Title: Representations of Lie groups

Yale University, USA, spring term, 2019.

Math 553 MW 1-2:15

[ Lecture 1 (Jan 23) | Lecture 2 (Jan 28) | Lecture 3 (Feb 4) | Lecture 4 (Feb 6) | Lecture 5 (Feb 11) | Lecture 6 (Feb 13) | Lecture 7 (Feb 18) | Lecture 8 (Feb 20) | Lecture 9 (Feb 25) | Lecture 10 (Feb 27) | Lecture 11 (Mar 4) | Lecture 12 (Mar 6) | Lecture 13 (Apr 1) | Lecture 14 (Apr 3) | Lecture 15 (Apr 8) | Lecture 16 (Apr 10) | Lecture 17 (Apr 15) | Lecture 18 (Apr 17) | Lecture 19 (Apr 22) | Lecture 20 (Apr 24) | Lecture 21 (Apr 29) | Lecture 22 (May 1) ]

Short description of the course:

An introduction to infinite-dimensional representations of Lie groups by geometric and analytic methods with basic examples, followed by some recent topics on symmetry breaking (restriction of representations).

Long description of the course:

This course gives an introduction to infinite-dimensional representations of real reductive Lie groups such asGL(n,R) by geometric and analytic methods.I begin with some basic concepts and techniques on real reductive Lie groups, their representations, and global analysis via representation theory, with a number of classical examples.

If time permits, I would discuss some recent developments on branching problems which ask "how irreducible representations of groups behave/decompose when restricted to subgroups". Possible topics include:

- Basic concepts for infinite-dimensional representations of Lie groups
- Structure theory of reductive groups
- Homogeneous bundles
- Symmetry breaking
- Idea of non-commutative harmonic analysis
- General theory of restriction to subgroups
- Branching laws (example)
- F-method and construction of symmetry breaking operators
- Symmetry breaking from conformal geometry

Inside: Analysis and synthesis — smallest objects and decompositions

finding smallest objects (irreducible reps, some other view points, too)

building up/decompositions

Induction ... e.g. global analysis on homogeneous spaces

Restriction ... e.g. tensor product

Various examples of classical analysis problems interpreted as special cases of the general problem of irreducible decompositions

Interpretation by "hidden symmetry"

Example: three viewpoints of spherical harmonics

- global analysis for Laplacian

- induced representations (analysis on homogeneous spaces)

- restriction of reps (conformal to isometry)

Reference:

A part of Today's lecture was based on Section 1 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., 1-37.

and

T. Kobayashi, Global analysis by hidden symmetry, In: Representation Theory, Number Theory, and Invariant Theory, In Honor of Roger Howe on the Occasion of His 70th Birthday, Progress in Mathematics, vol. 323 (2017), pp. 359-397.

Vector space which is controlled "well" by a group.

Learn from prototypes from finite-dimensional represenation theory

Shur's lemma (finite dimensiona case)

Completely reducible representationsSome comments about what might happen for infinite-dimensional representations. (Rigorous definitions are not for today.)

Change viewpoints.

Naive Question.

Is a vector space V controlled "well" by a group G?

For infinite-dimensional V, obstructions could be

- continuously many irreducibles may occur in V.
- irreducible representations might be of infinite dimension.
- multiplicity might be infinite.

Basic Question.

Given a representation (\pi, V) of G,(1)
when is the multiplicity m(\sigma) < \infty for all irreducible
\sigma?(2) when is the multiplicity m(\sigma) is uniformly bounded with respect to \sigma?

(1) (even better, (2)) will give us a nice framework for detailed studyof decompositions.

Typical Setting:

- (glaobal analysis) Given a G-space X, consider the regular representation on V=space of functions on X.
- (branching problem) Given an irreducible representation \Pi of \widetilde{G} and its subgroup G, consider the restriction \pi|_{G} as a representation of G.

Three viewpoint on spherical harmonics. It is multiplitity-free, hence spectral analysis of the Laplacian is essentiallyequivalent to the irreducible decomposition by Schur's lemma.

Remark.

Multiplicity depends on topology of representation
space

Example. regular representation of **R** on C^\infty(**R**) and L^2(**R**).

No lecture on January 30 (Wed)

"Analysis and synthesis" applied to Lie algebras

Definition 3.1. Simple Lie algebras

Classification theory of simle Lie algebras over \mathbb C or \mathbb R.

- Killing, Cartan (1894 over \mathbb C, 1914 over \mathbb R)
- invariants (root system, Dynking diagram, Satake diagram ([He,Kn])
- uniform construction (Serre, [Hu])

Eg. Classification of nilpotent Lie algebras only of low dimensions ([dG]).

Definition of semisimple or reductive Lie algebras.

Illustrate by examples a reason why we use "reductive" rather than simple Lie algebras.

Definition of nilpotent Lie algebras (three equivalent definitions, [Hu])

Definition 3.9 C^k Lie groups (k=0,1, ..., \infty, \omega)

Theorem 3.10.(an affirmative solution to Hilbert's 5-th problem [MZ])

The concept of C^k Lie groups is essentially the same.

Philosophy: algebraic structure raises topological assumptions to analytic results.

cf. Exotic spheres (Kervarie-Milnor [KeMi])

Exercise 3.11 : check the above phiosophy by a one-dimsnsional case.

Another example of this philosoph is:

Thereom 3.12 (von Neumann - Cartan) Any closed subgroup of GL(N,R)
carries a natural Lie group structure.

References for the 3rd lecture:

Textbooks:

- [He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001. xxvi+641 pp. ISBN: 0-8218-2848-7
- [Hu] J. Humphreys, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972. xii+169 pp.
- [Kn] A. Knapp, Lie Groups beyond an Introduction. Second edition. Progress in Mathematics, 140. Birkhauser Boston, Inc., Boston, MA, 2002. xviii+812 pp. ISBN: 0-8176-4259-5 22-01.

- [dG] W. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra 309 (2007), 640--653.
- [KeMi] M. Kervaire, J. Milnor, Groups of homotopy spheres. I. Ann. of Math. (2) 77 (1963), 504--537.
- [Ko]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.
- [MZ] D.Montgomery and L. Zippin, Topological Transformation Groups. Reprint of the 1955 original. Robert E. Krieger Publishing Co., Huntington, N.Y., 1974. xi+289 pp.

(cf. 3rd lecture... applied to Lie groups/Lie algebras)

This would give a motivation of unerstanding reductive groups that will be discussed from next week.

Definition of continuous representations on topological vector spaces (Def 4.1);

- purpose: to capture representations that appear naturally.

- delicate example: variants of definitions of continuity (Exer 4.2).

Definition of irreducible representations (Def 4.3)

- delicate example showing closedness condition is necessary
by an example of SU(1,1) action on L^2(S^1) (Ex 4.7).

Defintion of unitary representations.

The significance of this definition is a generalization of "complete reducibility" as follows.

Thereom 4.8 (Mautner-Teleman) Any unitary representations of a locally compact group can be decoposed into irreducible unitary representations.

Side remark: Universal covering of SL(2,R) is not a matrix group.

\widehat{G} = unitary dual

Example 4.9 ([P]) Suppose G is a locally compact abelian group.

\widehat{G} is discrete if G is compact.

\widehat{G} is compact if G is discrete.

\wieehat{G}\simeq \mathbb R if G = \mathbb R.

Theorem 4.10 (Duflo [D]) Classification problem of \widehat{G} for algebraic Lie group is reduced to that of reductive Lie groups.

Example 4.11 (Kirillov [Ki]) For a simply connected nilpotent Lie group,
there is a natural bijection between \widehat{G} and the set of coajoint
orbits.

(geometric quantization).

Status of the classification problem of the unitary dual for reductive
Lie groups (essentially simple Lie groups), I plan to mention later.

References

For the unitary dual of locally compact abelian groups, we refer to the textbook:

- [P] L.S. Pontryagin, Topological groups. Translated from the second
Russian edition by Arlen Brown Gordon and Breach Science Publishers, Inc.
, New York-London-Paris 1966 xv+543 pp.

- [GV4] I. M. Gel'fand and N. Ya. Vilenkin, Generalized functions, Vol. 4. Applications of harmonic analysis, Translated from Russian by Amiel Feinstein, Academic Press, 1964.
- [Wal88] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988.
- [Y] Kosaku Yosida, Functional analysis, Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, 1995.

- [Ma] F. I. Mautner, Unitary representation of locally compact groups I, II, Ann. Math., 51 (1950), 1-25; ibid., 52 (1950), 528-556.
- [Te] S. Teleman, On reduction theory. Rev. Roumaine Math. Pures Appl. 21 (1976), no. 4, 465-486.

- [D] M.Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math. 149 (1982), 153-213.

- [Ki] A.A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Math. Nauk 17, 1962, 57-110.

G: reductive Lie group

Successful example

G=SL(n,\mathbb R) Bargmann (1947) for n=2; Vakhutinsky (1968) for n=3;

Speh (1981) for n=3,4; Vogan (1986) for n general.

Successful example

G=SL(n,\mathbb C) Gelfand-Naimark (1947) for n=2; Tsuchikawa (1968) for n=3;

Duflo (1985) for n=4,5; Barbasch (1985) for n general.

Partially successful case: G=O(p,q): Hirai (1962) for q=1; Speh (1981) for (p,q)=(3,3), but not classified for general p,q.

Great progress has been made over 70 years both conceptually and computationally, but the final answer on the classification problem of the unitary dual for simple Lie groups has not been found.

We consider also irreducible representations that are not unitary.

\widehat{G} \subset Irr(G)

Definition of the set Irr(G) of "equivalence classes" of "irreducible representations" is subtle.

Usually, one assumes "admissibility" for the definition of irreduible representations for real reductive groups. One of several equivalent definitions for admissibility is that "Schur's type lemma" holds.

Example. There exist an infinite-dimensional irreducible representations of the abelian Lie group \mathbb R.

We shall exclude such representations.

Then Irr(G) \simeq \mathbb C, by \chi_\xi(x) = e^{i x \xi}
for the parameter \xi in \mathbb C.

Fact (Harish-Chandra) Any irreducible unitary representations of (linear) real reductive Lie groups are admissible.

Irreduible admissible reprensentations of real reductive groups are classified on the infinitesimal level.

(Infinitesimal level ... "weak equivalence classes"
illustrated by SL(2,R)-examples.

(They will be justified by using the concept
smooth representations or (\mathfrak g,K)-modules, which will
be explained later.)

There are three approached for the classification of Irr(G).

- (analytic approach) Langlands: Use the asymptoti behavior of matrix coefficients, combined with Knapp-Zuckerman's classification theory of tempered representations.
- (algebraic approach) Consider K-types (K= maximal compact subgroup of G), and focus on "minimal K-types" (Vogan's theory).
- (geometric approach) Consider K_C orbits on the flag variety of X=G_C. Use the ring of differential operators on X (Beilinson-Bernstein, Brylinski-Kashiwara)

Some of important results are derived from the case of GL(N,R).

We discuss first the GL(N,R) case, which is accessible by linear algebra,
but is also interacted with other areas of mathematics.

1. GL(N,R)

Cartan involution \theta

Proposition K= G^theta is a maximal compact subgroup.

Compactness: easy.

Maximality: global structure from infitesimal structure.

(infinitesimal) Cartan decomposition

gl(N,R)=Skey(N,R) + Symm(N,R)

Algebraicaly, such a decomposition is defined for any involutive
automorphism.

Key lemma: exp: Symm(N, R) --> Symm(N,R)_+

is C^\omega diffeomorphism.

Proof: bijection---cann be explained by linear algebra.

Differential geometric interpretation:

surjectivity fails for SL(2,R) but holds for SU(2) ---

a distinguishing feature of Lorentzian geometry from
Riemannian geometry.

C^\omega diffemorphism is more involved.

Proof uses an explicit formula for the differential of
exponential map.

Remark. Maximal compact subgroups are unique up to inner automorphism.

References (but different treatment from our approach):

- Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Chapter 10,
- Knapp, Lie Groups beyond Introduction, Chapter 6

* Formula of the differential of exponential map.

simple proof --- reduction techniques ([1]).

* X \in Symm(N,\mathbb R) v--> ad(X) \in Symm(N^2, \mathbb R)

Introduce an inner product on M(N, \mathbb R) by Trace ({}^t Y Z).

Proposition 7.1. (Cartan decomposition for GL(n, \mathbb R).

Symm(N,R) x O(N) is diffeomorhpic onto GL(N,R).

Corollary 7.2. GL(N,R) is homotopic to O(N).

Corollary 7.3. O(N) is a maximal compact subgroup of GL(N,R)

Background for our definition of real reductive Lie groups.

(Remark: there are small variations of the "definitions"

of real reductive groups, for connected compotents and coverings.)

Fact (Ado[1]-Iwasawa[2]) Any finite dimensional Lie algebra can be realized as a subalgebra of gl(N, R) for some N>0.

(Iwasawa: characteristic p>0)

Definition Representation of a Lie algebra $\mathfrak g$,
($mathfrak g$-module).

Complete reducibility

Example Ideals of a Lie algebra are nothing but submodules for the adjoint representation ad:\mathfrak g \to End(\mathfrak g).

Fact The following three conditions on a finite-imensional Lie algebra
are equivalent:

(i) \mathfrak g is reductive (Definition 3.2)

(ii) ad: \mathfrak g \to End(\mathfrak g) is completely reducible.

(iii) \matfrak g can be realized in gl(N, \mathbb R) for some N>0
such that \mathfrak g is stable under taking transpose of matrices.

We shall use (iii) in the next lecture.

- [1] I.D. Ado, "The representation of Lie algebras by matrices", American Mathematical Society Translations, 1949.
- [2] K.Iwasawa, On the representation of Lie algebras, Japanese Journal of Mathematics, 1948.
- [1] T. Kobayashi and T. Oshima, Lie Groups and Representation Theory, (in Japanese), Iwanami, 638 pages, 2005.

Definition 8.2. Identity componet of a topologial group G is denoted by G_0. It is a normal subgroup, and the quotient group G/G_0 is called the component group.

In view of Fact 7.5, we adopt the following definition for linear real reductive Lie group, or real reductive groups for short.

Definition 8.3. (reductive Lie group) A Lie group G is called a linear real reductive Lie group if G is realized as a \theta-stable closed subgroup of GL(N,\mathbb R) such that G/G_0 is finite.

The advantages of this definition incllude

(1) least requirement for connected components;

(2) classical groups (next week) are defined in this way;

(3) proof of structural results is parallel to that of GL(N,R).,

whereas an apparent disadvantage is that the definition looks
dependending ton a concrete realization of G in GL(N,R).
We shall discull later some universal properties of structura results
for real reductive groups.

Comments: Why did we assume that G is closed. I explained this by taking one-dimensional Lie group G= \mathbb R in GL(4,R).

Theorem 8.4 (Cartan decomposition) Any (linear) real reductive Lie group G has a decomposition \mathfrak p \times K \simeq G

Corollary 8.5 G is homotopic to K.

Corollary 8.7. K is a maximal compact subgroup of G.

Remark 1. Theorem works or covering groups which are not necessarily
linear.

2. Theorem may fail if we drop the assuption G/G_0 is finite.

Propositio 8.8. Suppoe A^2 is a nonzero multiple of the unit matrix. Then G(A) ={g GL(N,R): {}^tg A g = A} is real reductive.

Proposition 8.2 (Chevalley) F is a polynomial on M(N, R).
Suppose X is a symmetric matrix. Then (i) and (ii) are equivalent:

(i) F(exp (kX))= 0 for all k \in Z.

(ii) F(exp (sX))= 0 for all s \in R.

Counterexample 8.3 by dropping the assumption that X is symmetric.

Proposition 8.2 completes the proof of Proposition 8.8 by showing that G(A) has at most finitely many connected components.

In the third lecture, we discussed the classification of simple Lie algebras: there are 10 families of simple Lie algebras and 22 exceptional ones.

The next goal is to give a realization of the 10 families
of simple Lie groups (classical groups) by the combination of
the following three:

(1) G(A) and its variant G^*(A);

(2) change of fields; R, C, or H (quarternionic number field);

(3) det = 1.

We begin with

M(n,R) \subset M(n,C) \subset M(2n,R)

which are algebra homomorphisms commuting with the Cartan involution.

This consideration leads to

Proposition 9.5. Cartan decomposition for GL(n,C).

Let A \in GL(n,R) such that A^2 is a scalar multiple of I_n.

Proposition 9.6. The subgroups G_C(A) and G^*(A) of GL(n,C) are both reductive.

where G_C(A) = {g \in GL(n,C): {}^t g A g = A},

G^*(A) = {g \in GL(n,C): {}^t \bar{g} A = A}.

F=R, C or H = R + Ri + R j + Rk.

Regard H^n as an H-vector space by the right action, and
we can identify M(n,H) with the ring of H-endomorphisms of H^n.

The group of invertivel matrices is defined as GL(n,H).

We identify H^n with C^n+j C^n by x +jy to {}^t(x,y), which yields an injective algebra homomorphism \eta: M(n,H) to M(2n,C) with image consisting of T J = J \bar{T} where J is 0 -1 \\ 1 0.

The point is that \eta commutes with the transpose conjugation of matrices, and so the Caran involutions of M(n, H) and M(2n.C).

Definition (U^*(2n)) We define U^*(2n) as a subgoup of GL(2n,C) which is isomorphic to GL(n,H) via \eta.

More generally,

Fact (Mostow) Let G \subset G' are semisimple Lie groups. Then any Cartan involution of G' can be extended to that of G.

Remark. For F=R or C, the group GL(n,F) can be characterized by the condidition det(g) is nonzero, and the special linear group SL(n,F) is of codimension dim_R F = 1 or 2 because det: GL(n,F) \to F^x is a surjective group homomorphism when F= R or C. However, we can show:

Fact There does not exists a surjective group homomorphism h: GL(n, F) → F^x if n>1.

Proposition 10.7 GL(n,H) \simeq SL(n,H) x R_+ as a Lie group.

Point: The maximal compact subgroup of GL(n, H) is isomorphic to tht of SL(n,H), or equivalently, U^*(2n) \cap U(2n) \subset Sp(n) coincides with SU^*(2n) \cap U(2n).

Other classical subgoups

U(p,q:F) = {g \in GL(n,F): g^* I_{p,q} g= I_{p,q}

Def 10.8 O^*(2n) = O(2n,C) \cap U^*(2n).

Theorem 10.9 (classical groups) The 10 families of simple Lie groups are consructed in matrix forms as SL(n,F) and SU(p,q;F) for F=R,C,H and Sp(n,F) for F=R, C together with O(n,C) and O^*(2n).

The isotropy action of H on the tangent space T_o(G/H)
may be identified with the (quotient of) the adjoint representation

Ad_#:H \to GL(\mathfrak g/\mathfrak h).

The nondegenerate symmetric bilnear form

B(X, Y)= Trace (X Y) on M(N, R)

is positive definite on Symm(N,R) and negative
definite on Skew(N,R).

If H \subset G are both \theta-stable closed subgroups in GL(N,R), (hence reductive), then B induces an H-invariant symmetric bilinear form on \mathfrak g/\mathfrak h.

This consideration leads us to a construction of a family of pseudo-Riemannian manifolds on which the isometry groups act transitively.

Theorem 11.7 Suppose that H \subset G are pair of real reductive groups. Then the homogeneous space G/H carries a pseudo-Riemannian metric for which G acts as isometries.

An extension of the idea:

Theorem 11.8. Given pair of Lie groups H \subset G
and an H-manifold F, we form a G-equivariant fiber bundle
\mathcal F= G x_H F over the homogeneous space G/H.
Then there is a natural one-to-one correspondence
between

{G-invariant sections for \mathcal F \to G/H}

and

{H-invariant points in F}.

Reference

T. Kobayashi and T. Oshima. Lie Groups and Representation
Theory, (in Japanese), Iwanami, 2005.

Example 12.1 Riemannian symmetric space of noncompact type.

This is a special case of Theorem 11.7 by taking H to be
a maximal compact subgroup K.

Curvature formula for Riemannian symmetric space

Suppose G/K is a Riemannian symmetric space with induced
metric from the bilinear form B: g x g → R.

Then the sectional curvature at the origin is given by

K(X, Y) = B([X,Y],[X,Y])

up to normalizing factor B(X,X)B(Y,Y) - B(X,Y)^2
for any X, Y in \mathfrak p\simeq T_o(G/K).

This formula extends to reductive symmetric spaces (definition later)
where the metric is no more positive definite.

Example 12.3 (1) Poincare upper half plane by G=SL(2,R)
which can be generalized to three different ways.

(2) (Kahler manifold) Siegel upper half space for G=Sp(n,R).

(3) (constant curvature) Hyperbolic space for G=O(1,n).

(4) (pseudo-Riemannian symmetric space: later)

Example 12.4 Space form for pseudo-riemannian manifolds
X(p,q) = O(p,q)/O(p-1,q) has a pseudo-Riemannian metric
of signature (q,p-1), for which the sectional curvature
is constant -1.

Special cases:

Riemannian case: p=1 hyperboic manifold, q=0 sphere

Lorentzian case: p=2 anti-de Sitter space, q=1 de Sitter space

Back to Riemannian geometry

Fact 12.5 Any isometric action of a compact group on a simply
connected, complete, Riemannian manifold with nonpositive sectional curvature
has a fixed point.

Counterexamples by dropping one of the four assumptions.

Sketch of proof: Uses avaraging of distances.

Corollary. Any two maximal compact subgroups are conjugate to each other by inner automorphism.

No lectures on March 25, 27. Next lecture on April 1.

Aiming at

- a generalization of Theorem 11.7 to equivariant fiber bundles (Theorem 11.8)
- Geometric understanding of induced representations

Group actions on fiber bundle

Principal bundle

Homogeneus bundle

Example 13.2 G->G/H is a (GxH)-equivariant bundle.

It is a G-homogeneous bundle, and an H-principal bundle.

Action on the space of sections.

Proposition 13.3 An action of a subgroup H on a manifold gives rise to a G-homogeneous bundle over the homogeneous space G/H.

Definition 14.1 Let H be a closed subgroup of G.

An action of H on a manifold V gives rise
to a G-homogeneous bundle over G/H with typical fiber V.

This is called an associated bundle.

Theorem 14.3 Any G-homogeneous bundle is given as an associated bundle.

Theorem 14.4 There is a natural bijection between {sections for G-homogeneous bundle with typical fiber V} and {map from G to V with invariance condition by the isotropy subroup}.

A standard way in analysis is to decrease the number of variables,
whereas, increasing the number of variable sometimes clarifies the
structure of the objects (e.g. Gelfand's hypergeometric functions).

Theorem 14.4
is in the latter direction.

Corollary 14.5 (Theorem 11.8)

There is a natural one-to-one correspondence between
G-invariant sections for G-homogeneous bundles
and elements in the typical fiber that are invariant
by the istropy subgroup.

TX (tangent bundle), T^*X (cotangent bundle), L_vol(X) (density bundle, volume bundle), S^2(TX)_+, and S^2(TX)_reg,

for which sections are called vector fields, 1-forms, measures, Riemannian structures, and pseudo-Riemannian structures.

We also gave transition functions for each.

Remark. The point here is that we need "twist" for L_vol(X) in order to
define an integration of functions on X.

The regularity of "measures" is another issue, which is
a bit beyond differential geometry to define precisely
what "measures" mean. In fact, dependig on purposes,
we may treat different regularity such as
measures that are absolutely continuous to Lebesue measures,
or
Dirac delta function supported at one point (or subvariety).

These measures are examples of Radon measures, which are dual to
the space of compactly supported continuous functions on X when X
is a manifold.

Applying Theorem 14.2, the aforementioned bundles
are given as associated bundle if G acts on X transitively.

For example,
the density bundle L_vol(G/H), to be denoted by L_{2\rho}, is given by

Gx_H C_{2\rho},

where C_{2\rho} denotes the one-dimensinal representation

H -> GL(1,C), k -> |det(Ad(k):h->h)/det(Ad(k):g->g)|.

The line bundle associated to its square root represetation C_\rho
will be denoted by L_\rho, called the half density bundle.

This will be used later for "unitarily induced representation".

Remark. In the semisimple theory, usuall this is defined as half the sum of positive roots. In this lecture, we give a more intrinsic meaning which holds in a more general setting.

Application of Theorem 11.4 and Corollary 11.6, we obtain

Example 15.3. There is a natural one-to-one correspondence between {G-invariant vector fields on the homoeneous manifold G/H} and {H-invariant elements in the quotient g/h of Lie algebras}. The trivial case (H={e}) says that {G-invariant vector fields on G}=the Lie algebra g of G.

Example 15.4. G/H admits a G-invariant Radon measure
if and only if |det(Ad(k):h->h)/det(Ad(k):g->g)|=1 for any k \in H.

(1)(trivial case) H= {e}. This condition holds, giving rise to a left
Haar measure.

(2) (group manifold case) (GxG,\diag(G).

G is unimodular (i.e. left Haar measure is also right invariant)
if and only if |Ad(k):g->g|=1 for all k in G.

(3) If both G and H are unimodular, then G/H admits a G-invariant measure.

Example 15.5. Theorem 11.7 is recovered as a special case of the above general framework. That is, if G and H are real reductive groups, then G/H carries a pseudo-Riemannian structure for which G acts isometrically.

In analytic representation theory, we consider topology on representation spaces, e.g. Fréchet space, Hilbert space.

Begin with smooth induction.

Definition 16.1. Given a representation (\sigma, V) of a closed subgroup H
of a Lie group G, we define an induced representation

Ind_H^G(\sigma) on \Gamma(X, \mathcal V)

the space of smooth sections for the G-homogeneous bundle
\mathcal V=Gx_H V over X=G/H.

Example 16.2. The regular representation of G on C^\infty(G/H) is a special case of the induced representation, namely, Ind\H^G(1).

Theorem 16.3. (Frobenius reciprocity) If (\mu, U) is a finite-dimensional
representation of G, then there is a natural bijection

Hom_H(\mu|H, \sigma) = Hom_G(\mu, Ind_H^G(\sigma))

Remark 16.4. A special case of Theorem 16.3 is a special case of
Theorem 14.4. That is, if (\mu, U) is the trivial one-dimensional
representation, Theorem 16.3 says that

V^H = \Gamma(X, \mathcal V)^G,

which is a special case of Theorem 14.4 where V was not necessarily a
vector space. The proof of Theorem 16.3 goes similarly to Theorem
14.4.

Example 16.5. G=SL(2,R), P= lower triangular matrices.

Write the induced representation Ind_H^G(\sigma_\lambda)
in terms of the multiplier representation

f \mapsto |cx+d|^{\lambda} f(ax+b/cx+d)

by using the Bruhat cell.

More generally, for a G-action on X, a (matrix-valued) multilier is
a measurabe cocycle map

m:G x X -> GL(V).

Theorem 16.6. (Multiplier representation) A multplier of
a G-action on X gives rise to a representation of G on
V-valued functions f on X by

f \mapsto m(g, x) f(f^{-1} x)

The SL(2,R)-example shows that an induced representation
can be written as a "multiplier representation".

This idea works in a more general setting:

Induced representations can be expressed as multiplier representations.

This view has an advantage and a disadvantage, and we could use both, depending on purposes.

The following theorem shows that there are enough (infinite-dimensional) representations that can be constructed as induced representations of "finite-dimensional representations".

Theorem 16.7. (Harish-Chandra's subquotient theorem, Casselman's
subrepresentation theorem)

For any "irreducible representation" of a real reductive Lie group G
can be realized as a subrepresentation of an induced
representation of a (finite-dimensional) irreducible representation
\sigma of a minimal parabolic subgroup P of G.

Remark 16.8. (1) We need some careful consideration for "irreducible
representations" when they are infinite dimensional (e.g. admissible,
smooth...).

(2) Terminology such as minimal parabolic subgroup will be explained
in the 20th and 21th lectures.

[KO-2013] T. Kobayashi and T. Oshima. Finite multiplicity theorems for induction and restriction. http://dx.doi.org/10.1016/j.aim.2013.07.015 Advances in Mathematics, 248, (2013), pp.921-944.

[Wal88] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988.

Tensor product bundle, outer tensor product bundle.

L_{half}(X) \otimes L_{half}(X) = L_volume(X)

as line bundles over X.

Integration is defined for compactly supported continuous sections
for L_volume(X).

Denote by L^2(X, L_{half}(X)) for the completion of
\Gamma_c(X, L_{half}(X)) with the pre-Hilbert structure given
by integration.

Example When X= R, we have an isomorphism

L^2(R, L_{half}(R)) = L^2(R), f(x) \sqrt{dx} <--> f(x) (*)

where dx is the Lebesgue measure.

Later, we discuss these notions with group actions G. Then we may not have the standard measure such as the Lebesgue measure with respect to the transformation group G. In such a case, an isomorphism (*) is not canonical.

When X is a homogeneous space G/H, then

L_{2 \rho} = GxH C_{2\rho} is the density bundle

L_\rho = GxH C_{\rho} is the half density bundle

where C_{2\rho} is the one dimensional representation of H

k \maspto |det(Ad(k):h->h)/det(Ad(k):g->g)|.

Theorem 17.3. (1) Integration is a G-invariant functional
on \Gamma_c(X, L_{2 \rho}).

(2) The natural action of G on the Hilbert space
L^2(X, L_\rho) gives a unitary representation.

Example 17.4. G=SL(2,R).

\Gamma(X, L_\rho) is isomorphic to \pi_\lambda = Ind_P^G(\sigma_\lambda)
with \lambda=-1, cf. the notation in the 16th lecture.

For a unitary representation (\sigma, V) of H,
we get a G-equivariant Hermitian bundle \mathcal V=G x_H V with
the inner product at the fiber x=gH \in X=G/H given by

( [(g,v)], [(g,w)])_{\mathcal V_x} = (v,w).

Definition 17.5. L^2-Ind_H^G(1) denotes a unitary representation of H on the Hilbert space of square integrable sections for the half-density bundle on X.

If G/H carries a G-invariant measure, i.e. if \rho is trivial, then L^2-Ind_H^G(1) is identified with L^2(G/H).

Definition-Theorem 17.7. (unitary induction) For a unitary representation (\sigma, V) of H, we get a unitary representation, to be denoted by L^2-Ind_H^G(\sigma), of G on L^2(X, \mathcal V \otimes L_\rho) which is the Hilbert competition of compactly supported continuous sections of \mathcal V \otimes L_\rho.

Remark. The unitary induction L^2-Ind_H^G(\sigma_\nu)
of G=SL(2,R) induced unitarily from
the one dimensional representation \sigma_\nu (Re \nu =0),
which is identified with the multiplier representation on L^2(R)by

F(x) \mapsto |cx +d|^\lambda F(ax+b/cx+d).

Here \lambda = -1 + \nu.

The goal today is to give an analytic proof for
the irreducibility of any unitary sphecial principal
series representation of SL(2,R).

The irreducibility is delicate because a unitary
non-spherical principal series is not always irreducible.

Theorem 18.1. Let \sigma_\lambda (g) := |(0,1 ) g {}^t(0 1)|^{-\lambda}
be a one-dimensional representation of P (the group of lower triangular
matrices).

Then L^2_{P}^{G}(\sigma_\lambda) is an irreducible unitary
representation of G=SL(2,R) for all pure imaginary \lambda.

A usual proof is based on algebraic techniques.

In contrast, the strategy of our analytic proof uses the restriction to
subgroups ("branching laws").

There are three typical one-dimensional subgroups of G:

N=abelian sugbroup consisting of unipotent elements.

A=abelian subgroup consisting of hyperbolic elements.

K=abelian subgroup consisting of elliptic elements

Sketch of proof.

Suppose W is a closed G-invariant subspace of L^2_{P}^{G}(\sigma_\
lambda).

Obviously,

(1) W is N-invariant,

(2) W is A-invariant,

(3) W is K-invariant.

The condition (1) shows that W is a Wiener subspace in L^2(R), which
is a translation invariant closed subspace, or equivalently, the Fourier
transform of L^2(E) for some measurable set E in R.

Then the condition (2) shows that E is either empty, R>0, R<0 or R
up to measure zero set.

Finally, we define a natural unitary isomorphism between L^2(R) and L^
2(S^1),
and the condition (3) shows that E cannot be R>0 or R<0, whence
L^2_{P}^{G}(\sigma_\lambda) is irreducible.

Remark. The non-spherical principal unitary representation
L^2_{P}^{G}(\sigma_\lambda \otimes \sign) is not irreducible at \lambda=
0.

In fact, the Hardy space is an irreducible submodule at \lambda =0,
which corresponds to E=R>0.

Most of this topic can be found in the standard textbooks such as [He, Kn, War, Wal], but we plan to include some further materials (Wednesday).

For a maximal abelian subspace a in p, we set

g(a;\lambda):= {Z \in g: ad(X)Z=\lambda(X)Z}

Sigma\equive \Sigma(g,a):={\lambda| g(a;\lambda) is nonzero}-{0}.

Proposition 19.3. (1) The Lie algebra decomposes
into a direct sum of joint eigenspaces of ad(X) for X in a:

g= g(X;0)+ \Sum{\lambda \in\Sigma} g(a;\lambda).

(2) \Sigma satisfies the axiom of root system (see Remark)

(3) The Weyl group of \Sigma is a subgroup of M'/M=N_K(a)/Z_K(a).

They coincide if G is connected.

Remark 19.4. There are some differences from the complex reductive case.

(1) g(a;0) is not always abeian;

(2) dim g(a;\lambda) may be larger than one;

(3) \Sigma may contain \alpha and 2 \alpha.

Why do we consider a, not a Cartan subalgebra?

There are several aspects which include:

(1) For any (reasonable) representation of a real reductive
Lie group G on a topological space, we could define a differential
represetation of the complexified Lie algebras g_C. However, irreducible
representations of g_C are not always lift to a representation of
a real reductive Lie group G.

(2) Cartan subalgebra is unique in g_C up to inner automorphisms,
however, the observation (1) suggests that we need Cartan subalgebras
in g for studying a finer structure of representations of the Lie group G.

(3) a is a split part of the maximally split Catan subalgebra.

(Concerning (1), infinite-dimensional irreducible representationsof Lie algebras g_C are quite wild. The Bernstein-Gelfand-Gelfand category O or the category of Harish-Chandra modules ((g_C,K)-modules) focus on very small part of g_C modules. The latter is an algebraic countrpart of representations of G.)

Example. G=O(p,q)

Proposition 19.6. Any two maximal abelian subspaces a and a' in p is conjuate to each other by an element of K.

Proof. (Hunt) Give a numerical estimate about how a and a' are
"close" to each other by introducing a function on K by

F(k) := B(Ad(k)X, Y)

where X and Y are regular elements in a and a', respectively.

Then Ad(k_0)a=a' if F(k) attains its minimum at k=k_0.

[He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001. xxvi+641 pp. ISBN: 0-8218-2848-7

[Kn] A. Knapp, Lie Groups beyond an Introduction. Second edition. Progress in Mathematics, 140. Birkhauser Boston, Inc., Boston, MA, 2002. xviii+812 pp. ISBN: 0-8176-4259-5 22-01.

[War] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Chaptr 1, Springer.ISBN 978-3-642-50275-0.

[Wal] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988.

Corollary 20.1 Let a be a maximally abelian subspace in p.

(1) Ad(K) a =p.

(2) (Cartan decomposition) G=K exp(a) K.

Example 20.2. Corollary 20.1 applid to GL(n,R) and O(p,q) explain some well-known results about "normal forms" in linear algebras.

Definition. (symmetric pair, symmetric space)

(1) (group theoretic definition) Let G be a Lie group,
and \sigma an involutive automorphism. (G,H) is called
a symmetric pair, and G/H is a symmetric space
if H is an open subgroup of the group G^\sigma of fixed points by \sigma.

(2) (differetial geometric definition)
An affine manifold Mis called a symmetric space if the
geodesic symmetry at each point extends globally a diffeomoprhism
of M and preserves the original affine connection.

Fact. (E.Cartan) The geometric notion of symmetric spaces and the group theoretic notion of symmetric spaces are equivalent.

Example. Reductive symmetric spaces G/H carry a pseudo-Riemannian structure such that G acts isometrically, and the corresponding Levi-Civita connection makes G/H a (geometric) symmetric spaces.

Example. In the above example, if we take \sigma to be the Cartan involution then G/K becomes a Riemannian symmetric space of noncompact type. A hyperbolic space is given by when G=O(n,1).

Example. (group manifold) G=G_1 x G_1, \sigma(a,b):=(b,a), G^\sigma=diag G_1. The symmetric space G/G^\sigma is identified with a group manifold G_1 with both left and right action of G_1.

Theorem 20.10. Suppose that (G,H) is a reductive symmetric pair
defined by an involutive automorphism \sigma of G.

Let \theta be a Cartan involution of G commuting with \sigma,
g=k+p be the corresponding Cartan decomposition,
and fix X \in p.

Let P_+(X) be the parabolic subgroup associated X \in p,
and o=e P_+(X) \in G/P+(X).

(1)If \sigma(X) = -X, then H o is open in the real flag variety G/P(X).

(2)If \sigma(X) = X, then H o is closed in the real flag variety G/P(X).

Corollary 20.11. (elliptic orbit) Any elliptic orbit carries a complex structure on which G acts holomorphically.

Example. GL(n,R) is disconnected but contained in the connected complexification GL(n,C). Likewise, U(p,q) for p+q=n, GL(n/2,H) for even n are also examples of real reductive Lie groups having the same complexification GL(n,C).

Let g=k+p be a Cartan decomposition, and X \in p.

We write P_+(X) for the parabolic subgroup which normalizes the
parabolic subalgebra

p_+(X) = l(X) + n_+(X),

the sum of eigenspaces of ad(X) with 0, and positive eigenvalues.

The connectedness of G_C implies:

P_+(X) \cap P_-(X) = L(X).

Suppose (G,H) is a reductive symmetric pair defined by an involutive
automorphism \sigma of G commuting with a Cartan involution.

We note that the eigenvalues of the differential of \sigma is 1,and -1.

Theorem 21.2. If \sigma(X) = -X then H\dot o is open in G/P_+(X).

Moreover, H \dot o is a reductive homogeneous space H/Z_H(X).

Corollary. K/Z_K(X) = G/P_+(X)=G/P_-(X) as K-manifolds.

In particular, the real flag manifold G/P_+(X) is always compact.

Theorem 21.7. If \sigma(X) = X, then H\dot o is closed in G/P_+(X).

Remark. \sigma \theta is also an involutitive automorphism.Then

\sigma(X) = -X <==> \sigma\theta (X) = X

G^\sigma o is open <=> G^{\sigma\theta} o is closed.

This is a special case of the Matsuki duality [Ma79] between
G^\sigma orbits in G/P and G^{\sigma\theta} orbits in G/P
which reverse the closure relations, proved for a minimal parabolic
subgroup P of G.

Example. There are three SL(2,R) orbits in P^1 C (two open, and one closed), whereas there are three SO(2,C) orbits in P^1C (two closed, and one open).

Why does SL(2,R)/SO(2) carry a complex structure even though SL(2,R) and SO(2) do not have a complex Lie group structure?

The adjoint orbit goes through an element X in k is called an elliptic orbit.

Theorem 21.9. ([KoO90]) Any elliptic orbit D of a reductive Lie group carries a complex manifold structure such that G acts holomorphically on D.

Theorem 21.9 includes the following special cases

* G is compact ... D is a generalized flag variety,

* Z_G(X) coincides with K ... D=G/K is a Hermitian symmetric space,

but also includes indefinite-Kahler manifolds where Z_G(X) is noncompact.

[KoO90] T. Kobayashi and K. Ono, Note on Hirzebruch's proportionality principle, https://www.ms.u-tokyo.ac.jp/~toshi/pub/14.html J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), pp.71-87.

[Ma79] T.Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups.J. Math. Soc. Japan 31 (1979), no. 2, 331-357.

We consider basic questions for induction and restriction.

- （Induction) Analysis on homogeneous spaces G/G'
(1950-; long history, but still developing)

There is a natural differential operator (e.g. Lapacian) on G/G' as it carries a G-invariant pseudo-Riemannian structure (Thm 11.7).

Typical problems include:

* Construct eigenfunctions for differential operators on G/G'.

* Expand arbitrary functions into eigenfunctions.

They are closely related to representation theory of G, such as

* Plancherel type theorem = irreducible decomposition of L^2(G/G').

- (Restriction) Branching problem from G to G'.

(Systematic study has been started relatively recently)

Let \pi be an irreducible representation of G. The restriction $\pi|_{G'}$ is regarded as a representation of the subgroup G', which is no more irreducible in general.

Let \tau be another irreducibe rep of G'.

Typical problems for restriction (cf. [K-2015]) include

* Determine when Hom_{G'}(\pi|_{G'}, \tau)

* Construct intertwining operators (symmetry breaking operators) from \pi to \tau.

* (branching law) Decompose \pi|_{G'} into irreducible representations of the subgroup G'.

Theorem 22.2. ([KO-2013]) The following two conditions are equivalent:

(i) (representation theory) Hom_G(\pi, C^\infty(G/G')) is
finite-dimensional for all \pi \in Irr(G).

(ii) (geometry: real spherical) P has an open orbit in G/G'.

Theorem 22.4. ([KO-2013]) The following two conditions are equivalent:

(i) (representation theory) Hom_G(\pi, C^\infty(G/G')) is
uniformly bounded w.r.t. \pi \in Irr(G).

(ii) (complex geometry: spherical) B_C has an open orbit in G_C/G'_C.

Theorem 22.5. ([K-2014]) The following two conditions are equivalent:

(i) (representation theory) Hom_{G'}(\pi|_{G'}, \tau) is
finite-dimensional for all \pi \in Irr(G) and \tau \in Irr(G').

(ii) (geometry: real spherical) P has an open orbit in G/G'.

Theorem 22.6. ([K-2014]) The following two conditions are equivalent:

(i) (representation theory) Hom_{G'}(\pi|_{G'}, \tau) is uniformly bounded
w.t.t. \pi \in Irr(G) and \tau \in Irr(G').

(ii) (complex geometry: spherical) B_C has an open orbit in G_C/G'_C.

Remark 22.7. Can extend Theorems 22.2 and 22.4 to the sections of G-equivariant vector bundles of finite rank, and also to non-reductive subgroups G'.

Example 22.8. (1) Symmetric spaces for Theorems 22.2 and 22.4 (cf. [D-1998]).

(2) Whittaker models for Theorems 22.2 and 22.4 (quisi-split groups).

(3) Tensor product representations of O(n,1) (classification for
Theorem 22.3 [KM-2015])).

(4) The geometry (ii) in Theorem 22.6 singles out the pair,
(GL_n, GL_{n-1}) and (O_n, O_{n-1}), cf. the Gan Gross-Prasad Conjecture
[GP-1992][KS-2018]).

Theorem 22.10. (generalized Poisson transform 22.10 [K-1992])

Suppose w is an H-invariant element of \Gamma(X, \mathcal V^*_{2\rho}).

Then (T_w f)(g):= (\pi(g^{-1}) f, w) = (f, \pi(g) f, w) induces
a G-intertwining operator

T_w : Ind_P^G(\sigma) -> C^\infty(G/H).

In particular, the image satisfies a system of PDEs

D_z \phi = \lambda_\sigma(z) \phi for all z \in Z(g).

Here, D_z is a G-invariant differential operator on G/H,
and \lambda_\sigma(z) is the scalar determined by \sigma.

Example 22.11. G=SL(2,R) and K= SO(2).Keep notation as in Lecture 18th.

Define w by

w(g) = |g (0 1)|^{-\lambda -2}

Let H be the upper half plane identified with G/K.

Then T_w: Ind_P^G(\sigma_\lambda) -> C^\inft(G/K)
is written in the coordinate as

T_w: C_c^\infty(R) -> C^\infty(H),

f \mapsto \int_R f(t) (y/(x-t)^2+y^2)^{\lambda/2+1} dt

is the twisted Poisson transform, and the image satisfies

Δ g = λ(λ+2)/4 g

where Δ is the hyperbolic Laplacian y^2(∂^2/∂x^2 + ∂^2/∂ y^2)

Furthermore, T_w is injective if Re λ \ge -1.

The "converse map" of T_w is given as "boundary maps", which yields a shor proof of Casselman's subrepresentation theorem (Theorem 16.7) as well as the proof for (ii)->(i) in Theorem 22.2 ([KO-2014]).

[D-1998] P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. Math.147, (1998), pp.417--452.

[GP-1992] B.H.Gross and D.Prasad, On hte decomposition of SO_n when restricted to SO_{n-1}, Canad. J. Math. 44 (1992), pp.974-1002.

[K-1992] T.Kobayashi, Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds U(p,q;F)/U(p-m,q;F), Mem. Amer. Math. Soc. vol. 462, Amer. Math. Soc., 1992. v+106 pages, ISBN-13: 978-0-8218-2524-2.

[K-1998] T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. II. Micro-local analysis and asymptotic K-support. Ann. of Math. (2) 147 (1998), no. 3, 709–729.

[K-2015] T. Kobayashi. A program for branching problems in the representation theory of real reductive groups. http://dx.doi.org/10.1007/978-3-319-23443-4_10 In Representations of Lie Groups: In Honor of David A. Vogan, Jr. on his 60th Birthday, Progress in Mathematics 312, pages 277-322. Birkhäuser, 2015.

[KO-2013] T. Kobayashi and T. Oshima. Finite multiplicity theorems for induction and restriction. http://dx.doi.org/10.1016/j.aim.2013.07.015 Advances in Mathematics, 248, (2013), pp.921-944.

[K-2014] T. Kobayashi. Shintani functions, real spherical manifolds, and symmetry breaking operators. https://www.ms.u-tokyo.ac.jp/~toshi/pub/tk2013q.html Developments in Mathematics, 37, pages 127-159, 2014.

[KM-2014] T. Kobayashi and T. Matsuki. Classification of finite-multiplicity symmetric pairs. http://dx.doi.org/10.1007/s00031-014-9265-x Transformation Groups, 19, (2014), pp.457-493.

[KS-2018] T.Kobayashi and B.Speh, Symmetry breaking for orthogonal groups and a conjecture by B. Gross and D. Prasad. https://doi.org/10.1007/978-3-319-94833-1_8 In: Geometric Aspects of the Trace Formula, Simons Symposium on the Trace Formula, pages 245-266. Springer, Cham, 2018.

[W-1988] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics 132, Academic Press, 1988.

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Report for this lectures should be sent to TK by May 8th.

© Toshiyuki Kobayashi