Realizing smooth
representations
David Vogan,
David Jerison
Introduction
Realizing smooth representations
Case of R
SL(2, R) and the
hyperbolic
Laplacian
David Vogan
David Jerison
Department of Mathematics
Massachusetts Institute of Technology
Lie Groups and Geometry
Guanajuato
August 27–31 2012
Holomorphic
discrete series for
SL(2, R)
Conclusion
Outline
Realizing smooth
representations
David Vogan,
David Jerison
Introduction
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Case of R
SL(2, R) and the hyperbolic Laplacian
Holomorphic discrete series for SL(2, R)
Conclusion
Holomorphic
discrete series for
SL(2, R)
Conclusion
Gelfand’s abstract harmonic analysis
Lie group G acts on manifold X , have questions about X .
Realizing smooth
representations
David Vogan,
David Jerison
Step 1. Attach to X Hilbert space H(X ) (e.g. L2 (X )).
Questions about X
questions about H(X ).
Introduction
Step 2. Find finest G-invt decomp H = ⊕α Hα . Questions
about H(X )
questions about each Hα .
SL(2, R) and the
hyperbolic
Laplacian
Each Hα is irreducible unitary representation of G:
indecomposable action of G on a Hilbert space.
Holomorphic
discrete series for
SL(2, R)
Case of R
Conclusion
b u = all irreducible unitary
Step 3. Understand G
representations of G: unitary dual problem.
Step 4. Answers about irr reps
answers about X .
Today: technical problems in Steps 1 and 3. . .
Say Question ! eigenfns of G-invt diff op ∆X .
Problem with Step 1: eigenfunctions not in L2 (X ).
Problem with Step 3: try Hα =def eigenspace of ∆X .
But no Hilbert space structure.
How to address these problems
Realizing smooth
representations
David Vogan,
David Jerison
Problems arise because eigenspace
Introduction
Case of R
Vλ = {f ∈ C
−∞
(X ) | ∆X f = λf }
is inconveniently large.
For example, can’t complete Vλ to Hilbert space without
imposing addl growth conditions on f .
Solution: consider instead dual space
V λ = Cc∞ (X )/(∆X − λ)Cc∞ (X ).
Wasn’t that easy?
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
One concrete example
G = U(p, q) group of Herm form h, ip,q on Cn .
Y n = Y = {complete flags in Cn }
dimC (Y n ) = (n2)
n
= {0 = E0 ⊂ E1 ⊂ · · · ⊂ En = C | dim Ei = i}
Open orbits of U(p, q) on Y ↔
{(pi , qi ) | pi + qi = i, pi incr, qi incr} ↔
{S ⊂ {1, 2, . . . , n} | |S| = p}
XS = open orbit corr to S ⊂ {1, . . . , n}.
Complex mfld XS has cplx K = U(p) × U(q) orbit
ZS = {(Ei ) ∈ XS | Ei = (Ei ∩ Cp ) ⊕ (Ei ∩ Cq )} ' Y p × Y q .
Orbit method: L → XS
?
unitary reps of G.
Problem. Cpt cplx ZS ⊂ XS =⇒ XS not Stein; not
enough holom secs of L.
Soln: Dolbeault cohom H 0,s (XS , L), s = dimC (ZS ).
Problem. Dolbeault cohom too big for invt Herm form.
Soln: see below, we hope.
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Classical Fourier analysis
G = R acts on R by translation, H = L2 (R), ∆X =
Eigenspace representation
Vλ = {T ∈ C −∞ (R) |
d
dt .
dT
= λT }
dt
Of course Vλ is one-diml, basis eλt ; never in L2 (R).
Consider instead dual space
V λ = Cc∞ (R)/
dφ
∞
− λφ | φ ∈ Cc (R) .
dt
Subspace by which we divide is equal to
ψ ∈ Cc∞ (R) |
Z
ψ(t)etλ dt = 0 ,
R
so closed; so topology on V λ Hausdorff.
First advantage: have trivially quotient map
b(λ) : Cc∞ (R) → V λ ,
b :
φ 7→ φ(λ)
this is Fourier trans at λ on dense Cc∞ ⊂ L2 .
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Unitary structure
V λ = Cc∞ (R)/
dφ
− λφ | φ ∈ Cc∞ (R)
dt
David Vogan,
David Jerison
Introduction
(pre)Unitary structure is R-invt Hermitian form
λ
λ
λ
SL(2, R) and the
hyperbolic
Laplacian
h, i : V × V → C.
Schwartz kernel theorem: such pairing (lifted to
Cc∞ × Cc∞ ) given by distn kernel
Kλ ∈ C
−∞
(R × R),
λ
Kλ (s, t)φ(s)ψ(t).
R×R
∂Kλ
∂s
Holomorphic
discrete series for
SL(2, R)
Conclusion
Z
hφ, ψi =
Form descends to V λ ⇐⇒
Case of R
= λK ,
∂Kλ
∂t
= λK .
λs λt
Soln is Kλ (s, t) = Ce e ds dt, but we don’t care.
Form transl invt ⇐⇒ Kλ (s + x, t + x) = Kλ (s, t).
Transl invt compatible with diff eqn ⇐⇒ λ + λ = 0;
then it’s consequence of diff eqn.
Realizing smooth
representations
Plancherel theorem
David Vogan,
David Jerison
Introduction
V λ = Cc∞ (R)/
dφ
∞
− λφ | φ ∈ Cc (R)
dt
(λ ∈ C)
Fourier transform b(λ) : Cc∞ (R) → V λ .
h, iλ
V λ;
If λ ∈ iR there’s invt Herm form
on
normalize kernel to be ds dt near identity.
Plancherel theorem:
Z
Z
2
b
b
|φ(t)| dt = c
hφ(λ),
φ(λ)idλ.
R
iR
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Laplacian on H2
X = {z ∈ C | |z|2 < 1},
Realizing smooth
representations
ds2 = (1 − |z|2 )−2 (dx 2 + dy 2 )
unit disk model of two-diml hyperbolic space.
α β
G = SU(1, 1) =
| |α|2 − |β|2 = 1 ⊂ SL(2, C)
β α
acts on X by linear fractional transformations:
αz + β
α β
∈G .
g·z =
z ∈ X, g =
β α
βz + α
Laplace-Beltrami operator commutes with action of G:
2
∂
∂2
∆X = (1 − |z|2 )2
+
(z = x + iy ∈ X )
∂x 2
∂y 2
Long tradition of studying eigenspaces
Vλ = {T ∈ C −∞ (X ) | (∆X − λ)T = 0}
(λ ∈ C).
E.g. V0 = harm fns on X . Always Vλ = repn of G.
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Traditional boundary values
David Vogan,
David Jerison
Alg homog space X = G/H
proj alg X = X ∪ ∂X .
(noncpt analysis on X )
(easier cpt analyis on X ).
(
)
0 1
zC
2
Our X ' lines B
compactifies to
@ A | |z| − 1 < 0
1
(
)
(0 1)
0 1
iθ
z
2
C
Be C
X ' lines B
@ A | |z| − 1 ≤ 0 , ∂X '
@
A .
1
1
Idea of bdry value map: eigenfn φ(z) ∈ Vλ on X
φ∞ (eiθ ) = lim cλ (r )φ(reiθ ).
r →1
Problem: limit is terrible (hyperfunction); certainly
doesn’t exist pointwise, except under strong addl
hyps on eigenfn φ.
Eigenspace too big.
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Traditional harmonic analysis. . .
Describe eigenspaces like V0 = harmonic fns on disk
2
1−|z|
Poisson kernel P0 (z, eiθ ) = |z−e
iθ |2 writes harmonic
∞
φ(z) in terms of bdry value φ (θ):
2π
Z
P0 (z, eiθ )φ∞ (θ) dθ.
φ(z) =
0
iθ
Reason: P0 (·, e ) is harmonic with bdry value δeiθ .
Fourier analysis (using radially symm eigenfn φλ )
Z
bf (λ, hK ) =
f (gK )φλ (g −1 h) d(gK ) ∈ Vλ
X
extracts from nice f on X its “λ-eigenvalue part.”
Fourier synthesis reassembles (using Plancherel
measure dµ(λ)):
Z
f (gK ) =
bf (λ, gK )dµ(λ).
some λ
Realizing smooth
representations
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Casting out eigenspaces
Replace eigenspace Vλ by (smaller!) dual space
V λ = Cc∞ (X , dx)/(∆X − λ)Cc∞ (X , dx).
1st gain: Fourier analysis is trivial:
b ·) = φ + (∆X −
φ(λ,
λ)Cc∞ (X , dx)
(φ ∈
Any sesq pairing lifts to Cc∞ (X , dx) × Cc∞ (X , dx);
comes from Schwartz
Z distribution kernel:
λ
hφ, ψi =
K (s, t)φ(s)ψ(t).
(s,t)∈X ×X
Condition for pairing to descend to V λ :
(∆X (s) − λ)K λ (s, t) = 0,
Introduction
Case of R
Cc∞ (X , dx)).
2nd gain: V λ has G-invt sesq form. Reason. . .
λ
David Vogan,
David Jerison
(∆X (t) − λ)K λ (s, t) = 0.
Cond for G-invariance of pairing: K λ (g · s, g · t) = K λ (s, t).
THM. V λ has invt sesq form iff λ = λ; normalize
K λ (gK , hK ) = φλ (g −1 h) by K λ (0, 0) = 1.
Again φλ is unique radial eigenfn with φλ (0) = 1.
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Holomorphic line bundles on X
Isotropy at 0 for g · z =
action on unit disk is
„
α
0
0
α
«
| |α|2 = 1
„
α
β
β
α
«
ff
| |α|2 − |β|2 = 1 .
K =
⊂G=
αz+β
βz+α
ff
For n ∈ Z, eqvt line bundle
C ∞ (X , Ln ) ' {f ∈ C ∞ (G) | f (gk ) = α−n f (g)}.
Rightaction
of complexified Lie algebra element
E=
0
0
1
0
defines Cauchy-Riemann operator
∂ : C ∞ (X , Ln ) → C ∞ (X , Ln+2 ).
Get rep of G on holomorphic sections of Ln
Wn = {T ∈ C −∞ (X , Ln ) | ∂T = 0}.
Wn
David Vogan,
David Jerison
Introduction
Case of R
So X = {z ∈ C | |z|2 < 1} ' G/K .
Ln → X ,
Realizing smooth
representations
discrete spectrum of Laplacian on Ln+2k .
For that, need (pre)Hilbert space structure on Wn . . .
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Traditional
holomorphic
discrete
series
0
X '
1
0
1
zC
B C
⊂ CP1 , Ln (z) = CB
@ A
1
zC
B C
lines CB
@ A
1
⊗
.
0
1⊗n
z
Ln has nowhere zero holom sec τn (z) = C
.
A
1
So holom secs of Ln = (holom fns)·τn : Wn ' W0 · τn .
B
@
(g · τn )(z) =def g · (τn (g −1 · z))
n
γz−δ ⊗
B
C
B −δz+γ C
B
C
0
1
0
1
γ δC
C)
A
δ γ
0
1⊗n
γz − δ C
B
C
= (−δz + γ)−n · g · B
@
A
−δz + γ
=g·@
1
B
(g = B
@
A
0
1⊗n
zC
B C
= (−δz + γ)−n · B
@ A
1
Wn ' W0 · τn
= (−δz + γ)−n · τn (z).
(rep on Wn ) ' (multiplier rep on W0 ):
(g ·n f )(z) = (−δz + γ)−n f (g −1 · z)
Realizing smooth
representations
n
(f holom on X ).
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Unitary structure on Wn
Seeking unitary representation of G related to Wn .
1⊗n
0
2
David Vogan,
David Jerison
Bz C
= (1 − |z|2 )n .
C
G-invt Herm str on Ln , @ A
B
Introduction
Unitary structure
Z on Wn (?):
SL(2, R) and the
hyperbolic
Laplacian
1
?
kτ k2 =
kτ (z)k2
X
dz dz
(1 − |z|2 )2
(τ ∈ Wn ).
Using Wn ' W0 · τn , rewrite as
?
kf τn k2 =
Z
|f (z)|2 (1 − |z|2 )n−2 dz dz
X
R1
Polar coords
(1 − r )n−2 dr .
0
(f holomorphic).
Problem: if n ≤ 1, converges only for f = 0.
Problem: never converges for all f .
Solution: (Hermitian) DUAL SPACE. . .
W n = Cc∞ (X , Ln )/∂(Cc∞ (X , Ln+2 )).
Denominator is densities vanishing on hol secs of Ln ; ∃
lots because of Cauchy integral formula, etc.
Case of R
Holomorphic
discrete series for
SL(2, R)
Conclusion
Unitary structure on W n
n
W =
Realizing smooth
representations
Cc∞ (X , Ln )/∂(Cc∞ (X , Ln+2 )).
Sesq pairing on W n ! Schwartz distn kernel
K n (s, t) ∈ C −∞ (X Z× X , L−n × Ln )
n
n
hφ, ψi =
K (s, t)φ(s)ψ(t) ds dt.
X ×X
Pairing ↓ W n ⇐⇒ K n antiholom in s, holom in t.
Pairing G-invt ⇐⇒ K n (g · s, g · t) = K n (s, t).
fn κ on G ! hol sec of Ln on right and left.
Unique soln corrs to nowhere
zero sec τn :
κn
α
β
β
α
= α−n
THM. W n has invt sesq form; K n (gK , hK ) = τn (g −1 k ).
Form is pos def if n > 0; semidef if n ≥ 0.
Unitary rep W n
disc spec of Ln → X ⇐⇒ n > 1.
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
Realizing smooth
representations
Where can you go from here?
G reductive Lie ⊃ L = GT , cpt torus T .
X = G/L cplx mfld; Cauchy-Riemann eqns !
u = pos eigspaces of T ⊂ gC .
Holom bdle E → X ! smooth rep E of L.
Traditional rep of G: Dolbeault cohom of E:
WE = H 0,p (X , E) = H p (u, C −∞ (G, E)L )
Often unitary E for L
“ought-to-be-unitary” WE .
Problem: WE too big to carry invt sesq form.
Solution: (Hermitian) DUAL SPACE. . .
h
n,n−p
W E = Hcpt
(X , E h ) = Hp (u, Cc∞ (G, E h )L )
Easy: unitary E
invt Herm form on W E . OPEN:
Geometric proof of UNITARITY of W E .
?
FOURIER TRANSFORM: Cc∞ (G/H) → W E .
?
HARMONIC ANALYSIS: e.g., W E ,→ L2 (G))
David Vogan,
David Jerison
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion
References
Realizing smooth
representations
David Vogan,
David Jerison
Dual of holom fns: chap 8 in A. Andreotti, Nine
lectures on complex Analysis, in book Complex
Analysis, F. Gherardelli, ed, pub 1974.
Dolbeault cohom realizations of representations: W.
Schmid, Homogeneous complex manifolds and
representations of semisimple Lie groups, in book
Rep Theory and Harm Analysis on Semisimple Lie
Gps, P. Sally et al., eds, pub 1989.
Representations on compactly supported Dolbeault
cohomology: D. Vogan, Unitary representations and
complex analysis, in book Representation Theory
and Complex Analysis, A. D’Agnolo, ed, pub 2008.
Spectral theory on symm spaces: S. Helgason,
Geom Analysis on Symm Spaces, pub 1994.
Introduction
Case of R
SL(2, R) and the
hyperbolic
Laplacian
Holomorphic
discrete series for
SL(2, R)
Conclusion