Transversally elliptic operators and quantization

Transversally elliptic operators and quantization

work in progress of Paradan .

Michele Vergne

Juin 2006; Lusztig Anniversary

This report is on some common work in progress with Paradan.

I will discuss about an old topic: How to quantize a symplectic manifold.

Of course, this topic is one of the oldest topic in representation theory: the Stone Von-Neumann theorem (1930) asserts that given R ² with its canonical coordinates p, q and its Poisson structure {p, q} = 1, there exists a unique canonical quantization: An Hilbert space H with operators [P, Q] = ih, irre- ducible under the symmetry group coming from all affine symplectic transfor- mations of R ² , that is the semidirect product of the metaplectic group with the Heisenberg group. In particular, if we identify R ² with C, the Hilbert space can be taken as

H = F ock(C) = {f ∈ O(C);

Z

C

|f(z)| ² e ^−|z|

dxdy < ∞}

The rotation group acting by z 7→ e ^iφ z acts naturally in H, and the character of this representation is

Tr _H (e ^iφ ) = X ∞

n=0

e ^inφ .

The problem of geometric quantization is to associate to a symplectic

integral manifold N a Hilbert space Q(N ), where ”reasonable” symplectic

transformations of N will still acts on Q(N ). Despite many progress, this is still mysterious.

One example is N = T ^∗ B , where H = Q(N ) should be L ² (B): But as the example of R ² = T ^∗ R shows, a very large symmetry group of symplectic transformations of N still acts on H, even if they do not come from action on B lifted to T ^∗ B .

Here I restrict myself to the following situation. Let (N, Ω) be a symplec- tic pre-quantizable manifold with a Kostant line bundle L → N . Let K be a connected compact group acting on N in a Hamiltonian way. Let µ : N → k ^∗ be the moment map.

If φ ∈ k we denote by V φ = exp(−²φ) · m the vector field generated by the infinitesimal action of −φ. Then

ι(V φ)Ω = d(µ(m), φ)

where ι denotes contraction of a form by a vector field. Assume µ is proper (and another additional condition, satisfied if N is real algebraic).

Goals:

(a) Define a Hilbert (virtual) space Q(N, L, µ) with a canonical trace class representation of K.

(b) Give a character formula for

Tr _Q(N,L,µ) (k).

(c) Prove that ”quantization commute with reduction”.

We can answer (a), (b). Paradan has proven some cases of (c).

I recall first Guillemin-Sternberg definition of Q(N, L, µ) in the case of a compact symplectic manifold (N, Ω) with Kostant line bundle L. By defini- tion L → N is a bundle with connection ∇ of curvature −iΩ.

Choose K-invariant metrics on K ,k. Consider an almost complex struc- ture J on T N = T ^∗ N , compatible with Ω, that is Ω(v, Jv) > 0. Thus we obtain an Hermitian metric kξk ² on the complex vector bundle T ^∗ N . For [x, ξ] ∈ T ^∗ N , the symbol of the Dolbeaut-Dirac operator ∂ +∂ ^∗ is the Clifford multiplication c(ξ) on ΛT _x ^∗ N . It is invertible for ξ 6= 0, since c(ξ) ² = −kξk ² . Let L be the Kostant line bundle over N , then define

c _L ([x, ξ ]) := c(ξ) ⊗ Id _L

.

If the manifold N is compact, then c L gives us an element [c L ] of the equivariant K-theory of T ^∗ N. Indeed c _L (x, ξ) is invertible except on the zero section N of T ^∗ N , which is compact. Then Q(N, L) is defined by Guillemin- Sternberg as the equivariant index of the K-invariant operator twisted Dirac- Dolbeaut operator ∂ _L + ∂ ^∗ _L :

Q(N ) = index[c _L ].

Recall that this definition is the ”symplectic version” of the space Q(N ) = X

i

(−1) ⁱ H ⁱ (N, O(L)) in the algebraic geometry context.

That Guillemin-Sternberg definition is the correct one from the point of view of quantum mechanic has been ”certified” by the result of Meinrenken- Sjamaar : ”Quantization commutes with reduction”, which relates the mul- tiplicities of Q(N ) to the level sets of the moment map.

I recall this result.

Let T be the maximal torus of K, t ^∗ ₊ be a closed Weyl chamber, P ₊ the set of dominant weights, and for ξ ∈ P ₊ , let V _ξ the representation of K with highest weight ξ.

Assume that ξ is a regular value of µ. Let N _ξ = µ ⁻¹ (ξ)/K(ξ). Then N _ξ is a prequantizable symplectic orbifold, with Kostant line bundles L _ξ , and the preeceding construction (applied with K = {1} and suitably adapted to orbifolds and non regular values) gives us an integer Q(N _ξ , L _ξ ): the index of the twisted Dolbeaut-Dirac operator on the fiber.

Theorem:

Q(N, L) = X

ξ∈P

Q(N _ξ , L _ξ )V _ξ

Note that it implies in particular that if V ξ enters in the decomposition of Q(N, L) with non zero multiplicity, then the orbit Kξ is in the image µ(N ) ⊂ k ^∗ of the moment map.

It is most desirable to extend this definition of Q(N, L) to some non compact symplectic manifolds, like R ² or T ^∗ S ¹ . We give now a definition of quantization as an index of a transversally elliptic operator.

Recall Atiyah-Singer’s definition of transversally elliptic operators. As-

sume first that N is compact. Let N be a K-manifold and T _K ^∗ N be the

conormal bundle to K-orbits. A transversally elliptic pseudo-differential op- erator S is elliptic in the directions normal to the K -orbits. Thus S together with the action of the Casimir of k defines an elliptic system, and the space of solutions of S decomposes as a Hilbert direct sum of finite dimensional spaces of K-finite solutions. The symbol of S defines an element σ(S) of K _K (T _K ^∗ N ). The index of the operator S is the character of K in the virtual vector space obtained as difference of K-finite solutions of S and its adjoint.

This is an invariant generalized function on K . Reciprocally any element s in K _K (T _K ^∗ N ) arise this way. This definition makes sense for any symbol s ∈ K K (T _K ^∗ N ) even if N is non compact provided the support of s(x, ξ) (set of elements where s(x, ξ ) is not invertible ) intersected with T _K ^∗ N is com- pact. Then the index of s is well defined as a virtual character of K: a Fourier series of representations of K with integral coefficients with at most polynomial growth.

We return to the situation, where N is non necessarily compact symplec- tic manifold , with an Hamiltonian action of a compact group K, and the moment map µ is proper. Then µ gives rise to an invariant vector field κ on N . This is Kirwan vector field: At the point m of N , κ _m is the tangent vector to the curve exp(−²µ(m)).m. (Recall that µ(m) ∈ k ^∗ = k).

Let C be the set of zeroes of this vector field κ. Then C is also the set of critical points of the function kµk ² on N . Let us remark that one connected component of C is µ ⁻¹ (0), a compact set by definition if µ is proper. If µ is proper, and N real algebraic, then C is a compact set. Using the K- invariant metric, κ can also be considered as a K -invariant 1-form. Using an idea already present in Atiyah, we deform invariantly the zero section of T ^∗ N inside T ^∗ N by pushing [x, ξ] to [x, ξ − κ _x ].

Proposition 1 (Paradan) The element c _L,µ (x, ξ) = c(ξ − κ _x ) ⊗ Id _L

is an element of K _K (T _K ^∗ N ). It does not depend of the choices of metrics of T N .

Indeed to verify that c(ξ − κ _x ) is in K _K (T _K ^∗ N ), we have to verify that the set of points in T _K ^∗ N where this endomorphism of ΛT ^∗ N is not invertible is compact. This happens if ξ = κ x . But recall that ξ is normal to the K-orbit through x, while κ is tangent to the orbit. So ξ = κ = 0. Thus the support of c(ξ − κ _x ) is [C, 0] ⊂ T _K ^∗ N which is a compact set. The index of c _L,µ is then a virtual representation of K canonically associated to (N, L, µ).

We then propose to define:

Definition 2

Q(N, L, µ) = index(c _L,µ ) It is obvious now to formulate the conjecture

Conjecture: Quantization commutes with reduction

Q(N, L, µ) = X

ξ∈P

Q(N _ξ , L _ξ )V _ξ . Now I will

1) Give a meaningful formula for the character of Q(N, L, µ) as the inte- gral of an equivariant cohomology class on N

2) Explain some cases where Q(N, L, µ) can be identified and the conjec- ture holds.

1 Character formula

We consider equivariant forms: functions from k to the space A(N ) of differ- ential forms on N (invariant by K action on both sides). We denote (as the physicists do) by φ the variable in k and α(φ) such an equivariant form.

Consider the symplectic equivariant form Ω(φ) =< µ(m), φ > +Ω. This is a closed equivariant form for the equivariant differential

(Dα)(φ) = d(α(φ)) − ι(V φ)α(φ).

Indeed

(d − ι(V φ))(hφ, µi + Ω) = d(hφ, µi) − ι(V φ) · Ω + d(Ω) and this is equal to 0 as both equations

dΩ = 0, d(hφ, µi) = ι(V φ) · Ω hold.

I will use the notion of equivariant cohomology with generalized coef- ficients (introduced in Duflo-Kumar-Vergne), that is weak limits of C ^∞ - equivariant forms. We denote by H ^∞ (k, N ) and H ^−∞ (k, N ) the corresponding cohomology spaces.

For example if δ 0 (φ) is the Dirac mass at 0, and N = S ¹ = {e ^iθ }, then

α(φ) = δ (φ)dθ

is a closed equivariant form: Indeed

(d − ι(V φ))δ ₀ (φ)(dθ) = d(δ ₀ (φ)(dθ)) − ι(V φ) · δ ₀ (φ)(dθ)

= d(δ ₀ (φ)(dθ)) − φι(∂ _θ ) · δ ₀ (φ)(dθ) and this is equal to 0 as both equations:

d(dθ) = 0

φδ ₀ (φ)ι(∂ _θ )(dθ) = φδ ₀ (φ) = 0 hold.

If α : k → A _cpt (N ) is an equivariant generalized form with value in com- pactly supported differential forms on N , then the integral R

N α(φ) is well defined and is a generalized function on k. We just integrate a distribu- tion over a compact set.

The work of Witten on non abelian localization can be best formulated using equivariant cohomology on N with C ^−∞ -coefficients. We will see how it is in strong analogy with Atiyah’s definition of ”pushing” the zero section for obtaining transversally elliptic operators on non compact manifolds. In both cases, it says that, after ”localization ”, equivariant cohomology or equivariant K-theory of N are supported on C, the critical set of points of kµk ² . We will indeed use in parallel the maps:

H ^∞ (k, N ) 7→ H ^−∞ (k, N ) K _K (T ^∗ N ) 7→ K _K (T _K ^∗ N )

Let κ the Kirwan vector field and C its set of zeroes. Consider Dκ(φ) = dκ− < κ, V φ >. Remark that φ 7→< κ _x , V φ > is a linear form on k, non zero if x is not in C. It follows that the oscillatory integral

Z _∞

0

e ^iaDκ da

defines by limit a generalized equivariant form on N − C.

The following theorem is strongly inspired by Witten’s non abelian local-

ization formula, where this oscillatory integral is used to concentrate equivari-

ant integration on C.

Theorem 3 [?](Paradan) On N − C, the integral B := i

Z _∞

0

e ^−ia(Dκ) κda = ”κ/Dκ”

is a well defined element of A ^−∞ (k, N − C) and we have 1 = DB

on N − C. Thus (1, B) is in the relative cohomology H ^−∞ (k, N, N − C).

We identify H ^−∞ (k, N, N −C) with cohomology with support on C. More concretely: let χ be a C ^∞ function equal to 1 in a neighborhood of C and supported near C.

Define

P _µ = χ + dχB.

This is a closed generalized form, well defined on N ,supported near C. Its image H ^−∞ (k, N ) (without support conditions) is equal to 1.

To summarize: 1 is supported on C.

Multiplying by P _µ , any element α ∈ H ^∞ (k, N ) has a (rather explicit) representative in H ^−∞ (k, N) supported near C.

We are ready to formulate the character formula.

Theorem 4 (P-V)

Assume C is compact. Let T odd(φ, N ) the equivariant Todd class of the complex bundle T N → N . Then for φ small enough

Q(N, L, µ)(exp φ) = (2iπ) ^{− dim N/2} Z

N

e ^iΩ(φ) T odd(φ, N )P _µ (φ).

We give similar formula near any point s ∈ K. This is a well defined integral as P _µ is compactly supported near the set C of zeroes of the kirwan vector field κ. Of course if N is compact, we can replace P _µ by 1 (as P _µ = 1 in cohomology), then apply abelian localization and we find Atiyah-Bott-Segal fixed point formula for the equivariant index. However this formula is valid for ANY N with proper moment map for example for N = T ^∗ K or N = V a symplectic vector space with action of K provided the moment map is proper.

We proved this formula by identifying this formula to the Berline-Vergne

formula (which was a formula on T ^∗ N and had the disadvantage to be not

compactly supported)

2 Examples

Let us now analyze some cases where we can identify Q(N, L, µ) to some well known representations of K.

Example T ^∗ S ¹ = S ¹ × R. We obtain Q(N, L, µ)(exp φ) =

X ∞

n=−∞

e ^inφ .

Similarly we obtain

Q(C, L, µ)(exp φ) = φ 1 − e ^iφ

Z

C

e ^iφ|z|

dxdy = X ∞

n=0

e ^inφ .

Let G be a real Lie group with maximal compact group K. We assume that the rank of K is equal to the rank of G. Let t ^∗ ₊ be a Weyl chamber and P be the weight lattice for K. Let λ ∈ ρ(g) + P ∈ t ^∗ be a regular admissible elliptic element. Let Θ(λ) be Harish Chandra discrete series associated to λ.

Recall that its minimal K -type is the irreducible representation of K with highest weight v(λ) = λ − 2ρ(k) + ρ(g). Let L _v(λ) the Kostant line bundle on N = G · v(λ).

Assume that λ − t(2ρ(k) − ρ(g)) for t ∈ [0, 1] does not cross any wall for roots of t in g. Consider the symplectic manifold N = G · v(λ). The moment map N → k ^∗ is proper and the set C is compact. It is very easy to compute in this case: C = K · v(λ).

Theorem 5 (Paradan)

Let N = G · v(λ). Then Q(N, L _v(λ) , µ) = Θ(λ).

Furthermore, the conjecture that quantization commutes with reduction holds.

It follows in particular from Paradan’s result, and earlier result in the spin case, that if

Θ(λ) = X

m(ξ)V _ξ , then:

A) For any regular admissible λ, if m(ξ) is not zero then ξ + ρ(k) belongs to the interior of the Kirwan polyhedron µ(Gλ) ∩ t ^∗ ₊ .

B) If λ and v(λ) are regular and in the same chamber for the system of

all roots (g, t), then ξ belongs to the Kirwan polyhedron µ(Gv(λ)) ∩ t ^∗ ₊ .

(If is also true if v(λ) becomes singular for a compact root).

The hope is to prove similar theorems whenever H is a compact subgroup of G, and N a coadjoint orbit with proper moment map on h ^∗ .

References

[1] M.F. Atiyah, Collected works. Clarendon Press, Oxford (1988).

[2] M. F. Atiyah, Elliptic operators and compact groups. Lecture Notes in Mathematics 401, Springer, Berlin, (1974).

[3] M. F. Atiyah, R . Bott, A Lefschetz fixed-point formula for elliptic com- plexes. I: Ann. of Math. 86 (1967), 374-407. II: Ann. of Math. 88 (1968), 451-491.

[4] N. Berline, M. Vergne,L’indice ´equivariant des op´erateurs transversale- ment elliptiques. Invent. Math. 124 (1996) 51-101.

[5] M. Braverman, Index theorem for equivariant Dirac operators on non- compact manifolds. K-Theory 27 (2002), 61-101.

[6] M. Duflo, Shrawan Kumar, M. Vergne, Sur la cohomologie ´equivariante des vari´et´es diff´erentiables. Ast´erisque,215, S.M.F. Paris, (1993).

[7] B. Kostant, Quantization and unitary representations. I. Prequantiza- tion. Lectures in modern analysis and applications, III, pp 87-208. Lec- ture Notes in Math. 170, Springer,Berlin, (1970).

[8] E. Meinrenken, R. Sjamaar, Singular reduction and quantization. Topol- ogy 38 (1999), 699-762.

[9] P. E. Paradan, Spin ^c quantization and the K-multiplicities of the discrete series. Ann. Sci. Ecole Norm. Sup. 36 (2003), 805-845.

[10] M. Vergne, Geometric quantization and equivariant cohomology. First European Congress of Mathematics, vol.1 (Paris 1992), 249-295, Prog.

Math 119. Birkha¨user, Basel (1994).

[11] M. Vergne, Quantification g´eom´etrique et r´eduction symplectique.

S´eminaire Bourbaki. vol 2000/2001, Ast´erisque 282 (2002), 249-278.

[12] E. Witten,Two dimensional gauge theories revisited. J. Geom. Phys. 9

(1992), 303-368.