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

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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

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Supergeometry





Cω(Rm) f(x) =∑

k1,...,km≥0

ck1,··· ,kmxk1

1 · · ·xkm

m


k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}

ck1,...,km,l1,...,lnxk1

1 · · ·xkm

m · · · ξl11 · · · ξlnn

3 / 80

Supergeometry





Cω(Rm) f(x) =∑

k1,...,km≥0

ck1,··· ,kmxk1

1 · · ·xkm

m


k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}

ck1,...,km,l1,...,lnxk1

1 · · ·xkm

m · · · ξl11 · · · ξlnn

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Supergeometry





Cω(Rm) f(x) =∑

k1,...,km≥0

ck1,··· ,kmxk1

1 · · ·xkm

m


k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}

ck1,...,km,l1,...,lnxk1

1 · · ·xkm

m · · · ξl11 · · · ξlnn

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Supergeometry





Cω(Rm) f(x) =∑

k1,...,km≥0

ck1,··· ,kmxk1

1 · · ·xkm

m


k1, . . . , km ≥ 0l1, . . . , ln ∈ {0, 1}

ck1,...,km,l1,...,lnxk1

1 · · ·xkm

m · · · ξl11 · · · ξlnn

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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.

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Supergeometry






M = (M,OM)



(Uα,OM

∣

∣

Uα


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Supergeometry






M = (M,OM)



(Uα,OM

∣

∣

Uα


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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

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Lie supergroups

Lie Supergroups


µ : G× G → G ε : {∗} → G ι : G → G


µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε


[X, Y ] = −(−1)p(X)p(Y )[Y,X]


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Lie supergroups

Lie Supergroups


µ : G× G → G ε : {∗} → G ι : G → G


µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε


[X, Y ] = −(−1)p(X)p(Y )[Y,X]


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Lie supergroups

Lie Supergroups


µ : G× G → G ε : {∗} → G ι : G → G


µ ◦ (idG × ι) ◦∆G = µ ◦ (ι× idG) ◦∆G = ε


[X, Y ] = −(−1)p(X)p(Y )[Y,X]


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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 .

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Example


V = V0 ⊕ V1 = Cm ⊕ C

n


End(V )i ={


}

.




sl(m|n) ={

A ∈ gl(m|n) | str(A) = 0}

A=

m{ n{[L M

N P

] }m


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Example


V = V0 ⊕ V1 = Cm ⊕ C

n


End(V )i ={


}

.




sl(m|n) ={

A ∈ gl(m|n) | str(A) = 0}

A=

m{ n{[L M

N P

] }m


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Example


V = V0 ⊕ V1 = Cm ⊕ C

n


End(V )i ={


}

.




sl(m|n) ={

A ∈ gl(m|n) | str(A) = 0}

A=

m{ n{[L M

N P

] }m


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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)

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Example


osp(m|2n) ={




Further examples


[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′

• A = k[t±11 , . . . , t±1



−11 , θ] (θ2 = 0) (Kac–Todorov ’85)

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Example


osp(m|2n) ={




Further examples


[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′

• A = k[t±11 , . . . , t±1



−11 , θ] (θ2 = 0) (Kac–Todorov ’85)

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Example


osp(m|2n) ={




Further examples


[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′

• A = k[t±11 , . . . , t±1



−11 , θ] (θ2 = 0) (Kac–Todorov ’85)

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Example


osp(m|2n) ={




Further examples


[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′

• A = k[t±11 , . . . , t±1



−11 , θ] (θ2 = 0) (Kac–Todorov ’85)

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Example


osp(m|2n) ={




Further examples


[x⊗ a, x′ ⊗ a′] = [x, x′]⊗ (−1)p(a)p(x′)aa′

• A = k[t±11 , . . . , t±1



−11 , θ] (θ2 = 0) (Kac–Todorov ’85)

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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



G = (G,OG) (G, g)g = g0 ⊕ g1.

Lie(G) = g0.








ρ : g → End(V )




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.

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“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.

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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 .

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G : Lie group.



1

t

(π(etx)v − v



Example

DefineH

∞ ={v ∈ H : g 7→ π(g)v is C

∞}

andH

ω ={v ∈ H : g 7→ π(g)v is Cω

}.

The actions


are well-defined.



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G : Lie group.



1

t

(π(etx)v − v



Example

DefineH

∞ ={v ∈ H : g 7→ π(g)v is C

∞}

andH

ω ={v ∈ H : g 7→ π(g)v is Cω

}.

The actions


are well-defined.



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G : Lie group.



1

t

(π(etx)v − v



Example

DefineH

∞ ={v ∈ H : g 7→ π(g)v is C

∞}

andH

ω ={v ∈ H : g 7→ π(g)v is Cω

}.

The actions


are well-defined.



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G : Lie group.



1

t

(π(etx)v − v



Example

DefineH

∞ ={v ∈ H : g 7→ π(g)v is C

∞}

andH

ω ={v ∈ H : g 7→ π(g)v is Cω

}.

The actions


are well-defined.



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G : Lie group.



1

t

(π(etx)v − v



Example

DefineH

∞ ={v ∈ H : g 7→ π(g)v is C

∞}

andH

ω ={v ∈ H : g 7→ π(g)v is Cω

}.

The actions


are well-defined.



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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)

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An analytic unitary representation of a Harish–Chandra pair (G, g) isa triple (π, ρπ ,H ) where



(iii) ρπ : g → EndC(Hω) is a unitary representation of g.


∣

H ωfor every x ∈ g0.

Notation. Repω(G, g) denotes the category of analytic unitary rep.of (G, g).



ω(G, g)

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An analytic unitary representation of a Harish–Chandra pair (G, g) isa triple (π, ρπ ,H ) where



(iii) ρπ : g → EndC(Hω) is a unitary representation of g.


∣

H ωfor every x ∈ g0.

Notation. Repω(G, g) denotes the category of analytic unitary rep.of (G, g).



ω(G, g)

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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






Bλ : g1 × g1 → R , Bλ(X,Y ) = λ([X,Y ])

SetC (G) = {λ ∈ g∗0 | Bλ(x, x) ≥ 0 }


Theorem (S. ’10)




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,...

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(k) H(p, q) where p + q > 4.



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(k) H(p, q) where p + q > 4.



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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


Motivation




Rm|n

R∞|∞



Λn = (Λn)0 ⊕ (Λn)1





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.

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Category Gr


DeWitt topology


Superdomains


E : Gr → Man , E(Λn) = (E ⊗ Λn)0.




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Category Gr


DeWitt topology


Superdomains


E : Gr → Man , E(Λn) = (E ⊗ Λn)0.




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Category Gr


DeWitt topology


Superdomains


E : Gr → Man , E(Λn) = (E ⊗ Λn)0.




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Category Gr


DeWitt topology


Superdomains


E : Gr → Man , E(Λn) = (E ⊗ Λn)0.




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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.

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M =⋃

α∈I

Mα

where




morphisms.



Example


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M =⋃

α∈I

Mα

where




morphisms.



Example


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M =⋃

α∈I

Mα

where




morphisms.



Example


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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!

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x ∈ g ⇒ ρπ(x) : Hω→ H

ω ρπ(x) : H

∞→ H

∞




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x ∈ g ⇒ ρπ(x) : Hω→ H

ω ρπ(x) : H

∞→ H

∞




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x ∈ g ⇒ ρπ(x) : Hω→ H

ω ρπ(x) : H

∞→ H

∞




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x ∈ g ⇒ ρπ(x) : Hω→ H

ω ρπ(x) : H

∞→ H

∞




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x ∈ g ⇒ ρπ(x) : Hω→ H

ω ρπ(x) : H

∞→ H

∞




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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).

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Basic idea:



Proposition



for which the maps

g× · · · × g︸︷︷︸

n times

×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)


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Basic idea:



Proposition



for which the maps

g× · · · × g︸︷︷︸

n times

×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)


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Basic idea:



Proposition



for which the maps

g× · · · × g︸︷︷︸

n times

×G → C , (x1, . . . , xn, g) → F (x1 · · · xn)(g)


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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

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(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))


0(U(g),C∞(G,C)) f : S → C

f(g,D) = f(D)(g)


K : S × S → C , K(s, t) = f(s∗t)


λ1, . . . , λn ∈ C ⇒ ∑


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(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))


0(U(g),C∞(G,C)) f : S → C

f(g,D) = f(D)(g)


K : S × S → C , K(s, t) = f(s∗t)


λ1, . . . , λn ∈ C ⇒ ∑


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(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))


0(U(g),C∞(G,C)) f : S → C

f(g,D) = f(D)(g)


K : S × S → C , K(s, t) = f(s∗t)


λ1, . . . , λn ∈ C ⇒ ∑


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


Observation





Further directions




a gns construction for unitary representations of lie ... · a gns construction for unitary...

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