0. Representation Theory — inside problems and outside interactions
1. finding smallest objects (simple Lie algebras/homogeneous spaces/irreducible reps)
2. building up/decompositions
   1. Induction ... e.g. global analysis on homogeneous spaces
   2. Restriction ... e.g. tensor product
3. Examples of classical analysis problems interpreted as special cases of the general problem of irreducible decompositions

The aim of the second lecture is to get a flavor of multiplicity-free representations from many examples. Rigorous definitions of basic notations used in the second lecture will be explained in the course of lectures.

• Adjoint map (conjugation map)
• Elementary results for operator-valued reproducing kernels in the most general setup:
  X : set
  V : topological vector space, reflexive and complete (e.g. Hilbert space)
  V : vector bundle over X with fiber V
• Scalar valued case
• Example of the Bergman kernel
• Reproducing formula of the reproducing kernel

• Expression of reproducing kernel by o.n.b.
• Example of the Bergman kernel
• Uniqueness theorem

We continue the abstract theorem for operator valued reproducing kernels on the vector bundle over a set. We will bring geometric structure into this setting later.

• Definition (operator valued positive definite kernel)
• Theorem (Reconstruction of Hilbert space)
  This basic theorem establishes the one-to-one correspondence between the space of operator valued positive definite kernels and Hilbert spaces realized in the space of sections for vector bundles.
  • Step 1. Construction of sufficiently many sections
  • Step 2. Definition of sesqui-linear forms by positive definite kernel
  • Step 3. Proof of positivity of sesqui-linear form
  • Step 4. Realization of the completion of Hilbert spaces in the space of sections
  • Step 5. Continuity of point evaluation
  • Step 6. Coincidence of the given positive definte kernel with the operator valued reproducing kernel

1. Explicit estimate of the continuity of the point evaluation map (Hermitian vector bundle cases)
2. Hilbert spaces realized in the space of continuous sections
3. Hartogs theorem for several complex variables
4. Almost complex structure, conjugate complex manifolds, and anti-holomorphic maps
5. Reproducing kernels for Hilbert spaces realized in the space of holomorphic sections

Goal of this section: Give a proof of the following theorem:

Theorem 2.1: Holomorphically induced representation preserves irreducibility.

Two classically known applications:

1. Borel-Weil construction of all irreducible finite dimensional representations of compact Lie groups
   (a simpler proof for the Borel-Weil theorem)
   (algebraic counterpart = the Cartan-Weyl highest weight theory (e.g. [Hu])
2. geometric construction of all irreducible unitary highest weight modules
   (classified by Jakobsen [J1, J2] and Enright-Howe-Wallach [EHW] independently)

This is a subtle result in the sense that similar statements fail in general for L² induced representations, cohomologically induced representations, etc.

The above theorem may be interpreted as the propagation theorem of "irreducibility" from fibers to sections under the assumption that the group acts transitively on the base space.

Theorem 2.1 will be a prototype of our multiplicity-free theorems in the general setting where the group action on X is far from being transitive but is still "visible".

2.1	Holomorphic and anti-holomorphic vector bundle
2.2	Characterization of Hilbert spaces realized in holomorphic sections by means of operator-valued reproducing kernels (Theorem 2.2)

References:

Earier direction: ("inverse propagation results")
G-actions on the space of sections → G_x action on fivers

Today's setting: G-equivariant vector bundle V → X.

Today's goal: Prove the "inverse propagation theorem" on unitarity.

Theorem (characterization of unitary representation in terms of operator valued reproducing kernels).
For the Hilbert space H realized in the space of sections Γ(X, V), the following two conditions are equivalent:

1. G acts on H as a unitary representation.
2. The reproducing kernel is invariant under the diagonal G-action.

Definition. Equivalent definition of the effectiveness the realization of Hilbert space

References

[K]	T. Kobayashi, Propagation of multiplicity-free property for holomorphic vector bundles, math.RT/0607004.
[V]	D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), 141-187.
[W]	N. R. Wallach, Real reductive groups. I, II, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1988, 1992.

1. Casselman's subrepresentation theorem into non-unitary principal representations.
   Any irreducible unitary representation (or more generally, admissible represenations on complete locally convex topological spaces) can be realized as subrepresentations of hyperfunction valued principal series representations.
   Two proofs:
   (1-a) Usual algebraic proof based on the Jacque functor ([W]).
   (1-b) Micro-local analytic proof (due to Kashiwara) using boundary value maps [KKK] and the method of the proof of Helgason's conjecture (No literature, see [Progr 05, page 142] for a brief sketch of the ideas).
   Point: "Inverse propagation" of unitarizability fails.
   Reason: Point evaluation maps for hyperfunctions do not make sense.
2. Any irreducible admissible representation can be realized as subrepresentations of analytic sections for equivariant bundles over Riemannian symmetric spaces. (part of the proof for (1-b))
   "Inverse propagation theorem of unitarizability" is obvious.
   Point: Point evaluation maps are continuous.
3. Vogan's unitarizability theorem on G/L (L is the centralizer of a torus).
   Complex structure on G/L will be discussed soon.
   Under certain positivity, unitarity of fibers <-> unitarizability of sections.
   Proof: Algebraic proof by Vogan and Wallach based on Zuckerman's derived functor modules [V,W].
   Geometric result: Schmid, H.-W. Wong [Wo] on Dolbaulet cohomologies. For a survey for the both geometric and algebraic results, see [V87], [K98].
   

Totally real submanifold
Definition, examples

Invariant sections
Lemma For transitive base space, G-invariant sections <-> G_x invariant elements on the fiber

Example (de Rham cohomologies on compact symmetric spaces)

Theorem (Propagation of irreduciblity theorem)

Assumption: V → X: G-equivariant holomorphic bundle G_x on V_x is irreducible
Conclusion: Any unitary representations realized in the space of holomorphic sections Γ(X, V) is irreducible or zero (consequently, unique if it is non-zero).

Proof: All necessary results are already prepared.

References

[K98]	T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Translations, Series II, Selected Papers on Harmonic Analysis, Groups, and Invariants (K. Nomizu, ed.), vol. 183, Amer. Math. Soc., 1998, pp. 1-31, ISBN 0-8218-0840-0.
[KKK]	M. Kashiwara, T. Kawai and T. Kimura, Foundations of Algebraic Analysis, Princeton Math. Series, 37, Princeton Univ. Press, 1986.
[V]	D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), 141-187.
[V87]	D. A. Vogan, Jr., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, 118, Princeton University Press, Princeton, NJ, 1987.
[W]	N. R. Wallach, Real reductive groups. I, II, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1988, 1992.
[Wo]	H. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal. 129 (1995), 428-454.

The assumption of the last lecture was that the base space (complex manifold) has a transitive biholomorphic group action.
Today's goal: to analize this situation in terms of Lie algebras.

Remark 1. As an algebraic counterpart of 10th lecture, this condition leads us to a generalization of Verma modules for (not necessarily reductive) Lie algebras.

Remark 2. Later, we shall deal with a more general situation (foliations in complex manifolds)

Theorem 3.1. Characterization of G-invariant almost complex structure on G/H

Theorem 3.2. Characterization of G-invariant complex structures on G/H

Example 1. Almost complex structure on S⁶ with G₂ symmetry by using root diagrams of G₂

Example 2. (non)-existence of SL(2,R) invariant complex structure on hyperboloids, light cones
classifications of almost complex structures — module decompotions

cf. Geometric quantization
J²=id: ruled surface → para-Hermitian structure → (generalized) principal series represenations
J²=-id: Cauchy-Riemann → complex structure → discrete series, Zuckerman's derived functors

(continued)

Today's goal: to give a simple proof of classical results on G-invariant (almost) complex structures on homogeneous spaces

General scheme on homogeneous spaces:

2) <= V ≡ G ×_H V := (G × V)/\tilde H associated to the principal H-bundle G → G/H and H action on V

Lemma 3.3 Sections for V → X ⇔ map from G to V with functional equations

Proof of Theorem 3.1 for G-invariant almost complex structure
Use (3)

Proof of Theorem 3.2 for G-invariant complex structure
Use Ehresman, Newlander-Nirenberg for integrability of J + Writing the Lie algebra structures in the scheme of Lemma 3.3

Brief summary of the ideas of Theorems 3.1 and 3.2 (proof given last time) for G-equivariant complex structure on G/H.

Notion of (b,H)-module due to Lepowsky

Refinement of the following one-to-one correspondences
G-equivariant bundle over G/H ⇔ H-modules

Theorem 3.3. G-equivariant holomorphic bundle over G/H ⇔ (b^-,H)-modules

Griffiths-Schmid's formula (as a corollary)

(continued)

Proof of Key lemma: Definition of \bar{\mathfrak b}-action on fibers from \bar{∂}-operator,

Griffith-Schmid's lemma

Proof for sufficiency in Theorem 3.3

Step 1.
Local equivalence between "generalized Borel embedding" and "splitting of tangent bundle"

Step 2.
Construction of almost complex structure on the total space V = G ×_H V from (b^-, H)-module

Step 3.
Integrability (continued)

Final step of the proof.

Integrability of almost complex structure.

Point: avoid some global obstructions and some technical problems (G_C/B may not be Hausdorff if B is not closed, H may have connected compotents, G is not a subgroup of G_C, μ may not be lifted to B, etc.)

\mathfrak g: Lie algebra/R (not necessarily reductive)

Definition. Z is elliptic if ad(Z) is a semisimple endomorphism whose eigenvalues are all pure imaginary.

b₊ := sum of eigenspaces with non-negative eiganvalues of ad(Z)/\sqrt{-1} (generalized parabolic subalgebra)

H := normalizer of b₊ in G.

As a special case of Theorem 3.3, we get

Theorem 4.4.

1. G/H carries a G-invariant complex structure.
2. The adjoint orbit O_Z := Ad(G) Z is a covering of G/H, and thus carries a G-invariant complex structure.

Example. U(n) generalized flag variety

Example. GL(2n,R)/TⁿRⁿ, GL(2n,R)/GL(n,C)

Equivalent definitions of reductive Lie algebras

Cartan decomposition g = k + p

Elliptic elements ↔ conjugate to \mathfrak k
Hyperbolic elements ↔ conjugate to \mathfrak p

Elliptic orbits of real reductive Lie groups include

1. full/partial flag varieties
2. G/H where H is a fundamental Cartan subgroup
3. Hermitian symmetric spaces
4. 1/2 Kähler symmetric spaces (a la M. Berger [B])

Today's lecture forcuses on the most generic case, namely, on (2). On Friday, I will focus on the most degenerate case, namely on (3) and (4).

See [H], [F] for an exlanation of the general correspondence due to E. Cartan
Symmetric pair (algebra) ↔ Affine symmetric space (geometry)

The infinitesimal classification of semisimple symmetric pairs was accomplished in [B] (see also [Hm]).

• Characterization of Cartan involutions among all involutions (conjugation theorem ... a fixed point theorem of compact isometry groups)
• The stabilizer of a generic elliptic orbit is a fundamental Cartan subgroup [V]
• The stabilizer of a generic hyperbolic orbit is abelian iff \mathfrak g is quasi-split (e.g. o(p,q) where |p-q| = 0,1,2)
• There are finitely many conjugacy classes of Cartan subalgebras. Fundamental Cartan subalgebras, and maximally split Cartan subalgebras are two extremal cases. The classification was accomplished in Sugiura [S].

References

1. Symmetric space

Two equivalent definition:
(i) (geometry) affine symmetric space
(ii) (group) G/H H is an open subgroup of an involutive group automorphisms of G

E.g. GL(n,R)/O(n), GL(n,R)/O(p,q), GL(n,R)/GL(p,R)×GL(q,R), GL(2m,R)/GL(n,C)

2. Reductive symmetric speces with complex structure

1. G_C/K_C (defined by holomorphic involution)
2. Hermitian symmetric space
3. 1/2 Kähler symmetric space

3.

Generic elliptic orbit ... G/H where H is a fundamental Cartan subgroup
Most degenerate elliptic orbit ... G/H is either (2) or (3)
             equivalently, n₊ is abelian

(note: g=h+q, h is a subalgebra, and q is an h-stable complementary subspace. (g,h) is a symmetric pair iff [q,q] ⊂ h)

Then (2) ⇔ G/H has a bounded realization in n₊.

4. Example: U(p,q) case

Classical bounded domain
Borel embedding, Bruhat cell, Cartan decomposistion
U(p,q) case

Example of 1/2 Kähler symmetric space

U(p,q)/U(p₁,q₁) × U(p₂,q₂)
(in particular, p₁=1, q₁=0 case)

Characterization of Hermitian Lie algebras

Fact. Let X be a complete, connected, simply connected Riemannian space with non-positive sectional curvature. Then, any compact group consisting of isometries of X has a common fixed point.

Theorem. For any automorphism σ of a semisimple Lie algebra of finite order, there is a Cartan involution θ such that σ θ = θ σ.
(see [S] for an alternative proof when σ² = 1)

Theorem. G: a Lie group with at most finitely many connected components.

1. Maximal compact subgroups K are conjugate to each other.
2. G is homotopic to K.

Fact. For a connected semisimple Lie group, (i) and (ii) are equivalent:

1. Center of G is finite.
2. The analytic subgroup K is compact, where \mathfrak g= \mathfrak k + \mathfrak p is a Cartan decomposition.

References

1.

During the last lecture on Wednesday, one of the statements that I gave in answer to a question about faithful representations needs an assumption that the group is reductive.

Today, I begin with a general framework, and then give a correct statement (with a counterexample in the general case) with detailed proof.

2. Structure of symmetric spaces with indefinite Kähler metric

Definition (universal complexification).

Theorem. Universal complexification of a Lie group exists. It is unique in a obvious sense.

Proof. Construct it from simply connected, complex Lie group.

Remark. The Lie algebra of universal complexification is not determined by the Lie algebra of the original group G, but depends on the topology of G.

Lemmas on almost linear group and on linear groups.

Corollary. Necessary condition for G to have faithful representations.

Nilpotent case (counterexample)

Semisimple case: the necessary condition above is also sufficient.

3. Elliptic orbits which are also symmetric spaces

SO(2p,2q)/U(p,q), GL(2n,R)/GL(n,C), Sp(n,R)/U(p,q)

4. Involutions of holomophic type, anti-holomophic type

See [K, Tables 3.4.1 and 3.4.2] for the classification.

Reference

Recall:

1. Propagation theorem (irreducibility)
2. Complex structrue on X = G/H
3. Holomorphic bundle structure on G ×_H V

The evalutatin map at the origin leads us the the bijection between complex holomorphic objects and Lie algebra representations.

Theorem (holomorphic Frobenius theorem).

Hom_G(\mathcal H, O(X, \mathcal V) \simeq Hom_{b^-,H}(\mathcal H, V).

Point is that the left-hand side can be analyzed by complex analytic methods (e.g. reproducing kernels), while the right-hand side can be analyzed by algebraic methods (e.g. generalized Verma modules).

In the case of elliptic orbits, any G-equivariant vector bundle can be extended to a G-equivariant holomorphic vector bundle in a standard way (but such an extension is not unique).

Corollary. In the above setting,

Hom_G(\mathcal H, O(X, \mathcal V) \simeq Hom_H(\mathcal H/d\pi(n_-)(H), V).

Reproducing kernels at one point (review)
Stratification of the totality of positive definite matrices.

Theorem (propagation theorem of multiplicity-free property: transitive case)

Two mutually non-equivalent representations of H give rise to mutually non-equivalent representations of G (if they are not zero).

Summary: Transitive case (usual holomophic induction)

Smooth vectors of continuous representations on complete locally convex top. sp.

Density of Garding space
Proof: use a delta converging sequence.

Distribution vectors and the action of Lie algebras

Gelfand triple (H^∞ ⊂ H \simeq H^* ⊂ H^-∞)

Extension of "holomorphic Frobenius theorem" from finite dimensional representations to infinite dimensional representations defined on complete locally convex topogical space.

Special case: unitary representations on holomorphic vector bundles on elliptic orbits

G : general Lie group
H/Z_G compact

Theorem. There is a natural bijection:

\hat{H}		\hat{G}
∪		∪
\hat{H}(G)	\simeq	\hat{G}(n+)

Proof.
well-definedness:
  (a) propagation theoreom of irreduciblity
  (b) inverse propagation theorem of unitarity
injectivity:
  (c) propagation theorem of multiplicity-freeness (special case)
surjectivity:
  * Use again (a) and (c).

Example 1. G compact
 \hat{G} = \hat{G}(n₊).

1.1. G connected, Z generic
  → Borel-Weil theory
  ← Cartan-Weyl highest weight theory

1.2. G connected, Z general
  holomorphic induction by stages
  Example 1.2: criterion for the existence of non-zero holomorphic sections for holomorphic vector bundles over Grassmannian mfd

1.3. G disconnected case
  Cartan-Weyl theory for disconnected compact groups
  Example G = O(2n).

Example 2. G: simple non-compact, G/K Hermitian symmetric space (continued)

Example 2. G: non-compact, simple Lie group, G/K Hermitian

Wallach set,
Weil representation,

highest weight modules of scalar type
holomorphic discrete series representations

sp(n, R) (n = 5 case)

Some of features of these singular representations from analytic, geometric, and algebraic viewpoints will be discussed.

Weighted Bergman space

Theorem. (L² ∩ O)(D, m dz) is closed in L²(D, m dz)

Unitary highest weight modules for Sp(n,R)

Example: Upper half plane H with the measure y^a-2 dx dy

Theorem:

1. L^(H, y^a-2 dx dy) ∩ O(H) is non-zero if a > 1.
2. Its reproducing kernel is a scalar multiple of (w₁ - \bar{w₂})^-a

Remark: a = 1 is singular for (1), but analytically continued in (2) until a > 0.

Hilbert Space :

a = 2 :	Bergman space
a > 1 :	weighted Bergman space
a = 1 :	Hardy Space
a = 0 :	one dimensional Hilbert space

Classical fact on Hardy space: Fourier transform of L² functions on half line.

Twisted pull-back by biholomorphic transforms and gauge transforms.

When does the twisted pull-back gives a unitary map between two weighted Bergman space?
Its explicit condition and the formulas.

Example of unbounded domains with non-trivial, and trivial Bergman space

Definition of Fock spaces = weighted Bergman space with Gaussian kernel. Its reproducing kernel = e^(x,w)

Definition: Geometric Bergman space B²_X

Theorem: For a complex manifold Aut(X) acts unitarily on the Hilbert space B²_X.

Bergman kernel form, Bergman metric

Setting: X = G/H ⊂ G_C/B_- is a Borel embedding.
Assume G is the fixed point group of an anti-holomorphic involution of G_C.
V: standard holomorphic vector bundle associaged to an irreducible representation τ of H on V.

ψ: G B_- → H_C ,   g h n \mapsto h^-1 σ(h)

Theorem: K(x,x) = c τ ψ(x).

Examples for the unified treatment of U(n) and U(p,q)

some formula of the determinant
geometric Bergman space
(limit of) discrete series representation
first reduction point
singular highest weight modules
Shimura's differential operators and reproducing kernels

Two proofs for the meromorphic continuation of the Riesz potential.

Thereom (Bernstein-Gelfand, Atiyah). Let P(x) be a polynomial. Then, P^λ₊ extends meromorphically as a distribution.

Examples of Sato-Bernstein b function.

Shimura's differential operators on Hermitian symmetric domains.

Fourier transforms of relative invariants (Gindikin's Gamma function).

References:

[A]	M. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145-150.
[B]	J. Bernnstein, Analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Prilozen. 6 (1972), 26-40.
[G]	S. G. Gindikin, Analysis in homogeneous domains (in Russian), Uspekhi Math Nauk 19 (1964), 3-92.
[Sa]	M. Sato, Theory of prehomogeneous vector spaces (algebraic part) — the English translation of Sato's lecture from Shintani's note, Nagoya Math. J. 120 (1990), 1-34.
[Se]	H. Sekiguchi, The Penrose transform for certain non-compact homogeneous manifolds of U(n,n), J. Math. Sci. Univ. Tokyo 3 (1996), 655-697.
[Sh]	S. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math. 88 (1984), 463-488.

Point: Analysis on manifolds with infinitely many orbits
Idea: Control transversal directions to group orbits

Let V → X be a holomorphic Hermitian vector bundle on which a group G acts as automorhphisms.

Theorem: Assume G_x action on the fiber V_x is mutiplicity-free. Suppose there exists an anti-holomorphic bundle auto σ s.t.

1. (base space) σ(x) = g x (for some g ∈ G)
2. (compatibility) L_g¹ σ_x stablizes G_x irreducible compotents in V_x. Then any unitary representation realized in O(X, V) is multiplicity-free.

Reference:

Complete the proof of propagation theorem.

---

Direct integral of unitary representations (definition and examples)

Decompostion of unitary representations into irreducible representations (Mautner-Teleman)

Multiplicity-free representations (equivalent definition)

Definition: Lift of an anti-holomorphic automorphism to holomorphic bundles.

Definition: Lift of an anti-holomorphic automorphism to biholomorphic transformations groups.

Example 1. Complex Affine transformation group.

Example 2. Grassmannian manifolds, Siegel upper half plane.

Example 3. Elliptic orbits.

Definition: Visible action on a complex manifold. Slice.

Examples. 1 dimensional biholomorphic transforms on the unit disk.

Proposition: Conditions for Preserving/Stabilizing G-orbits on X (understanding of a global feature of visible actions).

Definition [K05]: A holomorphic action of a group G on a complex manifold is visible if (G X^σ)^o is not empty for some anti-holomorphic map σ. A slice is a submanifold S of X^σ such that (G S)^o is not empty.

Remark. In [K05], this is called "strongly visible", while infinitesimally visible action (see below for the definition is called "visible".

Theorem ([K07]). X = Hermitian symmetric space. Then, the G-action on X is visible if (Isom(X), G) symmetric pair.

E.g. X = Siegel upper half space, G = U(n), GL(n,R), U(p,q), Sp(p,R) × Sp(q,R). p + q = n, dim S = n, n, n. min(p,q), respectively.)