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 multiplicity-free 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 multiplicity-free branching rules,
- structure of semisimple symmetric spaces,
- Borel-Weil theorem — revisited,
- a generalized Hua-Kostant-Schmid formula,
- propagation of irreduciblity, unitarity, and multiplicity-free property,
- visible actions on complex manifolds, multiplicity-free spaces,
- some open problems.
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.
[ Lecture slides (pdf) ]Multiplicity-free 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 multiplicity-free representations as the underlying algebraic structure. Conversely, we may expect that multiplicity-free representations would yield natural and useful decompositions from unknown objects into well-understood objects.
Most of the examples of the second lecture were taken from an expository paper:
T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), 497-549, 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), 383-402.
I. M. Gel'fand, Spherical functions on symmetric spaces, Dokl. Akad. Nauk SSSR 70 (1950), 5-8.
V. Guillemin and S. Sternberg, Multiplicity-free spaces, J. Differential Geom. 19 (1984), 31-56.
T. Kobayashi, W. Schmid, and J.-H. Yang,
Representatin Theory and Automorphic Forms,
Progress in Mathematics 255, Birkhäuser, 2008, ISBN-13: 978-0-8176-4646-2.
A. Sasaki, Visible actions on multiplicity-free 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), 61-80.
J. R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb. 5 (2001), 113-121.
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.
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 semi-positive sesqui-linear forms. Conjugate complex spaces, dual spaces, and conjugate duals. Riesz's representatin 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.
Goal of this section: Give a proof of the following theorem:
Theorem 2.1: Holomorphically induced representation preserves irreducibility.
Two classically known applications:
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:
S. Kobayashi, Irreducibility of certain unitary representations, J. Math. Soc. Japan 20 (1968), 638-642. | |
[J1] | H. P. Jakobsen, The last possible place of unitarity for certain highest weight modules, Math. Ann. 256 (1981), 439-447. |
[J2] | H. P. Jakobsen, Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385-412. |
[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. 97-143. |
Earier direction: ("inverse propagation results")
G-actions on the space of sections → Gx 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:
Definition. Equivalent definition of the effectiveness the realization of Hilbert space
Theorem (inverse propagation theorem of unitarity).
Remark.
Let V → X is a G-equivariant bundle with irreducible isotoropy group actions.
If a unitary representation is realized effectively on Γ(X, V),
then V → X carries a G-invariant Hermitian bundle structure.
In particular Gx acts unitarily on Vx for every x.
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. |
Totally real submanifold
Definition, examples
Invariant sections
Lemma For transitive base space, G-invariant sections <-> Gx 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
Gx on Vx 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:
1) | G-transitive action on X with base space o ⇔ closed subgroup of G |
2) | G-equivariant fiber bundle ⇔ H-action on topological space |
2)' | G-equivariant vector bundle ⇔ representation of H |
3) | G-invariant sections ⇔ H-invariant elements in fibers |
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
Step 1. | Pull back to the principal bundle G → G/H. Write the \bar ∂ operator by means of the opearor \bar D acting on V-valued functions on G. |
Step 2. | Key formula: difference between \bar D and right differentiation. (continued) |
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 (GC/B may not be Hausdorff if B is not closed, H may have connected compotents, G is not a subgroup of GC, 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.
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
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]).
References
[B] | M. Berger, Les espaces symetriques noncompacts. (French) Ann. Sci. Ecole Norm. Sup. (3) 74 (1957) 85-177. |
[F] | M. Flensted-Jensen, Analysis on non-Riemannian symmetric spaces. CBMS Regional Conference Series in Mathematics, 61. AMS, 1986. x+77 pp. ISBN: 0-8218-0711-0. |
[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), 21-91. |
[S] | M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras. J. Math. Soc. Japan 11 (1959) 374-434. |
[V] | D. Vogan, Jr., The algebraic structure of the representation of semisimple Lie groups. I. Ann. of Math. (2) 109 (1979), 1-60. |
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)
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+.
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.
Fact. For a connected semisimple Lie group, (i) and (ii) are equivalent:
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. |
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.
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.
SO(2p,2q)/U(p,q), GL(2n,R)/GL(n,C), Sp(n,R)/U(p,q)
See [K, Tables 3.4.1 and 3.4.2] for the classification.
Reference
[K] | T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, Representation Theory and Automorphic Forms, Progr. Math. vol. 255, Birkhäuser, 2007, pp. 45-109. |
Recall:
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)
(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
--> a special case gives a new proof of the Cartan-Weyl and the Borel-Weil theorem (continued).
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/ZG 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
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 ya-2 dx dy
Theorem:
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 B2X
Theorem: For a complex manifold Aut(X) acts unitarily on the Hilbert space B2X.
Bergman kernel form, Bergman metric
Theorem: If geometric Bergman space is effectively realized then its Bergman metric is positive definite.
Cor. For a bounded domain, Aut(X) ⊂ Isom(X).
Examples of Berman metric.
Setting: X = G/H ⊂ GC/B- is a Borel embedding.
Assume G is the fixed point group of an anti-holomorphic involution of GC.
V: standard holomorphic vector bundle associaged to an irreducible representation
τ of H on V.
ψ: G B- → HC , 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, Pλ+ 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 Gx action on the fiber Vx is mutiplicity-free. Suppose there exists an anti-holomorphic bundle auto σ s.t.
Reference:
T. Kobayashi, Propagation of multiplicity-free property for holomorphic vector bundles, preprint. math.RT/0607004.
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 Xσ)o is not empty for some anti-holomorphic map σ. A slice is a submanifold S of Xσ 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.)
References:
[GS84] | Gullemin-Sternberg, 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 multiplicity-free 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. |
Visible action, polar action, and coisotoropic action
Theorem ([GS84]). For compact symplectic manifold X,
(a "classical limit" <-> geometric quantization : EndG(H) is commutative)
Recall the 33rd, 35th lectures:
Theorem. visible -> EndG(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:
E.g. GLm × GLn duality
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 multiplicity-free theorems
(e.g. weight mutliplicity-free reps, GLn to GLn-1, and Pieri law)
Multiplicity-free representations
Once we know the formula is multiplicity-free a priori, we could expect a beautiful formula will be there.
6-a) | Clebsh-Gordan, Pieri, Okada, Krattenthaler, Alikawa, etc (finite dimensional case), |
6-b) | Branching formula of Hua-Kostant-Schmid-K- [Ko07] (discrete in the infinite dimensional case), |
6-c) | singular (non-highest 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:
[GS84] | Gullemin-Sternberg, Jour. Diff. Geom. 1984. |
[HW90] | Huckleberry and T. Wurzbacher, Math. Ann. 1990. |
[Ka80] | V. Kac. J. Algebra 1980. |
[Ko04] | T. Kobayashi, Acta Appl. Math. 2004. |
[Ko05] | T. Kobayashi, Publ. RIMS 2005. |
[Ko07] | T. Kobayashi, Birkhäuser 2007. |
[BR97] | Benson-Ratcliff, J. Algebra 1997. |
[L98] | Leahy, J. Lie Theory 1998. |
[OZ] | Ørsted-Zhang. Canad. J. Math. 1997. |
[Sa08] | A. Sasaki (Ph.D. thesis 2008, March). |
[St01] | J. Stembridge, Ann. Comb. 2001. |
© Toshiyuki Kobayashi