[ 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 as GL(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
Reference:
Naive Question.
Basic Question.
(1) (even better, (2)) will give us a nice framework for detailed studyof decompositions.
Typical Setting:
Remark.
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.
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])
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:
References for the 3rd lecture:
Textbooks:
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);
Definition of irreducible representations (Def 4.3)
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.
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.
References
For the unitary dual of locally compact abelian groups, we refer to the
textbook:
G: reductive Lie group
Successful example
Successful example
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.
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.
There are three approached for the classification of Irr(G).
Some of important results are derived from the case of GL(N,R).
1. GL(N,R)
Compactness: easy.
(infinitesimal) Cartan decomposition
Key lemma: exp: Symm(N, R) --> Symm(N,R)_+
Proof: bijection---cann be explained by linear algebra.
C^\omega diffemorphism is more involved.
Remark. Maximal compact subgroups are unique up to inner automorphism.
References (but different treatment from our approach):
* X \in Symm(N,\mathbb R) v--> ad(X) \in Symm(N^2, \mathbb R)
Proposition 7.1. (Cartan decomposition for GL(n, \mathbb R).
Corollary 7.2. GL(N,R) is homotopic to O(N).
Background for our definition of real reductive Lie groups.
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).
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:
We shall use (iii) in the next lecture.
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
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.
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:
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:
We begin with
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.
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.
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
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
The nondegenerate symmetric bilnear form
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
Reference
Example 12.1 Riemannian symmetric space of noncompact type.
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.
Example 12.3 (1) Poincare upper half plane by G=SL(2,R)
which can be generalized to three different ways.
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.
Back to Riemannian geometry
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
Group actions on fiber bundle
Example 13.2 G->G/H is a (GxH)-equivariant 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.
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).
Corollary 14.5 (Theorem 11.8)
Remark. The point here is that we need "twist" for L_vol(X) in order to
define an integration of functions on X.
Applying Theorem 14.2, the aforementioned bundles
are given as associated bundle if G acts on X transitively.
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.
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.
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
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
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
Example 16.5. G=SL(2,R), P= lower triangular matrices.
More generally, for a G-action on X, a (matrix-valued) multilier is
a measurabe cocycle map
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
The SL(2,R)-example shows that an induced representation
can be written as a "multiplier representation".
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)
Remark 16.8. (1) We need some careful consideration for "irreducible
representations" when they are infinite dimensional (e.g. admissible,
smooth...).
[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.
Integration is defined for compactly supported continuous sections
for L_volume(X).
Example When X= R, we have an isomorphism
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
Theorem 17.3. (1) Integration is a G-invariant functional
on \Gamma_c(X, L_{2 \rho}).
Example 17.4. G=SL(2,R).
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
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
The goal today is to give an analytic proof for
the irreducibility of any unitary sphecial principal
series representation of SL(2,R).
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).
A usual proof is based on algebraic techniques.
In contrast, the strategy of our analytic proof uses the restriction to
subgroups ("branching laws").
Sketch of proof.
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.
Remark. The non-spherical principal unitary representation
L^2_{P}^{G}(\sigma_\lambda \otimes \sign) is not irreducible at \lambda=
0.
For a maximal abelian subspace a in p, we set
Proposition 19.3. (1) The Lie algebra decomposes
into a direct sum of joint eigenspaces of ad(X) for X in a:
Remark 19.4. There are some differences from the complex reductive case.
Why do we consider a, not a Cartan 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
[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.
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)
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.
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.
Lecture 2 (Jan 28 Mon, 2019)
Goal:
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.
Is a vector space V controlled "well" by a group G?
For infinite-dimensional V, obstructions could be
Actually, both a and b are harmless.
c is serious.
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?
Example.
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.
Multiplicity depends on topology of representation
space
Example. regular representation of R on C^\infty(R) and L^2(R).
Lecture 3 (Feb 4, 2019)
plan: structure of real reductive Lie groups (occasionally,
some advanced topic with way of thinkings)
But a description of consequtive extensions is nontrivial.
The concept of C^k Lie groups is essentially the same.
Thereom 3.12 (von Neumann - Cartan) Any closed subgroup of GL(N,R)
carries a natural Lie group structure.
Some Papers or research monograph:
Lecture 4 (Feb 6, 2019)
Analysis and synthesis applied to infinite-dimensional representations
(cf. 3rd lecture... applied to Lie groups/Lie algebras)
- purpose: to capture representations that appear naturally.
- delicate example: variants of definitions of continuity (Exer 4.2).
- delicate example showing closedness condition is necessary
by an example of SU(1,1) action on L^2(S^1) (Ex 4.7).
\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.
(geometric quantization).
Status of the classification problem of the unitary dual for reductive
Lie groups (essentially simple Lie groups), I plan to mention later.
Direct integral of Hilbert spaces and unitary representations on it. For
more details, we refer to the textbooks [GV4], [Y], and [Wal88]:
For Theorem 4.8,
For Theorem 4.10, we refer to the original paper
For Example 4.11 (actually a theorem), we refer to the original paper
Lecture 5 (Feb 11, 2019)
Current status of the classification of \widehat{G} (unitary dual)
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.
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.
Then Irr(G) \simeq \mathbb C, by \chi_\xi(x) = e^{i x \xi}
for the parameter \xi in \mathbb C.
(They will be justified by using the concept
smooth representations or (\mathfrak g,K)-modules, which will
be explained later.)
Lecture 6 (Feb 13, 2019)
Quick course on structural theory of real reductive Lie groups
We discuss first the GL(N,R) case, which is accessible by linear algebra,
but is also interacted with other areas of mathematics.
Cartan involution \theta
Proposition K= G^theta is a maximal compact subgroup.
Maximality: global structure from infitesimal structure.
gl(N,R)=Skey(N,R) + Symm(N,R)
Algebraicaly, such a decomposition is defined for any involutive
automorphism.
is C^\omega diffeomorphism.
Differential geometric interpretation:
surjectivity fails for SL(2,R) but holds for SU(2) ---
a distinguishing feature of Lorentzian geometry from
Riemannian geometry.
Proof uses an explicit formula for the differential of
exponential map.
Lecture 7 (Feb 18, 2019)
Comments on the last lecture:
* Formula of the differential of exponential map.
simple proof --- reduction techniques ([1]).
Introduce an inner product on M(N, \mathbb R) by Trace ({}^t Y Z).
Symm(N,R) x O(N) is diffeomorhpic onto GL(N,R).
Corollary 7.3. O(N) is a maximal compact subgroup of GL(N,R)
(Remark: there are small variations of the "definitions"
of real reductive groups, for connected compotents and coverings.)
Complete reducibility
(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.
Lecture 8 (Feb 20, 2019)
Definition 8.1. \theta-stable subset for GL(n,R) and the Lie algebra gl(N,R).
(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.
2. Theorem may fail if we drop the assuption G/G_0 is finite.Lecture 9 (Feb 25, 2019)
Cassical Lie groups I
(i) F(exp (kX))= 0 for all k \in Z.
(ii) F(exp (sX))= 0 for all s \in R.
(1) G(A) and its variant G^*(A);
(2) change of fields; R, C, or H (quarternionic number field);
(3) det = 1.
M(n,R) \subset M(n,C) \subset M(2n,R)
which are algebra homomorphisms commuting with the Cartan involution.
This consideration leads to
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}.Lecture 10 (Feb 27, 2019)
Classical Lie Groups II
The group of invertivel matrices is defined as GL(n,H).
U(p,q:F) = {g \in GL(n,F): g^* I_{p,q} g= I_{p,q}Lecture 11 (March 4, 2019)
Geometry of reductive homogeneous spaces
Ad_#:H \to GL(\mathfrak g/\mathfrak h).
B(X, Y)= Trace (X Y) on M(N, R)
is positive definite on Symm(N,R) and negative
definite on Skew(N,R).
{G-invariant sections for \mathcal F \to G/H}
and
{H-invariant points in F}.
T. Kobayashi and T. Oshima. Lie Groups and Representation
Theory, (in Japanese), Iwanami, 2005.
Lecture 12 (March 6, 2019)
Generalization of Theorem 11.7 to equivariant fiber bundles
(Theorem 11.8) will be discussed after spring break, and
we focus on some examples of Theorem 11.7 for today.
This is a special case of Theorem 11.7 by taking H to be
a maximal compact subgroup K.
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.
(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)
Special cases:
Riemannian case: p=1 hyperboic manifold, q=0 sphere
Lorentzian case: p=2 anti-de Sitter space, q=1 de Sitter space
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.Lecture 13 (April 1, 2019)
G-equivariant fiber bundles
Today, we prepare some basic notions.
Principal bundle
Homogeneus bundle
It is a G-homogeneous bundle, and an H-principal bundle.Lecture 14 (April 3, 2019)
Contiunation of "understaning sections for G-homogeneous bundles".
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 14.4
is in the latter direction.
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.
Lecture 15 (April 8, 2019)
On manifold X,
we may think of bundles
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.
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.
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
This will be used later for "unitarily induced representation".
(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.Lecture 16 (April 10, 2019)
Induced representations
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.
Hom_H(\mu|H, \sigma) = Hom_G(\mu, Ind_H^G(\sigma))
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.
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.
m:G x X -> GL(V).
f \mapsto m(g, x) f(f^{-1} x)
This idea works in a more general setting:
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.
(2) Terminology such as minimal parabolic subgroup will be explained
in the 20th and 21th lectures.Lecture 17 (April 15, 2019)
For a manifold X which is not necessarily orientable, one considers
its density bundle L_volume{X}, and half-density bundle L_{half}(X).
L_{half}(X) \otimes L_{half}(X) = L_volume(X)
as line bundles over 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.
L^2(R, L_{half}(R)) = L^2(R), f(x) \sqrt{dx} <--> f(x) (*)
where dx is the Lebesgue measure.
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)|.
(2) The natural action of G on the Hilbert space
L^2(X, L_\rho) gives a unitary representation.
\Gamma(X, L_\rho) is isomorphic to \pi_\lambda = Ind_P^G(\sigma_\lambda)
with \lambda=-1, cf. the notation in the 16th lecture.
( [(g,v)], [(g,w)])_{\mathcal V_x} = (v,w).
F(x) \mapsto |cx +d|^\lambda F(ax+b/cx+d).
Here \lambda = -1 + \nu.
Lecture 18 (April 17, 2019)
SL(2,R) example.
The irreducibility is delicate because a unitary
non-spherical principal series is not always irreducible.
Then L^2_{P}^{G}(\sigma_\lambda) is an irreducible unitary
representation of G=SL(2,R) for all pure imaginary \lambda.
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
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.
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.
In fact, the Hardy space is an irreducible submodule at \lambda =0,
which corresponds to E=R>0.Lecture 19 (April 22, 2019)
Quick review of (real) parabolic subalgebra of
real reductive groups.
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).
g(a;\lambda):= {Z \in g: ad(X)Z=\lambda(X)Z}
Sigma\equive \Sigma(g,a):={\lambda| g(a;\lambda) is nonzero}-{0}.
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.
(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.
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.
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.Lecture 20 (Apr 24, 2019)
The last proposition (Proposition 19.4) shows that
(1) Ad(K) a =p.
(2) (Cartan decomposition) G=K exp(a) K.
(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.
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).