Symplectic Induction, Prequantum Induction, and Prequantum Multiplicities
aa r X i v : . [ m a t h . S G ] J u l Symplectic versus
Prequantum Induction
Tudor S. Ratiu ∗ and François Ziegler † July 17, 2020
Abstract
This paper establishes two basic properties of the symplectic induction construction of Kazhdan,Kostant, Sternberg, and Weinstein: Induction in Stages and Frobenius Reciprocity. It then argues thata prequantum version of the construction, of which we prove the same two properties, is in fact theappropriate framework to geometrically model representation-theoretic phenomena.
Introduction
Beyond the mere parametrization of irreducible unitary representations by coadjoint orbits originat-ing in the work of Borel-Weil and Kirillov [S54, K62], there exists a certain well-known parallelism between representation theory and the symplectic theory of Hamiltonian G-spaces. To capture it withprecision, papers like [K78, W78, G82, G83] introduced purely symplectic constructions meant to mir-ror operations such as
Ind (inducing a representation from a subgroup) or
Hom G (forming the spaceof intertwining operators between two representations). In that setting, one of course expects basicproperties like induction in stages or Frobenius reciprocity to hold in symplectic geometry. A first goalof this paper is to spell out their proofs (§2, §3), fulfilling promises made in [Z96, p. 9] and [M07,p. 105].A second goal of the paper is to point out that, while these constructions fit their purpose whenthe correspondence from representations to coadjoint orbits is one-to-one [F15], they fall short whenit is many-to-one [S94]. To remedy this, we propose new versions of both constructions in the categoryof prequantum G -spaces (§5, §6) and establish the stages and Frobenius properties in that setting (§7,§8). Finally, we illustrate the need for our prequantized versions by what we believe is the simplestexample (§4, §9). Notation and conventions
We use a concise notation for the translation of tangent and cotangent vectors to a Lie group: for fixed g , q ∈ G,(0.1) T q G v →7→ T gq G gv , resp. T ∗ q G p →7→ T ∗ gq G gp will denote the derivative of q gq , respectively its contragredient, i.e., 〈 gp , v 〉 = 〈 p , g − v 〉 . Likewise,we define vg and pg with 〈 pg , v 〉 = 〈 p , vg − 〉 . ∗ School of Mathematical Sciences and Ministry of Education Key Lab on Scientific and Engineering Computing, ShanghaiJiao Tong University, Shanghai 200240, China and Section de Mathématiques, Université de Genève and Ecole PolytechniqueFédérale de Lausanne, Switzerland. [email protected], [email protected] † Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA. [email protected]
1y a
Hamiltonian G -space we mean the triple (X, ω , Φ ) of a manifold X on which G acts, a G-invariant symplectic form ω on it, and a G-equivariant momentum map Φ : X → g ∗ . We identifyspaces X , X which are isomorphic , i.e., related by a G-equivariant diffeomorphism which transforms ω into ω and Φ into Φ . If several are in play, we also use subscripts like ω X , Φ X . We recall twocardinal properties of the momentum map:(0.2) (a) Ker(D Φ ( x )) = g ( x ) ω (b) Im(D Φ ( x )) = ann( g x ).The first is the orthogonal relative to ω of the tangent space g ( x ) to the orbit G( x ), x ∈ X; the secondis the annihilator in g ∗ of the stabilizer Lie subalgebra g x ⊂ g . Given a closed subgroup H ⊂ G and a Hamiltonian H-space (Y, ω Y , Ψ ), [K78] constructs an induced Hamiltonian G-space as follows. Let ̟ denote the canonical 1-form on T ∗ G given by ̟ ( δ p ) = 〈 p , δ q 〉 ,where δ p ∈ T p (T ∗ G), δ q = π ∗ ( δ p ) ∈ T π ( p ) G, and π : T ∗ G → G is the canonical projection. Endow N : = T ∗ G × Y with the symplectic form ω : = d ̟ + ω Y and the G × H-action ( g , h )( p , y ) = ( gph − , h ( y )),where h ( y ) denotes the action of h ∈ H on y ∈ Y. This action admits the equivariant momentum map φ × ψ : N → g ∗ × h ∗ ,(1.1) (cid:26) φ ( p , y ) = pq − ψ ( p , y ) = Ψ ( y ) − q − p | h ( p ∈ T ∗ q G).The induced manifold is, by definition, the Marsden-Weinstein reduced space of N at 0 ∈ h ∗ , i.e.(1.2) Ind GH Y : = N // H = ψ − (0) / H.In more detail: the action of H is free and proper (because it is free and proper on the factor T ∗ G,where it is the right action of H regarded as a subgroup of the group T ∗ G [B72, §III.1.6]); so ψ is asubmersion (0.2b), ψ − (0) is a submanifold, and (1.2) is a manifold; moreover ω | ψ − (0) degeneratesexactly along the H-orbits (0.2a), so it is the pull-back of a uniquely defined symplectic form, ω N // H ,on the quotient. Furthermore, the G-action commutes with the H-action and preserves ψ − (0), and itsmomentum map φ is constant on H-orbits. Passing to the quotient, we obtain the required G-actionon Ind GH Y and momentum map Φ N // H : Ind GH Y → g ∗ . Note that since ψ is a submersion and H actsfreely, (1.2) has dimension equal to dim(N) − GH Y) = / H) + dim(Y). (2.1) Theorem (Stages). If H ⊂ K ⊂ G and H, K are closed subgroups of the Lie group G , then Ind GK Ind KH Y = Ind GH Y. Proof.
Let (N, ω , φ × ψ ) be as in §1 and consider M = T ∗ G × T ∗ K × Y with 2-form ω M = d ̟ T ∗ G + d ̟ T ∗ K + ω Y and G × K × H-action(2.2) ( g , k , h )( p , ¯ p , y ) = ( gpk − , k ¯ ph − , h ( y )).This admits the equivariant momentum map φ × ¯ φ × ¯ ψ : M → g ∗ × k ∗ × h ∗ :(2.3) φ ( p , ¯ p , y ) = pq − ¯ φ ( p , ¯ p , y ) = ¯ p ¯ q − − q − p | k ¯ ψ ( p , ¯ p , y ) = Ψ ( y ) − ¯ q − ¯ p | h ψ − (0) (¯ φ × ¯ ψ ) − (0) ψ − (0)M // H ¯ Φ − // H (0)(M // H) // K N // H rj π j s π j π j π t Figure 1: Construction of the isomorphism t .for ( p , ¯ p ) ∈ T ∗ q G × T ∗ ¯ q K. Define r : M → N by r ( p , ¯ p , y ) = ( p ¯ q , y ) and consider the commutativediagram in Fig. 1, where we have written j , j , j and π , π , π for the inclusion and projection mapsinvolved in constructing the reduced spaces M // H = T ∗ G × Ind KH Y, (M // H) // K = Ind GK Ind KH Y, andN // H = Ind GH Y; also j , π are the obvious inclusion and restriction, and ¯ Φ M // H is the momentum mapfor the residual K-action on M // H. The map r ◦ j ◦ j satisfies(2.4) ψ (( r ◦ j ◦ j )( p , ¯ p , y )) = ψ ( p ¯ q , y ) = Ψ ( y ) − ( q ¯ q ) − p ¯ q | h = Ψ ( y ) − ¯ q − ( q − p ) | k ¯ q | h = Ψ ( y ) − ¯ q − ¯ p | h since ¯ φ ( p , ¯ p , y ) = = ψ ( p , ¯ p , y ) = r ◦ j ◦ j takes values in ψ − (0), i.e., there is a map s as indicated in Fig. 1. Moreover, s is ontosince one verifies that ( p , y ) ( p , ( q − p ) | k , y ) provides a right inverse. The map s is equivariant:(2.5) s (( g , k , h )( p , ¯ p , y )) = r ( gpk − , k ¯ ph − , h ( y )) = ( gp ¯ qh − , h ( y )) = ( g , h )( p ¯ q , y ) = ( g , h )( s ( p , ¯ p , y )).Hence s descends to a G-equivariant surjection t as indicated in Fig. 1. Furthermore, one checkswithout trouble that the fibers of s are precisely the K-orbits in its domain. As π ◦ π collapses theseorbits to points, it follows that t is bijective, hence a diffeomorphism by [B67, 5.9.6]. The relation φ = r ∗ φ implies that t relates the momentum maps for G: Φ (M // H) // K = t ∗ Φ N // H , so there only remainsto see that ω (M // H) // K = t ∗ ω N // H . To this end we compute(2.6) ( s ∗ j ∗ ̟ T ∗ G )( δ p , δ ¯ p , δ y ) = ̟ T ∗ G ( δ [ p ¯ q ]) = 〈 p ¯ q , δ [ q ¯ q ] 〉 = 〈 p , δ q 〉 + 〈 q − p , [ δ ¯ q ]¯ q − 〉 since δ [ q ¯ q ] = [ δ q ]¯ q + q [ δ ¯ q ] = 〈 p , δ q 〉 + 〈 ¯ p , δ ¯ q 〉 since ¯ φ ( p , ¯ p , y ) = = ̟ T ∗ G ( δ p ) + ̟ T ∗ K ( δ ¯ p ).3aking exterior derivatives and adding ω Y we obtain s ∗ j ∗ ω N = j ∗ j ∗ ω M or, equivalently (by commu-tativity of the diagram and definition of the reduced 2-forms), π ∗ π ∗ t ∗ ω N // H = π ∗ π ∗ ω (M // H) // K . Since π ◦ π is a submersion, we are done. It is quite rare for an induced Hamiltonian G-space to be homogeneous or a fortiori a coadjoint orbit(by which we mean that its momentum map is 1-1 onto an orbit). In fact we have the following, where Φ N // H is the momentum map for the induced space (1.2). (3.1) Proposition. Let (Y, ω Y , Ψ ) be a Hamiltonian H -space. (a) A coadjoint orbit O of G intersects Im( Φ N // H ) ⇔ O | h : = { m | h : m ∈ O } intersects Im( Ψ ) . (b) If Ind GH Y is homogeneous , then Y is homogeneous. (c) If Ind GH Y is a coadjoint orbit , then Y is a coadjoint orbit.Proof. (a): This re-expresses Im( Φ N // H ) = φ ( ψ − (0)) (1.1). (b): Assume G is transitive on Ind GH Y andlet y , y ∈ Y. Pick m i ∈ g ∗ such that Ψ ( y i ) = m i | h . Then the H-orbits x i = H( m i , y i ) are points in(1.2). So transitivity says that x = g ( x ), i.e.(3.2) ( m , y ) = ( gm h − , h ( y )) for some h ∈ H.In particular y = h ( y ), as claimed. (c): Assume further that Φ N // H is injective and suppose Ψ ( y ) =Ψ ( y ). Then we can pick m = m above. Since Φ N // H ( x i ) = m i it follows, by injectivity, that x = x ,i.e., we have (3.2) with g = e . But then h = e and hence y = y , as claimed.If Y is a coadjoint orbit, (3.1a) says that Ind GH Y “involves” just those orbits O whose projectionin h ∗ contains Y. Guillemin and Sternberg [G82, §6] proposed to measure the “multiplicity” of thisinvolvement by the (possibly empty) space Hom G ( O , Ind GH Y), where we write suggestively(3.3) Hom G (X , X ) : = (X − × X ) // G,i.e., the Marsden-Weinstein reduction of X − × X at 0 ∈ g ∗ ; here X − is the Hamiltonian G-space X with its 2-form and momentum map replaced by their negatives. Then (3.1a) can be refined by thefollowing analog of Frobenius’s theorem [B85, III.6.2], already found in [G83, Thm 2.2] when both Xand Y are coadjoint orbits. (3.4) Theorem (Frobenius reciprocity). If X is a Hamiltonian G -space and Y a Hamiltonian H -space,then Hom G (X, Ind GH Y) = Hom H (Res GH X, Y). (3.5) Remarks.
Here Res GH X means X regarded as a Hamiltonian H-space, and “ = ” means only anatural bijection as sets. We believe (but haven’t proved) that both sides are automatically isomorphicas diffeological spaces with diffeological 2-forms as discussed in [S85, §2.5], [I13, §6.38], [K16].Note also that, by the symmetry of (3.3), we may equally write Frobenius reciprocity in the formHom G (Ind GH Y, Z) = Hom H (Y, Res GH Z).
Proof.
For bookkeeping reasons, soon to become clear, rename G also as K. Consider the spaces N = X − × Y with H-action h ( x , y ) = ( h ( x ), h ( y )), and M = X − × T ∗ K × Y with K × H-action ( k , h )( x , ¯ p , y ) = ( k ( x ), k ¯ ph − , h ( y )). Their equivariant momentum maps are ψ : N → h ∗ ,(3.6) ψ ( x , y ) = Ψ ( y ) − Φ ( x ) | h φ × ¯ ψ : M → k ∗ × h ∗ ,(3.7) (cid:26) ¯ φ ( x , ¯ p , y ) = ¯ p ¯ q − − Φ ( x )¯ ψ ( x , ¯ p , y ) = Ψ ( y ) − ¯ q − ¯ p | h (¯ p ∈ T ∗ ¯ q K)where Φ and Ψ are the equivariant momentum maps of X and Y, respectively. Defining r : M → N by r ( x , ¯ p , y ) = (¯ q − ( x ), y ) now sets us up for a proof using the same previous diagram (Fig. 1). Indeed,we have again this time(3.8) ψ (( r ◦ j ◦ j )( x , ¯ p , y )) = ψ (¯ q − ( x ), y ) = Ψ ( y ) − Φ (¯ q − ( x )) | h = Ψ ( y ) − ¯ q − Φ ( x )¯ q | h by equivariance = Ψ ( y ) − ¯ q − ¯ p | h since ¯ φ ( x , ¯ p , y ) = = ψ ( x , ¯ p , y ) = s as indicated in Fig. 1. Again, s is onto since ( x , y ) ( x , Φ ( x ), y ) provides a rightinverse, and s is equivariant:(3.9) s (( k , h )( x , ¯ p , y )) = r ( k ( x ), k ¯ ph − , h ( y )) = (( k ¯ qh − ) − ( k ( x )), h ( y )) = ( h (¯ q − ( x )), h ( y )) = h ( s ( x , ¯ p , y )).So the fibers of s are again the K-orbits and s descends again to a bijection t as required and indicatedin Fig. 1. The following example highlights a basic shortcoming in the analogy of (3.4) with representationtheory: it cannot mirror cases where more than one representation “quantizes” a given HamiltonianG-space or H-space.In the solvable group G ′ of all upper triangular matrices of the form(4.1) g ′ = e i a b e f a a , e , f ∈ R b ∈ C ,write G for the subgroup in which e = a ∈ π Z . Identify g ′∗ with R × C × R by writing ( p , q , s , t ) for the value at the identity of the 1-form(4.2) pda + Re(¯ qdb ) − sde − tdf .Likewise, identify g ∗ with triples ( p , q , t ) and h ∗ with pairs ( q , t ) so that the projections g ′∗ → g ∗ → h ∗ become ( p , q , s , t ) ( p , q , t ) ( q , t ). Then the coadjoint orbit X ′ = G ′ (0, 1, 0, 1) projects onto thecoadjoint orbit X = G(0, 1, 1) and is its universal covering:(4.3) X ′ = (cid:8) ( p , e i s , s , 1) : ( p , s ) ∈ R (cid:9) , ω X ′ = dp ∧ ds , ↓ X = (cid:8) ( p , q , 1) : ( p , q ) ∈ R × T (cid:9) , ω X = dp ∧ dq i q .5oreover, one checks (or finds by [Z14] applied to the normal subgroup H o ) that X = Ind GH Y, whereY is the point {(1, 1)}. So symplectic Frobenius reciprocity gives(4.4) Hom G (X, Res G ′ G X ′ ) = Hom H (Y, Res G ′ H X ′ ) = (X ′ → h ∗ ) − (1, 1) / Hwhich is a single point. But this fact is of little use for representation theory, as it fails to discriminatebetween the circle worth of representations attached to X, according to [A71] (where, we recall, theyare parametrized by the characters of the fundamental group π (X)). As one knows, this should befixed by working instead with prequantum spaces in the sense of the next section. G -spaces Following [S70], we call prequantum manifold a manifold ˜X with a contact 1-form ̟ whose Reeb vec-tor field generates a circle group action. We recall that ̟ contact means that Ker( d ̟ ) is 1-dimensionaland transverse to Ker( ̟ ); its Reeb vector field, i, on ˜X is defined by(5.1) i(˜ x ) ∈ Ker( d ̟ ) and ̟ (i(˜ x )) = ∀ ˜ x ∈ ˜X.Then (˜X, d ̟ ) is a presymplectic manifold whose null leaves are the orbits of the circle group T = U(1)acting on ˜X and d ̟ descends to a symplectic form ω on the leaf space X = ˜X / T . If a Lie group G actson ˜X and preserves ̟ , then it commutes with T and the equivariant momentum map Φ : ˜X → g ∗ ,(5.2) 〈 Φ (˜ x ), Z 〉 = ̟ (Z(˜ x )),descends to a momentum map Φ : X → g ∗ , making (X, ω , Φ ) a Hamiltonian G-space prequantized bythe prequantum G -space (˜X, ̟ ).We do not distinguish between two spaces ˜X , ˜X which are isomorphic , i.e., related by a G-equivariant diffeomorphism which transforms ̟ into ̟ . (If several are in play, we may also usesubscripts like ̟ ˜X , i ˜X , Φ ˜X , etc.) We recall three basic constructions in the prequantum category: (5.3) Prequantum dual. ([S70, 18.47].) We write ˜X − for the G-space equal to ˜X but with opposite1-form − ̟ ˜X (and consequently opposite Reeb field and T -action). It prequantizes the dual G-space(X − , − ω , − Φ ). (5.4) Prequantum product. ([S70, 18.52].) If ˜X and ˜X are prequantum G-spaces, then ˜X × ˜X (with diagonal G-action) is a T -space in which the action of the antidiagonal Δ = {( z − , z ) : z ∈ T }has as its orbits the characteristic leaves of the 1-form ̟ + ̟ . Hence this descends to the quotient˜X ⊠ ˜X : = (˜X × ˜X ) / Δ as a 1-form making it a prequantization of the symplectic product X × X . Inview of (5.3), the Δ -action on ˜X − × ˜X is z (˜ x , ˜ x ) = ( z (˜ x ), z (˜ x )). (5.5) Prequantum reduction. ([L01, Thm 2].) Assume G acts freely and properly on ˜X, and considerthe level L : = Φ − (0). By the very definition (5.2) of Φ and its being a momentum map, we have g (˜ x ) ⊂ Ker( ̟ | L ) ∩ Ker( d ̟ | L ). Since ̟ | L is also G-invariant, it follows (see [S70, 5.21]) that it de-scends to a contact 1-form on the quotient ˜X // G : = Φ − (0) / G. This prequantizes the symplectic re-duction X // G = Φ − (0) / G. Given a closed subgroup H ⊂ G and a prequantum H-space (˜Y, ̟ ˜Y ) whose momentum map (5.2)we denote Ψ , we propose to construct an induced prequantum G-space Ind GH ˜Y as follows. Consider6he prequantum (G × H)-space ˜N = T ∗ G × ˜Y with 1-form ̟ T ∗ G + ̟ ˜Y and action ( g , h )( p , ˜ y ) = ( gph − , h (˜ y )). This action has the equivariant momentum map φ × ψ : ˜N → g ∗ × h ∗ ,(6.1) (cid:26) φ ( p , ˜ y ) = pq − ψ ( p , ˜ y ) = Ψ (˜ y ) − q − p | h ( p ∈ T ∗ q G).The same arguments as with (1.2), then, show that(6.2) Ind GH ˜Y : = ˜N // H = ψ − (0) / H(prequantum reduction (5.5)) is naturally a prequantum G-space which prequantizes the symplecti-cally induced manifold (1.2). (7.1) Theorem. If H ⊂ K ⊂ G and H, K are closed subgroups of the Lie group G , then Ind GK Ind KH ˜Y = Ind GH ˜Y. Proof.
The proof is mutatis mutandis the same as for (2.1), only simpler. We just switch to workingwith restrictions and push-forwards of the 1-form ̟ ( δ p , δ ¯ p , δ ˜ y ) = 〈 p , δ q 〉 + 〈 ¯ p , δ ¯ q 〉 + ̟ ˜Y ( δ ˜ y ) on˜M = T ∗ G × T ∗ K × ˜Y instead of the 2-form ω on M. The three constructions (5.3–5.5) put together furnish us with a notion of the intertwiner space of twoprequantum G-spaces,(8.1) Hom G (˜X , ˜X ) : = (˜X − ⊠ ˜X ) // G.Freeness and properness of the last G-action are not assumed and we again regard (8.1) as just a set. (8.2) Theorem (Frobenius reciprocity). If ˜X is a prequantum G -space and ˜Y a prequantum H -space,then Hom G (˜X, Ind GH ˜Y) = Hom H (Res GH ˜X, ˜Y). Proof.
With Δ as in (5.4), define ˜˜ r in the following commutative diagram by ˜˜ r (˜ x , p , ˜ y ) = ( q − (˜ x ), ˜ y ),where p ∈ T ∗ q G:(8.3) ˜˜M : = ˜X − × T ∗ G × ˜Y ˜˜N : = ˜X − × ˜Y˜M : = ˜X − ⊠ T ∗ G × ˜Y ˜N : = ˜X − ⊠ ˜YM : = X − × T ∗ G × Y N : = X − × Y. ˜˜ r mod Δ mod Δ ˜ r mod ( T / Δ ) mod ( T / Δ ) r Then ˜˜ r descends, as indicated, to a map ˜ r and a map r which is the one in our proof of (3.4). Noweach floor of this diagram supports a horizontal copy of Fig. 1 giving rise to the appropriate tildedversions of s and t ; a straightforward diagram chase checks that ˜ t : ( ˜M // H) // G → ˜N // H is the requiredbijection. 7
An Example (Reprise)
Recall the coadjoint orbits X ′ ∼ = R and X ∼ = R × T of (4.3). Referring to [S70, 18.117, 18.133, 18.134]and performing direct verifications, one finds: • There are infinitely many prequantum G-spaces prequantizing X, namely all ˜X λ = Ind GH T λ where T λ is a single circle on which H acts by the character χ λ ( h ) = e − i λ a e i[Re( b ) − f ] (notation (4.1)).Explicitly(9.1) ˜X λ = R × T ∋ ( p , q , z ) with ̟ λ = ( p + λ ) dq i q + dz i z ,and λ , λ ∈ R give equivalent prequantizations iff they differ by an integer [A59, K06]. • There is a unique prequantum G ′ -space over X ′ , namely ˜X ′ = Ind G ′ H ′ T where H ′ is the subgroup a = ′ and T is a single circle on which H ′ acts by the character χ ( h ′ ) = e i[Re( b ) − f ] . Explicitly(9.2) ˜X ′ = R × T ∋ ( p , s , z ) with ̟ = pds + dz i z .Apply now (8.2) which replaces (4.4) in this case, to conclude(9.3) Hom G (˜X λ , Res G ′ G ˜X ′ ) = Hom H ( T λ , Res G ′ H ˜X ′ ).A direct verification shows that the right-hand side is a single circle for each λ . This identity alsoillustrates the power of Frobenius reciprocity: we can obtain the harder left-hand side from the easierright-hand side. Returning to the representation theoretical interpretation, note that (9.3) “predicts”that once restricted to G, the irreducible representation Ind G ′ H ′ χ (which quantizes X ′ ) splits into thedirect integral over λ ∈ R / Z of the irreducible representations Ind GH χ λ (which all quantize X) withmultiplicity 1; this prediction is correct and can be checked directly. References [A59] Yakir Aharonov and David Bohm, Significance of electromagnetic potentials in the quantum theory.
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