Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.


The classical Riemann-Roch Theorem
The classical Riemann-Roch Theorem is well-known in the function theory as [M.F. Atiyah and Hirzebruch, Riemann-Roch Theorem for differential manifolds, Bull. Amer. Math. Soc. 1959, Vol. 65, 276-281] Theorem 1 (The Riemann-Roch Theorem) where D is a fixed divisor of degree d(D) on a Riemann surface X of genus g, r(−D) is the dimension of the space of meromorphic functions of divisor ≥ −D on X, i(D) the dimension of the space of meromorphic 1-forms of divisor ≥ D on X.
This theorem can be considered as computing the Euler characteristics of the sheaf of germs of holomorphic sections of the holomorphic bundle, defined by the divisor D, over X. It plays also an important role in classical algebraic geometry.
Theorem 2 Let X be a nonsingular complex projective algebraic variety, c its first Chern class, ξ a holomorphic bundle over X.
Then the value on [X] of the cohomological class e c/2 .chξ.A −1 (p 1 (X), p 2 (X), . . .) equal to the Euler characteristic of the sheaf of holomorphic sections of the bundle ξ.

The Riemann-Roch Theorem in Algebraic Topology
In algebraic topology the Riemann-Roch theorem appeared as some measure of noncommutativity of some diagrams relating two generalized homology theories. Let us remind the most general setting of the Riemann-Roch Theorem.
Consider two generalized (co)homology theories k(X) and h(X) and τ : h → k be a multiplicative map, sending 1 ∈ h 0 (pt) to 1 ∈ k 0 (pt). Consider a vector bundle ξ, oriented with respect to the both theories, over the base X. Denote T (ξ) the Thom space of ξ, i.e. the quotient of the corresponding disk bundle D(ξ) modulo its boundary Sph(ξ). Let us consider an h-oriented vector bundle ξ = (E, X, V, p). Denote E ′ the complement to the zero section of E. It is easy to see thath r (T (ξ)) = h r (E, E ′ ).

Theorem 3 (Thom Isomorphism) The Thom homomorphism
is an isomorphism.

The Riemann-Roch Theorem and the Index Theorem of pseudodifferential operators
One of the consequences of the Riemann-Roch Theorem is the fact that the index of the Dirac operator on X is exactly the Euler characteristics of X. Let us review the classical results from algebraic topology and topology of pseudo-differential operators.
With the Riemann-Roch Theorem it is convenient to define the direct image map f ! as the special case of the composition map where h is the ordinary co-homology, X and Y are the oriented manifolds and g = f : X → Y .
Theorem 5 Let X and Y be two closed manifold, oriented with respect to the both generalized (co)homology theories h and k and f : X → Y a continuous map. Then the Todd classes T (X) and T (Y ) measure noncommutativity of the diagram and more precisely, The index of an arbitrary pseudo-differential operator D is re-Theorem 6 (Atiyah-Singer-Hirzebruch Index Theorem) where T (C τ (X)) is the Todd class of complexified tangent bundle, . As a consequence of the previous theorem we have the following result. Let X denote a 2n-dimensional oriented closed smooth manifold with spin structure, i.e. a fixed Hermitian structure on fibers, smoothly depending of points on the base X. Denote Ω k (X) the space of alternating differential k-forms on X, and d : Ω k (X) → Ω k+1 (X) the exterior differential, * the Hodge star operator and δ = * d * : Ω k−1 (X) → Ω k (X) the adjoint to d operator. The Dirac operator is d + δ is a first order elliptic operator and its index is just equal to the Euler characteristic χ(X) of the manifold.

Riemann-Roch Theorem in non-commutative geometry
Let us consider an arbitrary algebra A over the ground field of complex numbers C and G a locally compact group and denote dg the left-invariant Haar measure on G. The space C ∞ c (G) of all continuous functions with compact support on G with values in A under ordinary convolution involution and norm form a the so called cross-product A ⋊ G of A and G.
Theorem 8 (Connes-Thom Isomorphism) A consequence of this theorem is the existence of some noncommutative Todd class Theorem 9 There exists some Todd class which measures the noncommutativity of the diagram

Deformation quantization and periodic cyclic homology
Deformation quantization gives us some noncommutative algebras which are deformation of the classical algebras of holomorphic functions on X. For an arbitrary noncommutative algebra A there are at least two generalized homology theories: the Ktheory and periodic cyclic homology. The Connes-Thom isomorphism gives us a possibility to compare the two theories. There appeared some Todd class as the measure of noncommutativity. where f and g are two local sections of O M andf ,ĝ are their respective lifts in A h M , defines a Poisson structure associated to the deformation quantization A h M . It is well-known that all symplectic deformation quantization of M of dimension dim C M = 2d are locally isomorphic to the standard deformation quantization of C 2d , i.e. in a neighborhood U ′ of the origin in C 2d there is an isomorphism Theorem 10 (RRT for periodic cyclic cycles) The diagram For Dand E-modules of the ring of pseudo-differential operators, take M = T * X for a complex manifold X, and A h T * X is the deformation quantization with the characteristic class θ = 1 2 π * c 1 (X), Canonical coordinates on the upper half-planes. Recall that the Lie algebra g = aff(R) of affine transformations of the real straight line is described as follows, see for example [D2]: The Lie group Aff(R) of affine transformations of type It is well-known that this group Aff(R) is a two dimensional Lie group which is isomorphic to the group of matrices We consider its connected component of identity element. Its Lie algebra is The co-adjoint action of G on g * is given (see e.g. [AC2], [Ki1]) by −1 * Denote the co-adjoint orbit of G in g, passing through F by Because the group G = Aff 0 (R) is exponential (see [D2]), for F ∈ g * = aff(R) * , we have It is easy to see that It is easy therefore to see that For a general element U = αX + βY ∈ g, we have From this formula one deduces [D2] the following description of all co-adjoint orbits of G in g * : • If µ = 0, each point (x = λ, y = 0) on the abscissa ordinate corresponds to a 0-dimensional co-adjoint orbit • For µ = 0, there are two 2-dimensional co-adjoint orbits: the upper half-plane {(λ, µ) | λ, µ ∈ R, µ > 0} corresponds to the co-adjoint orbit 2. In the canonical coordinates (p, q) of the orbit Ω + , the Kirillov form ω Y * is just the standard form ω = dp ∧ dq.
Computation of generatorsl Z Let us denote by Λ the 2-tensor associated with the Kirillov standard form ω = dp ∧ dq in canonical Darboux coordinates. We use also the multi-index notation. Let us consider the well-known Moyal ⋆-product of two smooth functions u, v ∈ C ∞ (R 2 ), defined by . . , p n , q 1 , . . . , q n ) as multi-index notation. It is well-known that this series converges in the Schwartz distribution spaces S(R n ). We apply this to the special case n = 1. In our case we have only x = (x 1 , x 2 ) = (p, q).
Proposition 12 In the above mentioned canonical Darboux coordinates (p, q) on the orbit Ω + , the Moyal ⋆-product satisfies the relation Consequently, to each adapted chart ψ in the sense of [AC2], we associate a G-covariant ⋆-product.
Proposition 13 (see [G]) Let ⋆ be a formal differentiable ⋆-product Let us denote by F p u the partial Fourier transform of the function u from the variable p to the variable x, i.e.
∂ k p , with k ≥ 2. For each Z ∈ aff(R), the corresponding Hamiltonian function isZ = αp + βe q and we can consider the operator ℓ Z acting on dense subspace L 2 (R 2 , dpdq 2π ) ∞ of smooth functions by left ⋆-multiplication by iZ, i.e. ℓ Z (u) = iZ ⋆ u. It is then continued to the whole space L 2 (R 2 , dpdq 2π ). It is easy to see that, because of the relation in Proposition (12), the correspondence Z ∈ aff(R) → ℓ Z = iZ ⋆ . is a representation of the Lie algebra aff(R) on the space N [[ i 2 ]] of formal power series in the parameter ν = i 2 with coefficients in N = C ∞ (M, R), see e.g. [G] for more detail.
We study now the convergence of the formal power series. In order to do this, we look at the ⋆-product of iZ as the ⋆-product of symbols and define the differential operators corresponding to iZ. It is easy to see that the resulting correspondence is a representation of g by pseudo-differential operators.
Proposition 15 For each Z ∈ aff(R) and for each compactly supported C ∞ function u ∈ C ∞ 0 (R 2 ), we havê The associate irreducible unitary representations Our aim in this section is to exponentiate the obtained representationl Z of the Lie algebra aff(R) to the corresponding representation of the Lie group Aff 0 (R). We shall prove that the result is exactly the irreducible unitary representation T Ω + obtained from the orbit method or Mackey small subgroup method applied to this group Aff(R). • the representation U ε λ , where ε = 0, 1, λ ∈ R, realized in the 1-dimensional Hilbert space C 1 and is given by the formula U ε λ (g) = |a| iλ (sgn a) ε .
Let us consider now the connected component G = Aff 0 (R). The irreducible unitary representations can be obtained easily from the orbit method machinery.

By analogy, we have also
Theorem 18 The representation exp(l Z ) of the group G = Aff 0 (R) is exactly the irreducible unitary representation T Ω − of G = Aff 0 (R) associated following the orbit method construction, to the orbit Ω − , which is the lower half-plane H ∼ = R ⋊ R * , i. e.

The group of affine transformations of the complex straight line
Recall that the Lie algebra g = aff(C) of affine transformations of the complex straight line is described as follows, see [D]. It is well-known that the group Aff(C) is a four (real) dimensional Lie group which is isomorphism to the group of matrices: The most easy method is to consider X,Y as complex generators, X = X 1 + iX 2 and Y = Y 1 + iY 2 . Then from the relation This mean that the Lie algebra aff(C) is a real 4-dimensional Lie algebra, having 4 generators with the only nonzero Lie brackets: = Y 2 and we can choose another basic noted again by the same letters to have more clear Lie brackets of this Lie algebra: Remark 19 Let us denote: is a univalent sheet of the Riemann surface of the complex variable multi-valued analytic function Ln(w), (k = 0, ±1, . . .) Then there is a natural diffeomorphism w ∈ H k −→ e w ∈ C k with each k = 0, ±1, . . . . Now consider the map: with a fixed k ∈ Z. We have a local diffeomorphism This diffeomorphism ϕ k will be needed in the all sequel.
The corresponding symplectic matrix of ω is We have therefore the Poisson brackets of functions as follows. With each f, g ∈ C ∞ (Ω) Proposition 20 Fixing the local diffeomorphism ϕ k (k ∈ Z), we have: 1. For any element A ∈ aff (C), the corresponding Hamiltonian function A in local coordinates (z, w) of the orbit Ω F is of the form 2. In local coordinates (z, w) of the orbit Ω F , the symplectic Kirillov form ω F is just the standard form (1).

Computation of Operatorsl (k)
A . Proposition 21 With A, B ∈ aff(C), the Moyal ⋆-product satisfies the relation: For each A ∈ aff (C), the corresponding Hamiltonian function is and we can consider the operator ℓ

(k)
A acting on dense subspace L 2 (R 2 ×(R 2 ) * , dp 1 dq 1 dp 2 dq 2 (2π) 2 ) ∞ of smooth functions by left ⋆-multiplication by i A, i.e: ℓ From this it is easy to see that, the correspondence A ∈ aff(C) −→ ℓ (k) A =i A⋆. is a representation of the Lie algebra aff(C) on the space N[[ i 2 ]] of formal power series, see [G] for more detail.
Proposition 23 For each A = α β 0 0 ∈ aff(C) and for each i.el ,which provides a ( local) representation of the Lie algebra aff (C).

(k)
A is a representation of the Lie algebra Aff(C), we have: is just the corresponding representation of the corresponding connected and simply connected Lie group Aff (C). Let us first recall the well-known list of all the irreducible unitary representations of the group of affine transformation of the complex straight line, see [D] for more details.
Theorem 25 Up to unitary equivalence, every irreducible unitary representation of Aff (C) is unitarily equivalent to one the following one-to-another non-equivalent irreducible unitary representations: 1. The unitary characters of the group, i.e the one dimensional unitary representation U λ , λ ∈ C, acting in C following the 2. The infinite dimensional irreducible representations T θ , θ ∈ S 1 , acting on the Hilbert space L 2 (R × S 1 ) following the formula: In this section we will prove the following important Theorem which is very interesting for us both in theory and practice.
Theorem 26 The representation exp(l

Remark 27
We say that a real Lie algebra g is in the class M D if every K-orbit is of dimension, equal 0 or dim g.

MD 4 -groups
We refer the reader to the results of Nguyen Viet Hai [H3]- [H4] for the class of M D 4 -groups (i.e. 4-dimensional solvable Lie groups, all the co-adjoint of which are of dimension 0 or maximal). It is interesting that here he obtained the same exact computation for ⋆-products and all representations.

SO(3)
As an typical example of compact Lie group, the author proposed Job A. Nable to consider the case of SO (3). We refer the reader to the results of Job Nable [Na1]- [Na3].
In these examples, it is interesting that the ⋆-products, in some how as explained in these papers, involved the Maslov indices and Monodromy Theorem.

Compact groups
We refer readers to the works of C. Moreno [Mo].

Algebraic Noncommutative Chern Characters
Let G be a compact group, HP * (C * (G)) the periodic cyclic homology introduced in §2.
Lemma 28 Let {I N } N∈N be the above defined collection of ideals in C * (G). Then where T is the fixed maximal torus in G.
First note that the algebraic K-theory of C*-algebras has the stability property K * (A ⊗ M n (C)) ∼ = K * (C(T)). Hence, by Pontryagin duality. J. Cuntz and D. Quillen [CQ] defined the so called X-complexes of C-algebras and then used some ideas of Fedosov product to define algebraic Chern characters. We now briefly recall the their definitions. For a (non-commutative) associate C-algebra A, consider the space of even non-commutative differential forms Ω + (A) ∼ = RA, equipped with the Fedosov product ω 1 • ω 2 := ω 1 ω 2 − (−1) |ω 1 | dω 1 dω 2 , see [CQ]. Consider also the ideal IA := ⊕ k≥1 Ω 2k (A). It is easy to see that RA/IA ∼ = A and that RA admits the universal property that any based linear map ρ : A → M can be uniquely extended to a derivation D : RA → M . The derivations D : RA → M bijectively correspond to lifting homomorphisms from RA to the semi-direct product RA ⊕ M , which also bijectively correspond to linear mapρ :Ā = A/C → M given by a ∈Ā → D(ρa).
We quote the second main result of J. Cuntz and D. Quillen ( [CQ], Thm2), namely: We now apply this machinery to our case. First we have the is a lifting of e to an idempotent matrix in M n ( RA). Then the map [e] → tr(ẽ) defines the map K0 → H 0 (X( RA)) = HP 0 (A). To an element g ∈ GL n (A) one associates an element p ∈ GL( RA) and to the element g −1 an element q ∈ GL n ( RA) then put x = 1 − qp, and y = 1 − pq.
And finally, to each class [g] ∈ GL n (A) one associates Theorem 31 Let G be a compact group and T a fixed maximal compact torus of G. Then, the Chern character ch alg : K * (C * (G)) → HP * (C * (G)) is an isomorphism, which can be identified with the classical Chern character ch : K * (C(T)) → HP * (C(T)) which is also an isomorphism.