : H
∞→ H
∞
Question 2. Given a homomorphism (H, h) → (G, g), are thererestriction functors
Res∞ : Rep∞(G, g) → Rep∞(H, h) and Resω : Repω(G, g) → Repω(H, h)?
Answer. (Merigon, Neeb, S. ’11) Yes!
69 / 80
The GNS construction
G: Banach–Lie supergroup with corresponding Harish-Chandra pair (G, g).
Basic idea:
smooth positive definite functions on G ! cyclic objects of Rep∞(G, g)
C∞(G) := Nat(G,C) where C(Λn) = (C1|1 ⊗ Λn)0
Proposition
C∞(G) is isomorphic to the algebra of all
F ∈ Homg0(U(g), C∞(G,C))
for which the maps
g× · · · × g︸ ︷︷ ︸
n times
×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)
are smooth (for all n ≥ 0).
70 / 80
The GNS construction
G: Banach–Lie supergroup with corresponding Harish-Chandra pair (G, g).
Basic idea:
smooth positive definite functions on G ! cyclic objects of Rep∞(G, g)
C∞(G) := Nat(G,C) where C(Λn) = (C1|1 ⊗ Λn)0
Proposition
C∞(G) is isomorphic to the algebra of all
F ∈ Homg0(U(g), C∞(G,C))
for which the maps
g× · · · × g︸ ︷︷ ︸
n times
×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)
are smooth (for all n ≥ 0).
71 / 80
The GNS construction
G: Banach–Lie supergroup with corresponding Harish-Chandra pair (G, g).
Basic idea:
smooth positive definite functions on G ! cyclic objects of Rep∞(G, g)
C∞(G) := Nat(G,C) where C(Λn) = (C1|1 ⊗ Λn)0
Proposition
C∞(G) is isomorphic to the algebra of all
F ∈ Homg0(U(g), C∞(G,C))
for which the maps
g× · · · × g︸ ︷︷ ︸
n times
×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)
are smooth (for all n ≥ 0).
72 / 80
The GNS construction
G: Banach–Lie supergroup with corresponding Harish-Chandra pair (G, g).
Basic idea:
smooth positive definite functions on G ! cyclic objects of Rep∞(G, g)
C∞(G) := Nat(G,C) where C(Λn) = (C1|1 ⊗ Λn)0
Proposition
C∞(G) is isomorphic to the algebra of all
F ∈ Homg0(U(g), C∞(G,C))
for which the maps
g× · · · × g︸ ︷︷ ︸
n times
×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)
are smooth (for all n ≥ 0).
73 / 80
The involutive semigroup S
An antilinear antiautomorphism
(G, g) σ : gC → gC
σ(x) =
{
−x if x ∈ g0√−1x if x ∈ g1
σ : U(g) → U(g) , σ([x, y]) = [σ(y), σ(x)]
S = G⋉U(gC) (g1, D1)(g2, D2) = (g1g2, (g−12 ·D1)D2)
s 7→ s∗
, (g,D)∗ = (g−1, g · σ(D))
f ∈ C∞(G) ≃ Homg
0(U(g),C∞(G,C)) f : S → C
f(g,D) = f(D)(g)
A function f ∈ C∞(G) is called positive definite iff
K : S × S → C , K(s, t) = f(s∗t)
is positive definite, i.e., i.e., for every n,
λ1, . . . , λn ∈ C ⇒ ∑
1≤i,j≤nλiλjK(si, sj) ≥ 0
74 / 80
The involutive semigroup S
An antilinear antiautomorphism
(G, g) σ : gC → gC
σ(x) =
{
−x if x ∈ g0√−1x if x ∈ g1
σ : U(g) → U(g) , σ([x, y]) = [σ(y), σ(x)]
S = G⋉U(gC) (g1, D1)(g2, D2) = (g1g2, (g−12 ·D1)D2)
s 7→ s∗
, (g,D)∗ = (g−1, g · σ(D))
f ∈ C∞(G) ≃ Homg
0(U(g),C∞(G,C)) f : S → C
f(g,D) = f(D)(g)
A function f ∈ C∞(G) is called positive definite iff
K : S × S → C , K(s, t) = f(s∗t)
is positive definite, i.e., i.e., for every n,
λ1, . . . , λn ∈ C ⇒ ∑
1≤i,j≤nλiλjK(si, sj) ≥ 0
75 / 80
The involutive semigroup S
An antilinear antiautomorphism
(G, g) σ : gC → gC
σ(x) =
{
−x if x ∈ g0√−1x if x ∈ g1
σ : U(g) → U(g) , σ([x, y]) = [σ(y), σ(x)]
S = G⋉U(gC) (g1, D1)(g2, D2) = (g1g2, (g−12 ·D1)D2)
s 7→ s∗
, (g,D)∗ = (g−1, g · σ(D))
f ∈ C∞(G) ≃ Homg
0(U(g),C∞(G,C)) f : S → C
f(g,D) = f(D)(g)
A function f ∈ C∞(G) is called positive definite iff
K : S × S → C , K(s, t) = f(s∗t)
is positive definite, i.e., i.e., for every n,
λ1, . . . , λn ∈ C ⇒ ∑
1≤i,j≤nλiλjK(si, sj) ≥ 0
76 / 80
The involutive semigroup S
An antilinear antiautomorphism
(G, g) σ : gC → gC
σ(x) =
{
−x if x ∈ g0√−1x if x ∈ g1
σ : U(g) → U(g) , σ([x, y]) = [σ(y), σ(x)]
S = G⋉U(gC) (g1, D1)(g2, D2) = (g1g2, (g−12 ·D1)D2)
s 7→ s∗
, (g,D)∗ = (g−1, g · σ(D))
f ∈ C∞(G) ≃ Homg
0(U(g),C∞(G,C)) f : S → C
f(g,D) = f(D)(g)
A function f ∈ C∞(G) is called positive definite iff
K : S × S → C , K(s, t) = f(s∗t)
is positive definite, i.e., i.e., for every n,
λ1, . . . , λn ∈ C ⇒ ∑
1≤i,j≤nλiλjK(si, sj) ≥ 0
77 / 80
The GNS Construction
Observation
(π, ρπ,H ) smooth fv(D)(g) = 〈π(g)ρπ(D)v, v〉
unitary rep of (G, g) even positive definite
Theorem (Neeb, S. ’12)
For every even positive definite f ∈ C∞(G) there exists a unique (upto unitary equivalence) smooth unitary representation of (G, g) withmatrix coefficient f .
Further directions
Global representation theory of loop and superconformal currentgroups.
Unitary representations of direct limits.
Extension to Frechet–Lie groups (super Virasoro groups).78 / 80
The GNS Construction
Observation
(π, ρπ,H ) smooth fv(D)(g) = 〈π(g)ρπ(D)v, v〉
unitary rep of (G, g) even positive definite
Theorem (Neeb, S. ’12)
For every even positive definite f ∈ C∞(G) there exists a unique (upto unitary equivalence) smooth unitary representation of (G, g) withmatrix coefficient f .
Further directions
Global representation theory of loop and superconformal currentgroups.
Unitary representations of direct limits.
Extension to Frechet–Lie groups (super Virasoro groups).79 / 80
The GNS Construction
Observation
(π, ρπ,H ) smooth fv(D)(g) = 〈π(g)ρπ(D)v, v〉
unitary rep of (G, g) even positive definite
Theorem (Neeb, S. ’12)
For every even positive definite f ∈ C∞(G) there exists a unique (upto unitary equivalence) smooth unitary representation of (G, g) withmatrix coefficient f .
Further directions
Global representation theory of loop and superconformal currentgroups.
Unitary representations of direct limits.
Extension to Frechet–Lie groups (super Virasoro groups).80 / 80