THE PLANCHEREL THEOREM ON SL2 (R)
JON XU
1. Introduction
The purpose of this document is to
(1) Define a representation of a SL2 (R) on a Hilbert space H.
(2) Give (some?) irreducible representations of SL2 (R).
(3) State the Plancherel theorem on SL2 (R).
The document is a product of some reading and chats between myself and Alex
Amenta (Australian National University) during late November 2013. The main
reference is ‘Representation Theory of Semisimple Groups’ by Knapp, and all references in square brackets [. . .] refer to this book.
2. Preliminaries
A complex Hilbert space V is a complex vector space with an inner product h−, −i
such that V is complete with respect to the distance function
p
d(x, y) = kx − yk = hx − y, x − yi.
A bounded linear operator on a complex Hilbert space V is a linear transformation
L : V −→ V suth that there exists M > 0 with
kLvk ≤ M kvk
for v ∈ V .
The operator norm kLk of a bounded linear operator L is the infimum of all such
M . [§I.3] Let
G be a topological group,
V be a complex Hilbert space, and
Binv (V ) be the group of bounded linear operators on V with bounded inverses.
A representation of G on V is a topological group homomorphism
π : G −→ Binv (V )
such that
(1) If v ∈ V then the map
φv : G −→
V
g 7−→ π(g)v
is continuous at the point g = I.
(2) There exists ε > 0 and K ∈ Z≥1 such that
kπ(g)k < K
for g ∈ B(I, ε), where B(I, ε) is the open ball with center I and radius ε.
Date: December 2, 2013.
1
2
JON XU
Let Cc∞ (G) be the ring of smooth compactly supported functions on G. Define
π 0 : Cc∞ (G) −→ Binv (V )
where
Z
0
π (f )v =
f (x)π(x)vdx
G
for f ∈ Cc∞ (G).
Proposition 1. The representation π can be recovered from π 0 .
Proof. TO DO
Because of Proposition 1, we will sometimes abuse notation and write π(f ) instead
of π 0 (f ), where f ∈ Cc∞ (G). So the domain of π can be either G or Cc∞ (G) depending
on context.
An invariant subspace U ∈ V of a representation π is a vector subspace of V such
that
if g ∈ G then π(g)U ⊆ U.
A representation π is irreducible if it has no nontrivial closed invariant subspaces.
Proposition 2. If
L : V −→ V
is a bounded linear operator then there exists a unique bounded linear operator
L∗ : V −→ V
such that
hLx, yi = hx, L∗ yi
for x, y ∈ V .
Proof. Riesz representation theorem.
The adjoint L∗ of a bounded linear operator L is the operator L∗ from Proposition
2.
A unitary representation π is a representation such that
π(g)π(g)∗ = π(g)∗ π(g) = I
for g ∈ G.
Example 3 (§I.3). The map
π : SL2 (R) −→ Binv (L2 (R2 ))
defined by
π(g)f (x) = f (g −1 x)
for f ∈ L2 (R2 ), is a unitary representation.
3. Some irreducible representations of SL2 (R)
The reference is [§II.5].
THE PLANCHEREL THEOREM ON SL2 (R)
3
3.1. Discrete series representations. Let n ∈ Z≥2 . The n-dimensional Hardy
space is the Hilbert space with vector space
)
(
f is analytic on the upper half plane,
RR
+
Hn = f : C −→ C and kf k2 =
|f (z)|2 y n−2 dxdy < ∞
Imz>0
with inner product
ZZ
hf, gi =
f (z)g(z)
dxdy
y n−2
Imz>0
for f, g ∈ Hn+ .
The discrete series representations of SL2 (R) are
Dn+ : SL2 (R) −→ Binv (Hn+ )
where
Dn+
az − c
a b
n
,
f (z) = (−bz + d) f
c d
−bz + d
and Dn− is defined ‘by complex conjugation’, from Knapp, for n ∈ Zn≥2 .
Theorem 4. The discrete series representations Dn+ and Dn− for n ∈ Zn≥2 are
irreducible unitary representations.
3.2. Principle series representations. The principle series representations of
SL2 (R) are
+
Piv
: SL2 (R) −→ Binv (L2 (R))
−
Piv
: SL2 (R) −→ Binv (L2 (R))
defined by
a
c
a
−
Piv
c
+
Piv
ax − c
b
−1−iv
f (x) = | − bx + d|
f
,
d
−bx + d
b
a b
+
f (x) = sgn(−bx + d) Piv
f (x)
d
c d
for v ∈ R and f ∈ L2 (R).
+
−
Theorem 5. The principle series representations Piv
and Piv
for v ∈ R are irre−
ducible unitary representations, with the exception of P0
4. Trace class operators
Let V be a complex Hilbert space with inner product h−, −i. A set of vectors
{ei }i∈Z≥1 is a Hilbert basis [from Wolfram Mathworld, better reference?] of V if
(1) hei , ej i = δij for i, j ∈ Z≥1 (orthonormality),
(2) If v ∈P
V then there exists ai ∈ C for i ∈ Z≥1 such that
(a)
|ai |2 < ∞, and
i∈Z≥1
P
(b)
ai ei = v.
i∈Z≥1
4
JON XU
Let L be a bounded linear operator on V and let {ei }i∈Z≥1 be a Hilbert basis for V .
The trace of L is
X
tr(L) =
hLei , ei i.
i∈Z≥1
A trace class operator L is a bounded linear operator such that there exists a Hilbert
basis {ei }i∈Z≥1 for which tr(L) converges.
Proposition 6 (§X.1). If L is a trace class operator then tr(L) is independent of
the choice of Hilbert basis.
5. The Plancherel theorem for SL2 (R)
Proposition 7. The discrete series representations Dn+ , Dn− and principal series
+
−
representations Piv
, Piv
of SL2 (R) are trace class.
Proof. ???
Theorem 8 (Plancherel Theorem, §II. 7, eqn (2.25)). If h ∈ Cc∞ (SL(2, R)) then
Z∞
Z∞
πv
πv
+
−
tr(Piv
h(1) =
(h))vtanh( )dv +
tr(Piv
(h))vcoth( )dv
2
2
+
−∞
∞
X
−∞
4(n − 1)tr(Dn+ (h) + Dn− (h))
n=2
for a suitable normalisation of Haar measure.
6. To Do
(1) Clean up the referencing.
(2) Fill in the gaps.
(3) Calculate the traces of the Discrete Series representations and the Principal
Series representations.
(4) What is the link to the representation theory of the Lie algebra sl2 ?
(5) Link with the ‘Plancherel Theorem’ from representation theory of finite
groups, which says that the sums of the squares of the dimensions of the
irreducibles of a group is equal to the cardinality of the group (apply this
result to Sn for an adorable combinatorial identity).