Unitary Representations of Nilpotent Super Lie groups · Basic Deﬁnitions and Notation Let Gbe a...

Unitary Representations of Nilpotent SuperLie groups

Hadi Salmasian

February 6, 2010

Basic Definitions and Notation

Let G be a Lie group andH be a Hilbert space. A unitaryrepresentation π of G inH is a map

π : G→ U(H )

where U(H ) is the group of linear isometries ofH , suchthat :

π(g1g2) = π(g1)π(g2)π is strongly continuous, i.e., the map g 7→ π(g)v iscontinuous for every v ∈ H

Example : the Schrodinger model

Suppose (W,Ω) is a finite dimensional symplectic vectorspace, i.e.,

Ω is nondegenerate,Ω(v,w) = −Ω(w, v).




Set G = Hn where

Hn = (v, s) | v ∈W and s ∈ R

and the group law is defined by

(v1, s1) • (v2, s2) = (v1 + v2, s1 + s2 +1

2Ω(v1, v2)).




Set G = Hn where

Hn = (v, s) | v ∈W and s ∈ R

and the group law is defined by

(v1, s1) • (v2, s2) = (v1 + v2, s1 + s2 +1

2Ω(v1, v2)).

We know that dimZ(Hn) = 1 and Hn/Z(Hn) iscommutative (i.e., Hn is two-step nilpotent).

Example : the Schrodinger model (cont.)

Hn = (v, s) | v ∈W and s ∈ R


Hn = (v, s) | v ∈W and s ∈ R

Consider a polarization of (W,Ω), i.e., a direct sumdecomposition

W = X ⊕ Y such thatΩ(X,X) = Ω(Y,Y) = 0.


Hn = (v, s) | v ∈W and s ∈ R



SetH := L2(Y) := f : Y→ C |∫

Y| f |2dµ < ∞ .


Hn = (v, s) | v ∈W and s ∈ R



SetH := L2(Y) := f : Y→ C |∫

Y| f |2dµ < ∞ .

Fix a nonzero a ∈ R and define a representation πa of Hn onH via

(

πa(v, 0) f)

(y) = eaΩ(y,v)√−1f (y) if v ∈ X,

(

πa(0, v) f)

(y) = f (y + v) if v ∈ Y,(

πa(0, s) f)

(y) = eat√−1f (y) otherwise.


(

πa(v, 0) f)

(y) = eaΩ(y,v)√−1f (y) if v ∈ X,

(

πa(0, v) f)

(y) = f (y + v) if v ∈ Y,(

πa(0, s) f)



(

πa(v, 0) f)

(y) = eaΩ(y,v)√−1f (y) if v ∈ X,

(

πa(0, v) f)

(y) = f (y + v) if v ∈ Y,(

πa(0, s) f)


Facts:

For every a ∈ R, πa is an irreducible unitary representationof Hn. (i.e.,H does not have nontrivial Hn-invariant closed subspaces.)


(

πa(v, 0) f)

(y) = eaΩ(y,v)√−1f (y) if v ∈ X,

(

πa(0, v) f)

(y) = f (y + v) if v ∈ Y,(

πa(0, s) f)


Facts:


If a , b, the representations πa and πb are not (unitarily)equivalent.


(

πa(v, 0) f)

(y) = eaΩ(y,v)√−1f (y) if v ∈ X,

(

πa(0, v) f)

(y) = f (y + v) if v ∈ Y,(

πa(0, s) f)


Facts:


If a , b, the representations πa and πb are not (unitarily)equivalent.

(Stone-von Neumann, 1930’s)

Up to unitary equivalence, the irreducible unitaryrepresentations of Hn are :

1 one-dimensional representations (which factor throughHn/Z(Hn)),

2 The representations πa, a ∈ R×.

A bit of history...

Gelfand (1940’s) :

Unitary representationsof G

←−−−−−−−→ Quantization ofG − spaces

A bit of history...




Kirillov (1950’s) If G is a nilpotent simply connected Lie group,then there exists a bijective correspondence

Irreducible unitaryrepresentations of G

!

G − orbitsin g∗

A bit of history...




Kirillov (1950’s) If G is a nilpotent simply connected Lie group,then there exists a bijective correspondence

Irreducible unitaryrepresentations of G

!

G − orbitsin g∗

There is also a dictionary :

Algebraic operation Geometric operation

ResGHπ p(O) where p : g∗ → h∗IndG

Hπ p−1(O) where p : g∗ → h∗π1 ⊗ π2 O1 +O2

... ...

Note that the algebraic operations should be understood in the context of direct

integrals, i.e. : ResGHπ =∫

Hn(σ)σdµ(σ), etc.

Kirillov’s orbit method

Suppose G is nilpotent and simply connected. Set g = Lie(G).



Recipe to construct π from O :




1 Fix λ ∈ O. Consider the skew-symmetric form

Ωλ : g × g→ R

defined byΩλ(X,Y) = λ([X,Y]).





Ωλ : g × g→ R


2 Proposition. There exists a subalgebra m ⊂ g such that m isa maximal isotropic subspace of Ωλ.





Ωλ : g × g→ R



3 SetM = exp(m) and define χλ : M→ C× by

χλ(exp(X)) = eλ(X)√−1 for every X ∈ m.





Ωλ : g × g→ R



3 SetM = exp(m) and define χλ : M→ C× by

χλ(exp(X)) = eλ(X)√−1 for every X ∈ m.

4 Set π = IndGMχλ.

Example : Schrodinger model revisited

Recall that :Hn = (v, s) | v ∈W and s ∈ R

Set hn = Lie(Hn) and fix Z ∈ Z(hn).

Example : Schrodinger model revisited

Recall that :Hn = (v, s) | v ∈W and s ∈ R

Set hn = Lie(Hn) and fix Z ∈ Z(hn).

Hn-orbits in h∗n are :

λ where λ(Z) = 0 !one-dimensional

representations of Hn.

λ ∈ h∗n | λ(Z) = a ! the representation πa.

Lie superalgebras : introduction

g = g0 ⊕ g1 and [·, ·] : g × g→ g where

(−1)|x|·|z|[X, [Y,Z]] + (−1)|y|·|x|[Y, [Z,X]] + (−1)|z|·|y|[Z, [X,Y]] = 0


g = g0 ⊕ g1 and [·, ·] : g × g→ g where


Examples

gl(m|n) : V = V0 ⊕ V1 and g = End(V) = End0(V) ⊕ End1(V)with

[X,Y] = XY − (−1)|x|·|y|YX


g = g0 ⊕ g1 and [·, ·] : g × g→ g where


Examples


[X,Y] = XY − (−1)|x|·|y|YXsl(m|n), osp(m|2n), p(n), q(n),...


g = g0 ⊕ g1 and [·, ·] : g × g→ g where


Examples


[X,Y] = XY − (−1)|x|·|y|YXsl(m|n), osp(m|2n), p(n), q(n),...Heisenberg-Clifford Lie superalgebra.

Heisenberg-Clifford Lie superalgebra

Let (W,Ω) be a supersymplectic space, i.e.,

W =W0 ⊕W1.

Ω : W ×W → R satisfies

Ω(W0,W1) = Ω(W1,W0) = 0Ω|W1×W1

is a nondegenerate symmetric form.Ω|W0×W0

is a symplectic form.



W =W0 ⊕W1.


Ω(W0,W1) = Ω(W1,W0) = 0Ω|W1×W1



Set hW =W ⊕Rwhere

[(v1, s1), (v2, s2)] = (0,Ω(v1, v2))



W =W0 ⊕W1.


Ω(W0,W1) = Ω(W1,W0) = 0Ω|W1×W1



Set hW =W ⊕Rwhere

[(v1, s1), (v2, s2)] = (0,Ω(v1, v2))

hW is two-step nilpotent and dim(

Z(hW))

= 1.

Towards unitary representations : super Lie groups

• A super Lie group is a group object in the category ofsupermanifolds.



Proposition

The category of Super Lie groups is equivalent to a category ofHarish-Chandra pairs, i.e., pairs (G0, g) such that :

1 g = g0 ⊕ g1 is a Lie superalgebra over R.

2 G0 is a connected real Lie group with Lie algebra g0 wichacts on g smoothly via R-linear automorphisms.

3 The action of G0 on g0 is the adjoint action. The adjointaction of g0 on g is the differential of the action of G0 on g.



Proposition

The category of Super Lie groups is equivalent to a category ofHarish-Chandra pairs, i.e., pairs (G0, g) such that :

1 g = g0 ⊕ g1 is a Lie superalgebra over R.

2 G0 is a connected real Lie group with Lie algebra g0 wichacts on g smoothly via R-linear automorphisms.

3 The action of G0 on g0 is the adjoint action. The adjointaction of g0 on g is the differential of the action of G0 on g.

• For simplicity, from now on we assume that G0 is alwayssimply connected.

Unitary representaions of super Lie groups

• A unitary representation of (G0, g) is a triple (π, ρπ,H ) with thefollowing properties :



H =H0 ⊕H1 is a super Hilbert space. (Define it yourself!)




π : G0 → U(H ) is a unitary representation of G0 (in the usualsense).





ρπ : g→ End(H∞) is a super skew-Hermitian representationwhich satisfies

ρπ([X,Y]) = ρπ(X)ρπ(Y) − (−1)|X|·|Y|ρπ(Y)ρπ(X).







•HereH∞ is the space of smooth vectors of (π,H ).








Reason: Domain issue. If X ∈ g1, thenρπ([X,X]) = ρπ(X)ρπ(X) + ρπ(X)ρπ(X) = 2ρπ(X)2, but ρπ([X,X]) is an

unbounded, densely defined operator.








Reason: Domain issue. If X ∈ g1, thenρπ([X,X]) = ρπ(X)ρπ(X) + ρπ(X)ρπ(X) = 2ρπ(X)2, but ρπ([X,X]) is an

unbounded, densely defined operator.

ρπ|g0 = π∞ and ρπ(Ad(g)(X)) = π(g)ρπ(X)π(g−1).

Unitary equivalence and parity

Unitary equivalence

Two irreducible unitary representations (π, ρπ,H ) and(π′, ρπ

′,H ′) are said to be unitarily equivalent if there exists a

linear isometry T : H →H ′ such that :

T preserves the Z/2Z-grading.


Unitary equivalence





T(H∞) ⊂ H ′∞


Unitary equivalence





T(H∞) ⊂ H ′∞

π′(g) T = T π(g) and ρπ′(X) T = T ρπ(X).


Unitary equivalence





T(H∞) ⊂ H ′∞


Parity

By tensoring (π, ρπ,H ) with the trivial representation on C0|1

we obtain (π, ρπ,ΠH ).ΠH0 = H1 and ΠH1 =H0 .


Unitary equivalence





T(H∞) ⊂ H ′∞


Parity

By tensoring (π, ρπ,H ) with the trivial representation on C0|1

we obtain (π, ρπ,ΠH ).ΠH0 = H1 and ΠH1 =H0 .

• (π, ρπ,H ) and (π, ρπ,ΠH ) are not necessarily unitarily equivalent.

Remark on Harish-Chandra’s method

The general method to study unitary representations of areductive Lie group is to look at the (g,K)-module obtainedby K-finite analytic vectors, where K ⊂ G is the maximalcompact subgroup. (Harish-Chandra modules)



This approach has been extended to the super case when g0is reductive (e.g., sl(m|n), osp(m|2n), ...) by H. Furtsu, T.Hirai, K. Nishiyama, S. J. Cheng, R. B. Zhang, ...



This approach has been extended to the super case when g0is reductive (e.g., sl(m|n), osp(m|2n), ...) by H. Furtsu, T.Hirai, K. Nishiyama, S. J. Cheng, R. B. Zhang, ...

Nevertheless, it is not applicable to the cases where g0 isnot reductive (e.g., the nilpotent or solvable case).

Nilpotent super Lie groups

A super Lie group (G0, g) is called nilpotent if the lower central series ofg has finitely many nonzero terms (equivalently, if g appears in its ownupper central series).



Unlike Lie groups, certain super Lie groups do not have any faithfulunitary representairons!




Lemma

If X1, ...Xm ∈ g1 such thatm∑

i=1

[Xi,Xi] = 0

then for every unitary representation (π, ρπ,H) we haveρπ(X1) = · · · = ρπ(Xm) = 0.




Lemma

If X1, ...Xm ∈ g1 such thatm∑

i=1

[Xi,Xi] = 0

then for every unitary representation (π, ρπ,H) we haveρπ(X1) = · · · = ρπ(Xm) = 0.

Proof. Observe that∑m

i=1 ρπ(Xi)

2 = 0 and for every i, the operator eπ4

√−1ρπ(Xi)

is symmetric. For every v ∈ H∞ we have :m∑

i=1

〈e π4√−1ρπ(Xi)v, e

π4

√−1ρπ(Xi)v〉 = 〈v, e

π2

√−1

m∑

i=1

ρπ(Xi)2v〉 = 0.

Reduced form

Set a(1) = 〈X ∈ g1 | [X,X] = 0〉. Then a(1) lies in the kernel ofevery unitary representation of (G0, g). We say that g is

reduced if a(1) = 0.

Reduced form


reduced if a(1) = 0.Set

a(2) = 〈X ∈ g1 | [X,X] ∈ a(1)〉a(3) = 〈X ∈ g1 | [X,X] ∈ a(2)〉

...

Reduced form



a(2) = 〈X ∈ g1 | [X,X] ∈ a(1)〉a(3) = 〈X ∈ g1 | [X,X] ∈ a(2)〉

...

We have

a(1) ⊂ a(2) ⊂ a(3) ⊂ · · ·

Reduced form



a(2) = 〈X ∈ g1 | [X,X] ∈ a(1)〉a(3) = 〈X ∈ g1 | [X,X] ∈ a(2)〉

...

We have

a(1) ⊂ a(2) ⊂ a(3) ⊂ · · ·Set a =

⋃

j≥1a(j). One can see that ρπ(a) = 0 for every unitary

representation (π, ρπ,H ).

Reduced form



a(2) = 〈X ∈ g1 | [X,X] ∈ a(1)〉a(3) = 〈X ∈ g1 | [X,X] ∈ a(2)〉

...

We have

a(1) ⊂ a(2) ⊂ a(3) ⊂ · · ·Set a =

⋃

j≥1a(j). One can see that ρπ(a) = 0 for every unitary

representation (π, ρπ,H ).a is graded and hence it corresponds to a sub-supergroup(A0, a) of (G0, g). The quotient g/a is reduced.

Kirillov’s Lemma for super Lie groups

Lemma

Let (G0, g) be a nilpotent super Lie group such that g is reduced anddimZ(g) = 1. Then exactly one of the following statements is true :


Lemma


There exists a graded decomposition

g = RX ⊕RY ⊕RX ⊕w

such that SpanX,Y,Z is a three-dimensional Heisenbergalgebra, Z ∈ Z(g),

g′ := RY ⊕RZ ⊕wis a subalgebra, and Y ∈ Z(g′).


Lemma






g is isomorphic to a Heisenberg-Clifford superalgebra.


Lemma






g is isomorphic to a Heisenberg-Clifford superalgebra.

• A Heisenberg-Clifford superalgebra hW is reduced if and onlyif Ω|W0

is a definite form.

Unitary representations as induced representations

Let (G0, g) be a nilpotent super Lie group such that

g is reduced,

dimZ(g) = 1,

g is not a Heisenberg-Clifford superalgebra.



g is reduced,

dimZ(g) = 1,


Let g′ be as in Kirillov’s lemma, and let (G′0, g′) be the corresponding

sub-supergroup of (G0, g).

One can see that dim g′1= dim g1.

It follows that unitary induction from (G′0, g′) to G0, g) is defined.



g is reduced,

dimZ(g) = 1,


Let g′ be as in Kirillov’s lemma, and let (G′0, g′) be the corresponding

sub-supergroup of (G0, g).

One can see that dim g′1= dim g1.

It follows that unitary induction from (G′0, g′) to G0, g) is defined.

Theorem (codimension one induction)

Let (π, ρπ,H ) be an irreducible unitary representation of (G0, g)whose restriction toZ(G0) is nontrivial. Then there exists anirreducible unitary representation (π′, ρπ

′,H ′) of (G′0, g′) such that

(π, ρπ,H ) = Ind(G0 ,g)

(G′0,g′)

(π′, ρπ′,H ′)

Unitary representations of hW

• Recall that hW =W ⊕R where

[(v1, s1), (v2, s2)] = (0,Ω(v,w))

Set g = hW and let (G0, g) be the corresponding super Lie group.



[(v1, s1), (v2, s2)] = (0,Ω(v,w))


Theorem (generalized Stone-von Neumann)

Let χ : R→ C× be defined by χ(t) = eat√−1 where a > 0. (The case a < 0 is

similar.)

IfΩ|W0×W0is positive definite, then up to unitary equivalence and

parity there exists a unique unitary representation with centralcharacter χ.

IfΩ|W0×W0is not positive definite, then (G0, g) does not have any

unitary representations with central character χ.



[(v1, s1), (v2, s2)] = (0,Ω(v,w))


Theorem (generalized Stone-von Neumann)

Let χ : R→ C× be defined by χ(t) = eat√−1 where a > 0. (The case a < 0 is

similar.)

IfΩ|W0×W0is positive definite, then up to unitary equivalence and

parity there exists a unique unitary representation with centralcharacter χ.

IfΩ|W0×W0is not positive definite, then (G0, g) does not have any

unitary representations with central character χ.

•When dim g1 is even, parity change yields two non-unitary equivalentrepresentations, whereas when dim g1 is odd, the two representations that areobtained by parity change are isomorphic. (Similar to Clifford modules.)

The general case

Let (G0, g) be a nilpotent super Lie group.

The general case


For every λ ∈ g∗0one can define a symmetric bilinear form

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

The general case




Setg⋆0 = λ ∈ g∗0 | Bλ is nonnegative definite

The general case





Observe that g⋆0is G0-invariant.

The general case





Observe that g⋆0is G0-invariant.

THEOREM (S.’09)

There exists a bijective correspondence

Irreducible unitaryrepresentations of (G0, g)

!

G0 − orbitsin g⋆

0

Polarizing systems


Polarizing systems


A polarizing system of (G0, g) is a 6-tuple

(M0,m,Φ,C0, c, λ)

such that :

Polarizing systems



(M0,m,Φ,C0, c, λ)

such that :

dimm1 = dim g1.

Polarizing systems



(M0,m,Φ,C0, c, λ)

such that :

dimm1 = dim g1.

λ ∈ g∗0and m0 is a maximally isotropic subalgebra of g0

with respect to the skew symmetric formΩλ(X,Y) = λ([X,Y]).

Polarizing systems



(M0,m,Φ,C0, c, λ)

such that :

dimm1 = dim g1.



(C0, c) is a Heisenberg-Clifford super Lie group such thatdimC0 = 1.

Polarizing systems



(M0,m,Φ,C0, c, λ)

such that :

dimm1 = dim g1.




Φ : (M0,m)→ (C0, c) is an epimorphism.

Polarizing systems



(M0,m,Φ,C0, c, λ)

such that :

dimm1 = dim g1.




Φ : (M0,m)→ (C0, c) is an epimorphism.

m0 ∩ kerΦ = m0 ∩ kerλ.

Proposition

Every irreducible representation (π, ρπ,H ) of (G0, g) isinduced from a polarizing system (M0,m,Φ,C0, c, λ) i.e.,

(π, ρπ,H ) = Ind(G0 ,g)(M0,m)

(σ Φ, ρσΦ,K)

where λ(W) = ρσ Φ(W).

(M0,m)Φ−−−→ (C0, c)d (σ, ρσ,K )

Proposition

Every irreducible representation (π, ρπ,H ) of (G0, g) isinduced from a polarizing system (M0,m,Φ,C0, c, λ) i.e.,

(π, ρπ,H ) = Ind(G0 ,g)(M0,m)

(σ Φ, ρσΦ,K)

where λ(W) = ρσ Φ(W).

(M0,m)Φ−−−→ (C0, c)d (σ, ρσ,K )

Moreover, if (π, ρπ,H ) is induced from two differentpolarizing systems

(M0,m,Φ,C0, c, λ) and (M′0,m′,Φ,C′0, c

′, λ′)

then1 (C0, c) ≃ (C′

0, c′)

2 λ′ = Ad∗(g)(λ) for some g ∈ G0.

Nonnegativity condition

If (π, ρπ,H ) = Ind(G0,g)(M0,m)

(σ Φ, ρσΦ,K) then from

λ(W) = ρσ Φ(W) and properties of Clifford modules wehave :

for every X ∈ g1,Bλ(X,X) = λ([X,X]) = ρσ Φ([X,X])

= [ρσ Φ(X), ρσ Φ(X)] ≥ 0






= [ρσ Φ(X), ρσ Φ(X)] ≥ 0

which implies that λ ∈ g⋆0.






= [ρσ Φ(X), ρσ Φ(X)] ≥ 0

which implies that λ ∈ g⋆0.

Conversely, we should show that every λ ∈ g⋆0fits into a

polarizing system (M0,m,C), c,Φ, λ).

Proposition

For every λ ∈ g⋆0there exists a polarizing system

(M0,m,Φ,C0, c, λ).

Proposition

For every λ ∈ g⋆0there exists a polarizing system

(M0,m,Φ,C0, c, λ).

The proof is based on the following lemma :

Lemma

There exists a subalgebra p0 ⊂ g0 such that :

p0 is a maximal isotropic subalgebra for the skewsymmetric form Ωλ,

p0 ⊃ [g1, g1].

Proof of the lemma

Lemma


p0 is a maximal isotropic subalgebra for the skew symmetric formΩλ,

p0 ⊃ [g1, g1].

Proof of the lemma

Lemma



p0 ⊃ [g1, g1].

1 i = [g1, g1] is an ideal of g0, hence there exists a sequence

0 = i0 ⊂ i1 ⊂ i2 ⊂ · · · ⊂ is = [g0, g0] ⊂ is+1 ⊂ · · · ⊂ ir = g0of ideals such that dim (ik/ik−1) = 1 for every k ≥ 1.

Proof of the lemma

Lemma



p0 ⊃ [g1, g1].



2 (M. Vergne) Define p0 to be

p0 :=

r∑

k=1

rad(Ωλ | ik×ik ).

Then p0 is a maximal isotropic subalgebra forΩλ.

Proof of the lemma

Lemma



p0 ⊃ [g1, g1].



2 (M. Vergne) Define p0 to be

p0 :=

r∑

k=1

rad(Ωλ | ik×ik ).

Then p0 is a maximal isotropic subalgebra forΩλ.

3 One can show that Ωλ([g1, g1], [g1, g1]) = 0, which implies that[g1, g1] ⊂ p0.

Immediate consequences

For every unitary representation (π, ρπ,H ) of (G0, g) wehave ρπ([g1, [g1, g1]]) = 0.



(π, ρπ,H )|G0= πλ ⊕ · · · ⊕ πλ︸︷︷︸

2l time

where πλ is the irreducible unitary representation of G0

corresponding to G0 · λ.



(π, ρπ,H )|G0= πλ ⊕ · · · ⊕ πλ︸︷︷︸

2l time

where πλ is the irreducible unitary representation of G0

corresponding to G0 · λ.One can see that if (π, ρπ,H ) is induced form thepolarizing system (M0,m,C0, c,Φ, λ) then :

dim c =

2l if (π, ρπ,H ) and (π, ρπ,ΠH )are unitarily equivalent,

2l + 1 otherwise.

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