irreducible harish chandra modules for SL(2, R): reference guide
label
condition
name
C
F (d)
d ∈ Z, d > 1
trivial
d-dimensional
Dn+
n ∈ Z, n ≥ 1
Dn−
n ∈ Z, n ≥ 1
D0+
D0−
0
d − 1, . . . , −(d − 1)
holomorphic dis- n + 1, n + 3, n + 5, . . .
crete series
anti-holomorphic −(n + 1), −(n + 3), . . .
discrete series
limit of discrete
1, 3, 5, . . .
series
limit of discrete
−1, −3, −5, . . .
series
spherical unitary
0, ±2, ±4, . . .
principal series
nonspherical uni±1, ±3, ±5, . . .
tary principal series
complementary
0, ±2, ±4, . . .
series
P+ (ν)
ν ∈ iR≥0
P− (ν)
ν ∈ iR>0
P+ (ν)
ν ∈ (0, 1)
P+ (ν)
ν ∈ C: ℜ(ν) > principal series
0, ν ∈
/ (0, 1),
ν 6= 1, 3, . . .
ν ∈ C: ℜ(ν) > principal series
0, ν 6= 2, 4, . . .
P− (ν)
K-types
Ω action
0
− 1)
1 2
4 (d
infinitesimal
character
ρ
dρ
leading exps. unitary? comments
0
1
2 (−d
+ 1)
yes
no
unipotent
1
2
4 (n
− 1) nρ
1
2 (n
+ 1)
yes
discrete series
1
2
4 (n
− 1) nρ
1
2 (n
+ 1)
yes
discrete series
− 41
0
1
2
yes
− 41
0
1
2
yes
1
2
4 (ν
− 1)
νρ
1
2 (±ν
+ 1)
yes
tempered,
tent
tempered,
tent
tempered
1
2
4 (ν
− 1)
νρ
1
2 (±ν
+ 1)
yes
tempered
1
2
4 (ν
− 1)
νρ
1
2 (±ν
+ 1)
yes
Selberg’s 1/4 Conjecture: not automorphic
0, ±2, ±4, . . .
1
2
4 (ν
− 1)
νρ
1
2 (±ν
+ 1)
no
±1, ±3, ±5, . . .
1
2
4 (ν
− 1)
νρ
1
2 (±ν
+ 1)
no
Definition of principal series: Pε (ν) is the Harish-Chandra module of the normalized induced representation Ind∞ (ε ⊗ eν ⊗ 1|P, G); so Pε (ν) ≃ Pε (−ν) if Pε (ν) is irreducible.
1 (XY + Y X)
Canonical normalization of Casimir: Ω = 1
H2 + 2
4
Leading exponent: η a leading exponent means there is a matrix coefficient whose restriction to A grows like
!
a1/2
0
η
7→ a
0
a−1/2
as a tends to 0. Since a → 0 corresponds to going to infinity in G, ℜ(η) > 0 means “decay at infinity on G”.
unipounipo-
Reducibility of principal series. Fix a nonnegative integer n and set ε equal to the parity of n.
We have the following exact sequences:
0 −→ Dn− ⊕ Dn+ −→ Pε (n) −→ Fn −→ 0;
and
0 −→ Fn −→ Pε (−n) −→ Dn− ⊕ Dn+ −→ 0.
Character Formulas:
Principal series. If π = Ind∞ (ε ⊗ eν ⊗ 1|P, G), the character is given by the following function:
cos θ
sin θ
Θε,ν (g) = 0
if g is conjugate to
;
− sin θ cos θ
and
ε(eνt + e−νt )
Θε,ν (g) =
|et − e−t |
t
e
if g is conjugate to
0
0
e−t
.
Sums of discrete series. The character of Dn+ ⊕ Dn− (for n ≥ 1) is given by:
−einθ − e−inθ
cos θ sin θ
if g is conjugate to
;
Θn (g) =
− sin θ cos θ
eiθ − e−iθ
and
t
2e−n|t|
e
0
Θn (g) = ǫn+1 t
if
g
is
conjugate
to
ǫ
.
0 e−t
|e − e−t |
Individual discrete series. The character of Dn± (n ≥ 1) is given by:
∓e−inθ
cos θ sin θ
±
Θn (g) = iθ
if g is conjugate to
;
− sin θ cos θ
e − e−iθ
and
n+1
Θ±
n (g) = ǫ
e−n|t|
t
|e − e−t |
if g is conjugate to ǫ
t
e
0
0
e−t
.