MultiplicityFree Representations: Complex Geometric Methods in
Representation Theory.
Harvard University, USA, spring term (JanuaryMay) 2008.
Catalog Number: 0818
Mondays, Wednesdays & Fridays, 11:0012:00
Exam Group: 4
Room: Science Center 310
Office hour: 14:0015:00 Wed
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Summary
Explanation of complex geometric methods such as reproducing kernels and
''visible actions'' for the study of infinite dimensional representations.
From this viewpoint, various examples of multiplicityfree representations
of Lie groups will be discussed.

This course gives an introduction to infinite dimensional representations
of Lie groups with emphasis on geometric and analytic methods, followed by
a focus on developing current research topics.
Possible topics include:
 operator valued reproducing kernels,
 unitary representation theory of real reductive Lie groups,
 classical examples of multiplicityfree branching rules,
 structure of semisimple symmetric spaces,
 BorelWeil theorem — revisited,
 a generalized HuaKostantSchmid formula,
 propagation of irreduciblity, unitarity, and multiplicityfree property,
 visible actions on complex manifolds, multiplicityfree spaces,
 some open problems.
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First week: Warming up and course guide
Jan 30 (Wed) Analysis and synthesis — decomposition into irreducibles
Feb 1 (Fri) Examples of multiplityfree representations, and "visible actions"
Lecture 1 (Jan 30): Analysis and synthesis — smallest objects and decompositions
 Representation Theory — inside problems and outside interactions
 finding smallest objects (simple Lie algebras/homogeneous
spaces/irreducible reps)
 building up/decompositions
 Induction ... e.g. global analysis on homogeneous spaces
 Restriction ... e.g. tensor product
 Examples of classical analysis problems
interpreted as special cases of the general problem of irreducible decompositions
Reference:
Today's lecture was based on §0 of the following paper:
T. Kobayashi,
Theory of discrete decomposable branching laws of unitary representations
of semisimple Lie groups and some applications,
Sugaku Expositions
18 (2005), Amer. Math. Soc., 137.
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Lecture 2 (Feb 1): Examples of multiplityfree representations, and "visible actions"
The aim of the second lecture is to get a flavor of multiplicityfree
representations from many examples.
Rigorous definitions of basic notations used in the second lecture
will be explained in the course of lectures.
[ Lecture slides (pdf) ]
* Meaning of multiplicityfree representations
Multiplicityfree representations give the "canonical" decomposition.
It is just the same as digging the rock out of the earth.
"Canonical decompositions" used in various places of mathematics
(e.g. Fourier series, Taylor expansions, expansions by spherical functions, etc.) are often explained by multiplicityfree representations as the
underlying algebraic structure.
Conversely, we may expect that multiplicityfree representations would
yield natural and useful decompositions from unknown objects into wellunderstood objects.
* Examples of multiplicityfree representations
 FourierPeterWeyl theorem
 Kac's multiplicityfree spaces
 GLGL duality
 HuaKostantSchmid formula and its generalization
 Pieri's rule
 Tensor product representations that are not multiplicityfree
 Classification of multiplicityfree tensor product reprsentations
 Plancherel formula for Riemannian symmetric space and semisimple symmetric spaces
* Visible actions (complex manifoles),
coisotropic actions (symplectic manifold), polar actions (Riemannian manifolds)
Most of the examples of the second lecture were taken from an expository paper:
T. Kobayashi, Multiplicityfree representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), 497549, special issue commemorating the fortieth anniversary of the founding of RIMS.
For individual topics, I list a few number of closely related references:
J. Faraut and E. G. F. Thomas, Invariant Hilbert spaces of holomorphic functions, J. Lie Theory 9 (1999), 383402.
I. M. Gel'fand, Spherical functions on symmetric spaces, Dokl. Akad. Nauk SSSR 70 (1950), 58.
V. Guillemin and S. Sternberg, Multiplicityfree spaces, J. Differential Geom. 19 (1984), 3156.
T. Kobayashi, W. Schmid, and J.H. Yang,
Representatin Theory and Automorphic Forms,
Progress in Mathematics 255, Birkhäuser, 2008, ISBN13: 9780817646462.
A. Sasaki, Visible actions on multiplicityfree spaces, to be submitted as a Ph.D. thesis of Waseda University (2008, March).
W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 6180.
J. R. Stembridge, Multiplicityfree products of Schur functions, Ann. Comb. 5 (2001), 113121.
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Section 1. Operator valued reproducing kernel
Goal of this week:
To develop the general theory of operator valued reproducing kernel
in the framework of vector bundles.
Special cases include the classic theory of reproducing kernels.
Lecture 3 (Feb 4): Operator valued reproducing kernel  1
1.1 
Comparison of the Cauchy kernel and the Bergman kernel 
 Point: the former is independent of the domain, whereas
the latter depends on (in fact, characterizes) the domain. 
1.2 
Elementary linear algebra and functional analysis 
 Basic properties of semipositive sesquilinear forms.
Conjugate complex spaces, dual spaces, and conjugate duals.
Riesz's representatin theorem. 
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Lecture 4 (Feb 6): Operator valued reproducing kernel  2
 Adjoint map (conjugation map)
 Elementary results for operatorvalued 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
References:
K.H. Neeb, Holomorphy and convexity in Lie theory,
de Gruyter Expositions in Mathematics, 28.
Walter de Gruyter & Co., Berlin, 2000. xxii+778 pp.
ISBN: 3110156695
T. Kobayashi, Propagation of multiplicityfree property for holomorphic vector bundles, math.RT/0607004.
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Lecture 5 (Feb 8): Operator valued reproducing kernel  3
 Expression of reproducing kernel by o.n.b.
 Example of the Bergman kernel
 Uniqueness theorem
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Lecture 6 (Feb 11): Operator valued positive definite kernels
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 onetoone 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 sesquilinear forms by positive definite kernel
 Step 3. Proof of positivity of sesquilinear 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
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Lecture 7 (Feb 13): Hilbert spaces of continuous/holomorphic sections
The last lecture dealt with the broadest setting
for the onetoone correspondence
between the operatorvalued positive definite kernels
and Hlibert spaces realized in the space of sections
for vector bundles.
From now, we will specify the settings,
and give a refinement of the above correspondence.
 Explicit estimate of the continuity of the point evaluation map
(Hermitian vector bundle cases)
 Hilbert spaces realized in the space of continuous sections
 Hartogs theorem for several complex variables
 Almost complex structure, conjugate complex manifolds, and
antiholomorphic maps
 Reproducing kernels for Hilbert spaces realized in the space of
holomorphic sections
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Lecture 8 (Feb 15):
Section 2. Propagation Theorem of irreducible representations
Goal of this section: Give a proof of the following theorem:
Theorem 2.1: Holomorphically induced representation preserves irreducibility.
Two classically known applications:
 BorelWeil construction of all irreducible finite dimensional
representations of compact Lie groups
(a simpler proof for the BorelWeil theorem)
(algebraic counterpart = the CartanWeyl highest weight theory (e.g. [Hu])
 geometric construction of all irreducible unitary highest weight modules
(classified by Jakobsen [J1, J2] and EnrightHoweWallach [EHW] independently)
This is a subtle result in the sense that similar statements fail in general for L^{2} 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 multiplicityfree theorems
in the general setting where the group action on X is far from being
transitive but is still "visible".
2.1 
Holomorphic and antiholomorphic vector bundle 
2.2 
Characterization of Hilbert spaces realized in holomorphic sections
by means of operatorvalued reproducing kernels (Theorem 2.2) 
References:

S. Kobayashi, Irreducibility of certain unitary representations, J. Math. Soc. Japan 20 (1968), 638642. 
[J1] 
H. P. Jakobsen, The last possible place of unitarity for certain highest weight modules, Math. Ann. 256 (1981), 439447. 
[J2] 
H. P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385412. 
[EHW] 
T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97143. 
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Lecture 9 (Feb 20): Unitary representation realized in Γ(X, V)
Earier direction: ("inverse propagation results")
Gactions on the space of sections → G_{x} action on fivers
Today's setting: Gequivariant 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:
 G acts on H as a unitary representation.
 The reproducing kernel is invariant under the diagonal Gaction.
Definition. Equivalent definition of the effectiveness the realization of Hilbert space
{x: K(x,x)=0} = \emptyset <=> {x: f(x)=0 (∀x ∈ X)} = \emptyset
Theorem (inverse propagation theorem of unitarity).
Let V → X is a Gequivariant bundle with irreducible isotoropy group actions.
If a unitary representation is realized effectively on Γ(X, V),
then V → X carries a Ginvariant Hermitian bundle structure.
In particular G_{x} acts unitarily on V_{x} for every x.
Remark.
 No assumption on X in the above theorem.
 Casselman's subrepresentation theorem into nonunitary
principal representations (see [W]).
 Vogan's theorem for the unitarizability of Zuckerman's
derived functor modules (see [V]).
(continued)
References
[K] 
T. Kobayashi, Propagation of multiplicityfree 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), 141187. 
[W] 
N. R. Wallach, Real reductive groups. I, II, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1988, 1992. 
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Lecture 10 (Feb 22): Propagation theorem for irreducibility
Remarks on the "inverse propagation of unitarizability theorem in the 9th
lectures) for real reductive groups:
 Casselman's subrepresentation theorem into nonunitary 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:
(1a) Usual algebraic proof based on the Jacque functor ([W]).
(1b) Microlocal 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.
 Any irreducible admissible representation can be realized as
subrepresentations of analytic sections for equivariant bundles over Riemannian symmetric spaces.
(part of the proof for (1b))
"Inverse propagation theorem of unitarizability" is obvious.
Point: Point evaluation maps are continuous.
 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, Ginvariant sections <> G_{x} invariant elements on the fiber
Example (de Rham cohomologies on compact symmetric spaces)
Theorem (Propagation of irreduciblity theorem)
Assumption: V → X: Gequivariant 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 nonzero).
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. 131, ISBN 0821808400. 
[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), 141187. 
[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), 428454. 
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Lecture 11 (Feb 25): Complex manifolds with transitive holomorphic actions
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 Ginvariant almost complex structure on G/H
Theorem 3.2. Characterization of Ginvariant complex structures on G/H
Example 1. Almost complex structure on S^{6} with G_{2} symmetry by using root diagrams of G_{2}
Nonexistence of homogeneous complex structure on S^{6}
cf. Longstanding problem: whether S^{6} admits a complex structure or not
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
J^{2}=id: ruled surface → paraHermitian structure → (generalized) principal series represenations
J^{2}=id: CauchyRiemann → complex structure → discrete series, Zuckerman's derived functors
(continued)
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Lecture 12 (Feb 27): Complex manifolds with transitive holomorphic actions
Today's goal: to give a simple proof of classical results
on Ginvariant (almost) complex structures on homogeneous spaces
General scheme on homogeneous spaces:
1) 
Gtransitive action on X with base space o ⇔ closed subgroup of G 
2) 
Gequivariant fiber bundle ⇔ Haction on topological space 
2)' 
Gequivariant vector bundle ⇔ representation of H 
3) 
Ginvariant sections ⇔ Hinvariant elements in fibers 
2) <= V ≡ G ×_{H} V := (G × V)/\tilde H associated to the principal Hbundle 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 Ginvariant almost complex structure
Use (3)
Proof of Theorem 3.2 for Ginvariant complex structure
Use Ehresman, NewlanderNirenberg for integrability of J
+ Writing the Lie algebra structures in the scheme of Lemma 3.3
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Lecture 13 (Feb 29): Complex manifolds with transitive holomorphic actions
Brief summary of the ideas of Theorems 3.1 and 3.2 (proof given last time)
for Gequivariant complex structure on G/H.
Notion of (b,H)module due to Lepowsky
Refinement of the following onetoone correspondences
Gequivariant bundle over G/H ⇔ Hmodules
Theorem 3.3. Gequivariant holomorphic bundle over G/H
⇔ (b^{},H)modules
Step 1. 
Pull back to the principal bundle G → G/H.
Write the \bar ∂ operator by means of the opearor \bar D
acting on Vvalued functions on G. 
Step 2. 
Key formula: difference between \bar D and right differentiation.
(continued) 
GriffithsSchmid's formula (as a corollary)
(continued)
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Lecture 14 (Mar 3): Characterization of holomorphic vector bundles with transitive base space actions 1
Proof of Key lemma: Definition of \bar{\mathfrak b}action on fibers from
\bar{∂}operator,
GriffithSchmid's lemma
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Lecture 15 (Mar 5): Characterization of holomorphic vector bundles with transitive base space actions 2
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)
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Lecture 16 (Mar 7): Characterization of holomorphic vector bundles with transitive base space actions 3
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.)
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Lecture 17 (Mar 10): Elliptic orbts
\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 nonnegative 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.
 G/H carries a Ginvariant complex structure.
 The adjoint orbit O_{Z} := Ad(G) Z is a covering of G/H, and thus carries a Ginvariant complex structure.
Example. U(n) generalized flag variety
Example. GL(2n,R)/T^{n}R^{n}, GL(2n,R)/GL(n,C)
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Lecture 18 (Mar 12): Elliptic orbits for real reductive Lie groups 1
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
 full/partial flag varieties
 G/H where H is a fundamental Cartan subgroup
 Hermitian symmetric spaces
 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 quasisplit (e.g. o(p,q) where pq = 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
[B] 
M. Berger, Les espaces symetriques noncompacts. (French)
Ann. Sci. Ecole Norm. Sup. (3) 74 (1957) 85177. 
[F] 
M. FlenstedJensen, Analysis on nonRiemannian symmetric spaces. CBMS Regional Conference Series in Mathematics, 61. AMS, 1986. x+77 pp. ISBN: 0821807110. 
[H] 
S. Helgason, Differential geometry, Lie groups, and symmetric spaces.
Corrected reprint of the 1978 original. Graduate Studies in Mathematics,
34. American Mathematical Society, Providence, RI, 2001. xxvi+641 pp. 
[Hm] 
A. Helminck, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces.
Adv. in Math. 71 (1988), 2191. 
[S] 
M. Sugiura, Conjugate classes of Cartan subalgebras in real semisimple Lie algebras. J. Math. Soc. Japan 11 (1959) 374434. 
[V] 
D. Vogan, Jr., The algebraic structure of the representation of
semisimple Lie groups. I.
Ann. of Math. (2) 109 (1979), 160. 
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Lecture 19 (Mar 14): Most degenerate elliptic orbits
— Hermitian symmetric spaces, and 1/2 Kähler symmetric spaces
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
 G_{C}/K_{C} (defined by holomorphic involution)
 Hermitian symmetric space
 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 hstable 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
(continued)
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Lecture 20 (Mar 17): Examples of most degenerate elliptic orbits
Classical bounded domain
Borel embedding, Bruhat cell, Cartan decomposistion
U(p,q) case
Example of 1/2 Kähler symmetric space
U(p,q)/U(p_{1},q_{1}) × U(p_{2},q_{2})
(in particular, p_{1}=1, q_{1}=0 case)
Characterization of Hermitian Lie algebras
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Lecture 21 (Mar 19): Fixed point theorem and structural results of Lie groups
Fact. Let X be a complete, connected, simply connected Riemannian space with nonpositive 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 σ^{2} = 1)
Theorem. G: a Lie group with at most finitely many connected components.
 Maximal compact subgroups K are conjugate to each other.
 G is homotopic to K.
Fact. For a connected semisimple Lie group, (i) and (ii) are equivalent:
 Center of G is finite.
 The analytic subgroup K is compact, where \mathfrak g= \mathfrak k + \mathfrak p is a Cartan decomposition.
* Lie group with faithful representations (i.e. Lie groups which are
realized as a subgroup of GL(n).)
cf. Mp(n,R) = the metaplectic group = the two fold covering of Sp(n,R).
The universal covering of SL(2,R)
References
[H] 
S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001, Corrected reprint of the 1978 original. 
[S] 
H. Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston Inc., Boston, MA, 1984. 
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Lecture 22 (Mar 21): Universal complexification
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, antiholomophic type
See [K, Tables 3.4.1 and 3.4.2] for the classification.
Reference
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Lecture 23 (Mar 31): Holomorphic Frobenius Theorem
Recall:
 Propagation theorem (irreducibility)
 Complex structrue on X = G/H
 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 lefthand side can be analyzed by complex analytic methods
(e.g. reproducing kernels), while the righthand side can be analyzed
by algebraic methods (e.g. generalized Verma modules).
In the case of elliptic orbits, any Gequivariant vector bundle
can be extended to a Gequivariant 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).
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Lecture 24 (April 2):
Reproducing kernels at one point (review)
Stratification of the totality of positive definite matrices.
Theorem (propagation theorem of multiplicityfree property: transitive case)
Two mutually nonequivalent representations of H give rise to
mutually nonequivalent representations of G (if they are not zero).
Summary: Transitive case (usual holomophic induction)
 Irreducibility propagates (Lecture 10).
 Multiplicityfreeness propagates (today).
(Later, we shall consider the case where G acts on X with infinitely many orbits)
Back to the setting of elliptic orbits (or its quotients by discontinuous groups)
Some algebraic lemmas for the understanding of
 irreducible representations of H such that they give nonzero unitary representations
 irreducible unitary representation of G that come from transitive G complex manifolds
(i.e. holomporphically induced representations of G)
> a special case gives a new proof of the CartanWeyl and the BorelWeil
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Lecture 25 (April 4): Holomorphic Frobenius theorem for intinite dimensional representations
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
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Lecture 26 (April 7): Generalization of CartanWeyl & and BorelWeil theory
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.
welldefinedness:
(a) propagation theoreom of irreduciblity
(b) inverse propagation theorem of unitarity
injectivity:
(c) propagation theorem of multiplicityfreeness (special case)
surjectivity:
* Use again (a) and (c).

Example 1. G compact
\hat{G} = \hat{G}(n_{+}).
1.1. G connected, Z generic
→ BorelWeil theory
← CartanWeyl highest weight theory
1.2. G connected, Z general
holomorphic induction by stages
Example 1.2: criterion for the existence of nonzero holomorphic sections for holomorphic vector bundles over Grassmannian mfd
1.3. G disconnected case
CartanWeyl theory for disconnected compact groups
Example G = O(2n).
Example 2. G: simple noncompact, G/K Hermitian symmetric space (continued)
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Lecture 27 (April 9): Singular highest weight modules
Example 2. G: noncompact, 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.
(L^{2} ∩ O)(D, m dz) is closed in L^{2}(D, m dz)
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Lecture 28 (April 11): Elementary analytic aspect for the continuous part (until the first reduction point)
Unitary highest weight modules for Sp(n,R)
Example: Upper half plane H with the measure y^{a2} dx dy
Theorem:
 L^(H, y^{a2} dx dy) ∩ O(H) is nonzero if a > 1.
 Its reproducing kernel is a scalar multiple of (w_{1}  \bar{w_{2}})^{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 L^{2} functions on half line.
Twisted pullback by biholomorphic transforms and gauge transforms.
When does the twisted pullback gives a unitary map between two weighted
Bergman space?
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Lecture 29 (April 14): Geometric Bergman space
Example of unbounded domains with nontrivial, and trivial Bergman space
Definition of Fock spaces = weighted Bergman space
with Gaussian kernel. Its reproducing kernel = e^{(x,w)}
Definition: Geometric Bergman space B^{2}_{X}
Theorem: For a complex manifold Aut(X) acts unitarily on
the Hilbert space B^{2}_{X}.
Bergman kernel form, Bergman metric
Theorem: If geometric Bergman space is effectively realized
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Cor. For a bounded domain, Aut(X) ⊂ Isom(X).
Examples of Berman metric.
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Lecture 30 (April 16): Explicit formula for reproducing kernels
Setting: X = G/H ⊂ G_{C}/B_{} is a Borel embedding.
Assume G is the fixed point group of an antiholomorphic 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).
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Lecture 31 (April 18): Explicit formula for the reproducing kernel 2
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
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Lecture 32 (April 21): Singular representations and differential equations
Two proofs for the meromorphic continuation of the Riesz potential.
Thereom (BernsteinGelfand, Atiyah). Let P(x) be a polynomial.
Then, P^{λ}_{+} extends meromorphically as a distribution.
Examples of SatoBernstein 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), 145150. 
[B] 
J. Bernnstein, Analytic continuation of generalized functions with
respect to a parameter,
Funkcional. Anal. i Prilozen. 6 (1972), 2640. 
[G] 
S. G. Gindikin, Analysis in homogeneous domains (in Russian), Uspekhi Math Nauk 19 (1964), 392. 
[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), 134. 
[Se] 
H. Sekiguchi, The Penrose transform for certain noncompact
homogeneous manifolds of U(n,n),
J. Math. Sci. Univ. Tokyo 3 (1996), 655697. 
[Sh] 
S. Shimura, On differential operators attached to certain
representations of classical groups,
Invent. Math. 88 (1984), 463488. 
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Lecture 33 (April 23): Propagation theorem of multiplicityfree property I
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 mutiplicityfree.
Suppose there exists an antiholomorphic bundle auto σ s.t.
 (base space) σ(x) = g x (for some g ∈ G)
 (compatibility) L_{g}^{1} σ_{x} stablizes G_{x} irreducible compotents in V_{x}.
Then any unitary representation realized in O(X, V) is
multiplicityfree.
Reference:
T. Kobayashi,
Propagation of multiplicityfree property for holomorphic vector bundles, preprint. math.RT/0607004.
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Lecture 34 (April 25): Propagation theorem of multiplicityfree property II
Complete the proof of propagation theorem.

Direct integral of unitary representations
(definition and examples)
Decompostion of unitary representations into irreducible representations
(MautnerTeleman)
Multiplicityfree representations (equivalent definition)
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Lecture 35 (April 28): Visible action on complex manifolds I
Definition: Lift of an antiholomorphic automorphism to holomorphic bundles.
Definition: Lift of an antiholomorphic 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 Gorbits on X (understanding of a global feature of visible actions).
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Lecture 36 (April 30): Visible action on complex manifolds II
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 antiholomorphic 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 Gaction 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.)
 Propagation theorem of multiplicityfree property
Theorem (reformulation of the Theorem in the 33rd lecture).
V → X is Gequivariant holomorphic vector bundle, for which the Gaction on X is visible.
Multiplicity freeness on the fiber over S ⇒ Multiplicity free on
sections O(X, V)
 Local structure on visible actions [Ko05]
Definition (infinitesimally visible actions) — without antiholomorphic diffeo.
 Visible action, Polar action, and coistoropic action.
Visible action — complex manifold [K05]
Polar action — Riemannian manifold cf. [HPTT94]
Coisotoropic action — symplectic manifold [GS84, HW90]
Relations among these three (continued)
References:
[GS84] 
GulleminSternberg, Jour. Diff. Geom. 1984. 
[HPTT94] 
E. Heintze, R. Palais, C. L. Terng, and G. Thorbergsson, J. reine angew Math. 1994. 
[HW90] 
Huckleberry and T. Wurzbacher, Math. Ann. 1990. 
[K05] 
T. Kobayashi, Publ RIMS 2005. 
[K06] 
T. Kobayashi, Propagation of multiplicityfree property for holomorphic vector bundles, math.RT/0607004. 
[K07] 
T. Kobayashi, Transformation Groups 2007. 
[S08] 
A. Sasaki, Ph. D thesis at Waseda University 2008, March. 
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Lecture 37 (May 2): Visible actions and multiplicityfree theorems

Visible action, polar action, and coisotoropic action
U(n) = GL(n,C) ∩ O(2n) ∩ Sp(n,R)
Theorem ([GS84]). For compact symplectic manifold X,
coisotoropic <=> P(X)^{G} is commutative
(a "classical limit" <> geometric quantization : End_{G}(H) is commutative)
Recall the 33rd, 35th lectures:
Theorem. visible > End_{G}(H) is commutative.
Theorem ([HW90, Ko05]).
For Kähler manifolds, polar action with totally real slice >
infinitesimally visible and coisotoropic.
Theorem ([HW90, Ko05, Sa08]).
For Linear actions, the three notions are equivalent:
 visible action (complex geoemtry)
 multiplicityfree space (classified by [Ka, BR, L])
 coitotropic action (symplectic geometry)
E.g. GL_{m} × GL_{n} duality

Five multiplicityfree examples of SL(2,R) representations, and two examples of SU(2) by a simple geometry
(both continuous and discrete cases)

Triunity principle [K04]
Visible actions for the following three cases:
L on G/H, H on G/L, G on (G×G)/(H×L)
=> Three different multiplicityfree theorems
(e.g. weight mutliplicityfree reps, GL_{n} to GL_{n1}, and Pieri law)

Stembridge's classification for the multiplicityfree tensor product [St01]
(conceptual explanation by using visible actions [Ko04])

A new proof of multiplicityfree theorem of old results by E. Cartan,
I. M. Gelfand
(symmetric spaces)

Multiplicityfree representations
Once we know the formula is multiplicityfree a priori, we could expect a
beautiful formula will be there.
6a) 
ClebshGordan, Pieri, Okada, Krattenthaler, Alikawa, etc (finite
dimensional case), 
6b) 
Branching formula of HuaKostantSchmidK [Ko07] (discrete in the
infinite dimensional case), 
6c) 
singular (nonhighest wt) rep in the discrete spectrum for visible actions
(Ørsted, G. Zhang, etc) (continuous spectrum in the infinite dimensional case)
(see [Ko05] for references therein) 
References:
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© Toshiyuki Kobayashi