Unitary
representations
Vogan
SL(2, R)
Unitary representations of
reductive Lie groups
David Vogan
Workshop on Unitary Representations
University of Utah
July 1–5, 2013
b
Picture of G
Formulas for
signatures
Outline
Unitary
representations
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
SL(2, R)
What’s a (unitary) dual look like?
Computing signatures of Hermitian forms
Gelfand’s abstract harmonic analysis
Unitary
representations
Vogan
Topological grp G acts on X , have questions about X .
Step 1. Attach to X Hilbert space H (e.g. L2 (X )).
Questions about X
questions about H.
Step 2. Find finest G-eqvt decomp H = ⊕α Hα .
Questions about H
questions about each Hα .
Each Hα is irreducible unitary representation of G:
indecomposable action of G on a Hilbert space.
b u = all irreducible unitary
Step 3. Understand G
representations of G: unitary dual problem.
Step 4. Answers about irr reps
answers about X .
This week: Step 3 for reductive Lie group G.
SL(2, R)
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Picture of G
Formulas for
signatures
Example: SL(2, R) on the upper half plane
Unitary
representations
Vogan
SL(2, R) acts on upper half plane H; ∆H = Laplacian.
SL(2, R)
repn E(ν) on ν 2 − 1 eigenspace of Laplacian ∆H
b
Picture of G
ν ∈ C parametrizes line bdle on circle where bdry values live.
Most E(ν) irreducible; always unique irr subrep J(ν) ⊂ E(ν).
Spherical reps for SL(2, R) ! C/±1
r1
−i∞
i∞
r−1
Spectrum of self-adjt ∆H on L2 (H) is (−∞, −1].
unitary principal series ! {E(ν) | ν ∈ iR}.
E(±1) = [harm fns on H] ⊃ [const fns on H] = J(±1) = triv rep.
J(ν) is Herm. ⇔ J(ν) ' J(−ν) ⇔ ν ∈ iR ∪ R.
By continuity, signature stays positive from 0 to ±1.
complementary series reps ! {E(t) | t ∈ (−1, 1)}.
Formulas for
signatures
Unitary
representations
The moral[s] of the picture
Vogan
Spherical unitary dual for SL(2, R) ! C/±1
r1
SL(2, R)
b
Picture of G
−i∞
i∞
r−1
SL(2, R)
E(ν), ν ∈ C
E(ν), ν ∈ iR
J(ν) ,→ E(ν)
[−1, 1]
G(R)
I(ν), ν ∈
a∗C
I(ν), ν ∈ ia∗R
I(ν) J(ν)
polytope in a∗R
Will deform Herm forms
unitary axis ia∗R
real axis a∗R .
Deformed form pos
unitary rep.
Reps appear in families, param by ν in cplx vec space a∗ .
Pure imag params ! L2 harm analysis ! unitary.
Each rep in family has distinguished irr piece J(ν).
Difficult unitary reps ↔ deformation in real param
Formulas for
signatures
Unitary
representations
Principal series for SL(2, R)
Vogan
Want to understand more explicitly analysis of repns E(ν)
for SL(2, R). Use different picture
SL(2, R)
I(ν, ) = {f : (R2 − 0) → C | f (tx) = |t|−ν−1 sgn(t) f (x)},
functions homogeneous of degree (−ν − 1, ).
Formulas for
signatures
The −1 next to −ν makes later formulas simpler.
Lie algs easier
basis
thanLie gps write
sl(2, R)
action,
D=
1
0
0
,
−1
[D, E] = 2E,
E=
0
0
1
,
0
[D, F ] = −2F ,
F =
0
1
0
,
0
[E, F ] = D.
action on functions on R2 is by
D = −x1
∂
∂
+ x2
,
∂x1
∂x2
E = −x2
∂
,
∂x1
F = −x1
∂
.
∂x2
Now want to restrict to homogeneous functions. . .
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Picture of G
Principal series for SL(2, R) (continued)
Study homog fns on R2 − 0 by restr to {(cos θ, sin θ)}:
I(ν, ) ' {w : S 1 → C | w(−s) = (−1) w(s)}, f (r , θ) = r −ν−1 w(θ).
Compute Lie algebra action in polar coords using
∂
∂
∂
∂
∂
∂
= −x2
= x1
+ x1 ,
+ x2 ,
∂x1
∂θ
∂r
∂x2
∂θ
∂r
∂
= −ν − 1,
x1 = cos θ,
x2 = sin θ.
∂r
Plug into formulas on preceding slide: get
ρν (D) = 2 sin θ cos θ
∂
+ (cos2 θ − sin2 θ)(ν + 1),
∂θ
∂
+ (cos θ sin θ)(ν + 1),
∂θ
∂
ρν (F ) = − cos2 θ
+ (cos θ sin θ)(ν + 1).
∂θ
ρν (E) = sin2 θ
Hard to make sense of. Clear: family of reps analytic
(actually linear) in complex parameter ν.
Big idea: see how properties change as function of ν.
Unitary
representations
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
A more suitable basis
Have family ρν, of reps of SL(2, R) defined on functions
on S 1 of homogeneity (or parity) :
∂
+ (cos2 θ − sin2 θ)(ν + 1),
ρν (D) = 2 sin θ cos θ
∂θ
∂
ρν (E) = sin2 θ
+ (cos θ sin θ)(ν + 1),
∂θ
∂
ν
2
ρ (F ) = − cos θ
+ (cos θ sin θ)(ν + 1).
∂θ
Problem: {D, E, F } adapted to wt vectors for diagonal Cartan
subalgebra; rep ρν, has no such wt vectors.
But rotation matrix E − F acts simply by ∂/∂θ.
Suggests new basis of the complexified Lie algebra:
H = −i(E − F ),
X =
1
(D + iE + iF ),
2
Y =
1
(D − iE − iF ).
2
Same commutation relations [H, X ] = 2X , [H, Y ] = −2Y ,
[X , Y ] = H, but cplx conj is different: H = −H, X = Y .
1 ∂
ρν (H) =
,
i ∂θ
e2iθ ∂
−e−2iθ ∂
ρν (X ) =
+ i(ν + 1) ,
ρν (Y ) =
− i(ν + 1) .
2i
∂θ
2i
∂θ
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
Matrices for principal series
Vogan
Have family ρν, of reps of SL(2, R) defined on functions
on S 1 of homogeneity (or parity) :
ρν (H) =
ρν (X ) =
e2iθ
2i
∂
+ i(ν + 1) ,
∂θ
b
Picture of G
1 ∂
,
i ∂θ
ρν (Y ) =
Formulas for
signatures
−e−2iθ
2i
∂
− i(ν + 1) .
∂θ
These ops act simply on basis wm (cos θ, sin θ) = eimθ :
ρν (H)wm = mwm ,
ρν (X )wm =
1
(m+ν+1)wm+2 ,
2
ρν (Y )wm =
1
(−m+ν+1)wm−2 .
2
Suggests reasonable function space to consider:
I(ν, )K = fns homog of deg (ν, ), finite under rotation
' trig polys on S 1 of parity 
SL(2, R)
= span({wm | m ≡ (mod 2)}).
Space I(ν, )K has beautiful rep of g: irr for most ν, easy
submods otherwise. Not preserved by rep of G = SL(2, R).
Invariant forms on principal series by hand
Unitary
representations
Vogan
Write I(ν) = I(ν, 0) = even fns homog of deg −ν − 1
SL(2, R)
Need “signature” of invt Herm form on inf-diml space.
b
Picture of G
Basis {wm | m ∈ 2Z}, wm ! eimθ , H · wm = mwm ,
Formulas for
signatures
X · wm =
1
(ν + m + 1)wm+2 ,
2
Y · wm =
1
(ν − (m − 1))wm−2 .
2
Requirements for invariant Hermitian form h, iν :
hH·w, w 0 iν = hw, H·w 0 iν ,
hX ·w, w 0 iν +hw, Y ·w 0 iν = 0.
Apply first requirement to w = wm , w 0 = wm0 ; get
mhwm , wm0 iν = m0 hwm , wm0 iν ,
and therefore hwm , wm0 iν = 0 for m 6= m0 .
So only need hwm , wm iν
(m ∈ 2Z). Second reqt says
((m + 1) + ν)hwm+2 , wm+2 iν = ((m + 1) − ν)hwm , wm iν .
Easy solution: ν imaginary, all hwm , wm iν equal
THM: For ν ∈ iR, L2 (S 1 /{±1})
I(ν, 0) unitary rep of G.
Invariant forms on I(ν) by hand, continued
Recall I(ν) = even functions on R2 , homog deg −ν − 1;
seeking invt Herm form h, iν , specified by values on basis
wm (r , θ) = r
−ν−1 imθ
e
Vogan
SL(2, R)
b
Picture of G
(m ∈ 2Z).
((m + 1) + ν)hwm+2 , wm+2 iν = ((m + 1) − ν)hwm , wm iν .
Non-imag ν: nonzero (real) solns exist iff ν ∈ R:
((m+1)+ν)hwm+2 , wm+2 iν = ((m+1)−ν)hwm , wm iν
Unitary
representations
(ν ∈ R).
Natural to normalize hw0 , w0 iν = 1, calculate
(1 − ν)
(1 − ν)(3 − ν)
hw±2 , w±2 iν =
, hw±4 , w±4 iν =
(1
+
ν)
(1 + ν)(3 + ν)
..
.
(1 − ν)(3 − ν) · · · (2m − 1 − ν)
hw±2m , w±2m iν =
(1 + ν)(3 + ν) · · · (2m − 1 + ν)
If ν ∈ (2m − 1, 2m + 1), sign alternates on w0 , w2 , . . . w2m .
pos def for 0 ≤ ν < 1; for ν > 1, sign diff on w0 , w2 .
h, iν “meromorphic” in (real) ν
One K -type-at-a-time calc too complicated to generalize.
Formulas for
signatures
Unitary
representations
Deforming signatures for SL(2, R)
Here’s representation-theoretic picture of deforming h, iν .
2
Vogan
ν = 0, I(0) “⊂” L (H): unitary, signature positive.
SL(2, R)
0 < ν < 1, I(ν) irr: signature remains positive.
b
Picture of G
ν = 1: form finite pos on quotient J(1) ! SO(2) rep 0.
Formulas for
signatures
ν = 1: form has simple zero, pos residue on ker(I(1) → J(1)).
1 < ν < 3, across zero at ν = 1, signature changes.
ν = 3: form finite − + − on quotient J(3).
ν = 3: form has simple zero, neg residue on ker(I(3) → J(3)).
3 < ν < 5, across zero at ν = 3, signature changes. ETC.
Conclude: J(ν) unitary, ν ∈ [0, 1]; nonunitary, ν ∈ (1, ∞).
···
−6
−4
−2
0
+2
+4
+6
···
SO(2) reps
···
···
···
···
···
···
+
+
+
−
−
+
+
+
+
−
−
+
+
+
+
−
−
−
+
+
+
+
+
+
+
+
+
−
−
−
+
+
+
−
−
+
+
+
+
−
−
+
···
···
···
···
···
···
ν=0
0<ν<1
ν=1
1<ν<3
ν=3
3<ν<5
From SL(2, R) to reductive G
Unitary
representations
Vogan
SL(2, R)
Calculated signatures of invt Herm forms on
spherical reps of SL(2, R).
Seek to do “same” for real reductive group. Need. . .
List of irr reps = ctble union (cplx vec space)/(fin grp).
reps for purely imag points “⊂” L2 (G): unitary!
Natural (orth) decomp of any irr (Herm) rep into fin-diml
subspaces
define signature subspace-by-subspace.
Signature at ν + iτ by analytic cont tν + iτ , 0 ≤ t ≤ 1.
Precisely: start w unitary (pos def) signature at t = 0; add
contribs of sign changes from zeros/poles of odd order in
0≤t ≤1
signature at t = 1.
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Picture of G
Formulas for
signatures
How to think about the unitary dual.
Unitary
representations
Vogan
Know a lot about complex repns of Γ algebraically.
Want to study unitarity of repns algebraically.
Helpful to step back, ask what we know about the
set of representations of Γ.
Short answer: it’s a complex algebraic variety.
Then ask Felix Klein question: what natural
automorphisms exist on set of representations?
Short answer: from auts of Γ and from lin alg.
Try to relate unitary structure to these natural things.
Short answer: they’re related to R-rational
structure on complex variety of repns.
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
What’s a set of irr reps look like?
Γ fin gen group, gens S = {σ}, relations R = {ρ}.
Relation is a noncomm word ρ =
m
σ1 1
· · · σnmn (σi
∈ S, mi ∈ Z).
N-dim rep π ! N × N matrices {π(σ) | σ ∈ S} subject to
alg rels π(ρ) = I for ρ ∈ R: π(σ1 )m1 π(σ2 )m2 · · · π(σn )mn = I.
Conclude: {N-dim reps of Γ} = aff alg var in GL(N, C)S .
S
Reduc reps are closed 0(W (CN {π | π(σ)W = W (σ ∈ S)}, so
irr N-dimls reps are open-in-affine alg variety.
Reps up to equiv: divide by GL(N, C) conj; still more or less alg
variety. (Possibly not separated, etc.)
Thm. Set b
Γfin of equiv classes of fin-diml reps of fin-gen Γ
is (approx) disjt union of complex alg vars.
Similar ideas apply to (g, K )-modules: reps containing fixed rep
of K with mult N are N-diml modules for a fin-gen cplx algebra.
[ of equiv classes of irr (g, K )-mods is
Thm. Set G(R)
(approx) disjt union of complex alg vars.
Langlands identifies alg vars as a∗ /W δ .
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
Group automorphisms acting on reps
Unitary
representations
Vogan
Γ fin gen group, τ ∈ Aut(Γ), (π, V ) rep of Γ
(π τ , V )
τ
new rep on same space, π (γ) =def π(τ (γ)).
Gives (right) action of Aut(G) on b
Γ.
Inner auts act trivially: linear isom π(γ0 ) intertwines π
and π Int(γ0 ) since π Int(γ0 ) (γ)π(γ0 ) = π(γ0 )π(γ).
(Easy) Thm. Out(Γ) =def Aut(Γ)/ Int(Γ) acts by
algebraic variety automorphisms on b
Γfin .
(Easy) Thm. Out(G(R)) acts by algebraic variety
[
automorphisms on G(R).
Main technical point: each aut of G(R) can be
modified by inner aut so as to preserve K ; so get
action on (g, K )-modules.
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
Bilinear forms and dual spaces
V cplx vec space (or (g, K )-module).
Dual of V V ∗ = {ξ : V → C additive | ξ(zv ) = zξ(v )}
(V alg K -rep
require ξ K -finite; V topolog.
∗
N
V = C N × 1 column vectors
Vogan
require ξ cont.)
N
t
V = C , ξ(v ) = ξv .
Bilinear pairings between V and W
Bil(V , W ) = {h, i : V × W → C, lin in V , lin in W }
Bil(V , W ) ' Hom(V , W ∗ ), hv , wiT = (Tv )(w).
Exchange vars in forms to get linear isom
Bil(V , W ) ' Bil(W , V ).
Corr lin isom on maps is transpose:
Hom(V , W ∗ ) ' Hom(W , V ∗ ),
(TS)t = S t T t ,
(T t w)(v ) = (Tv )(w).
(zT )t = z(T t ).
Bil form h, iT on V (! T ∈ Hom(V , V ∗ )) orthogonal if
hv , v 0 iT = hv 0 , v iT ⇐⇒ T t = T .
Bil form h, iT on V (! T ∈ Hom(V , V ∗ )) symplectic if
hv , v 0 iS = −hv 0 , v iS ⇐⇒ S t = −S.
SL(2, R)
b
Picture of G
Formulas for
signatures
Defining contragredient repn
Unitary
representations
Vogan
(π, V ) (g, K )-module; had (K -finite) dual space V ∗ of V .
SL(2, R)
Want to construct functor
cplx linear rep (π, V )
b
Picture of G
cplx linear rep (π ∗ , V ∗ )
using transpose map of operators.
Because transpose is antiaut REQUIRES twisting by
antiaut of (g, K ).
X 7→ −X is Lie alg antiaut, and k 7→ k −1 group antiaut
Define contragredient (g, K )-module π ∗ on V ∗ ,
π ∗ (Z ) · ξ=def [π(−Z )]t · ξ
(Z ∈ g, ξ ∈ V ∗ ),
π ∗ (k ) · ξ=def [π(k −1 )]t · ξ
(k ∈ K , ξ ∈ V ∗ ).
Thm. If Γ is a fin gen group, passage to contragredient is
an involutive automorphism of the algebraic variety b
Γ.
Thm. If G(R) real reductive, passage to contragredient is
[
an involutive automorphism of the algebraic variety G(R).
Formulas for
signatures
Unitary
representations
Invariant bilinear forms
V = (g, K )-module, τ involutive aut of (g, K ).
Vogan
An invt bilinear form on V is bilinear pairing h, i such that
hZ · v , wi = hv , −Z · wi,
hk · v , wi = hv , k
−1
· wi
(Z ∈ g; k ∈ K ; v , w ∈ V ).
Proposition
Invt bilinear form on V ! (g, K )-map T : V → V ∗ :
hv , wiT = (Tv )(w).
Form is orthogonal ⇐⇒ T t = T .
Form is symplectic ⇐⇒ T t = −T .
Assume from now on V is irreducible.
V ' V ∗ ⇐⇒ ∃ invt bilinear form on V
Invt bil form on V unique up to real scalar mult.;
non-deg whenever nonzero.
Invt bil form must be either orthogonal or symplectic.
T → T ∗ ! involution of cplx line Homg,K (V , V ∗ ).
[
Existence of invt bil form ! compute V 7→ V ∗ on G(R).
Deciding orth/symp usually somewhat harder.
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
Hermitian forms and dual spaces
Vogan
V cplx vec space (or (g, K )-module).
Herm dual of V V h = {ξ : V → C additive | ξ(zv ) = zξ(v )}
(V alg K -rep
require ξ K -finite; V topolog.
V = CN N × 1 column vectors
require ξ cont.)
V h = CN , ξ(v ) = t ξv .
Sesquilinear pairings between V and W
Sesq(V , W ) = {h, i : V × W → C, lin in V , conj-lin in W }
Sesq(V , W ) ' Hom(V , W h ),
hv , wiT = (Tv )(w).
Cplx conj of forms is (conj linear) isom
Sesq(V , W ) ' Sesq(W , V ).
Corr (conj lin) isom on maps is Hermitian transpose:
Hom(V , W h ) ' Hom(W , V h ),
(TS)h = S h T h ,
(T h w)(v ) = (Tv )(w).
(zT )h = z(T h ).
Sesq form h, iT on V (! T ∈ Hom(V , V h )) Hermitian if
hv , v 0 iT = hv 0 , v iT ⇐⇒ T h = T .
Could define “skew Hermitian” by analogy with orth/symp bil
forms. Exercise: why is this boring?
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
Defining Herm dual repn(s)
(π, V ) (g, K )-module; Recall Herm dual V h of V .
Want to construct functor
cplx linear rep (π, V )
Vogan
SL(2, R)
cplx linear rep (π h , V h )
using Hermitian transpose map of operators.
Definition REQUIRES twisting by conj lin antiaut of g, gp
antiaut of K .
Since g equipped with a real form g0 , have natural conj-lin aut
σ0 (X + iY ) = X − iY (X , Y ∈ g0 ). Also X 7→ −X is Lie alg
antiaut, and k 7→ k −1 gp antiaut.
Define Hermitian dual (g, K )-module π h on V h ,
π h (Z ) · ξ=def [π(−σ0 (Z ))]h · ξ
(Z ∈ g, ξ ∈ V h ),
π h (k ) · ξ=def [π(k −1 )]h · ξ
(k ∈ K , ξ ∈ V h ).
Need also a variant: suppose τ inv aut of G(R) preserving
K . Define τ -herm dual (g, K )-module π h,τ on V h ,
π h,τ (X ) · ξ = [π(−τ (σ0 (Z ))]h · ξ
(Z ∈ g, ξ ∈ V h ),
π h,τ (k ) · ξ = [π(τ (k )−1 )]h · ξ
(k ∈ K , ξ ∈ V h ).
b
Picture of G
Formulas for
signatures
Unitary
representations
Invariant Hermitian forms
For τ an inv aut of (G(R), K ), defined τ -herm dual
π h,τ (X ) · ξ = [π(−τ (σ0 (Z ))]h · ξ
(Z ∈ g, ξ ∈ V h ),
π h,τ (k ) · ξ = [π(τ (k )−1 )]h · ξ
(k ∈ K , ξ ∈ V h ).
[
Thm. τ -herm dual is Galois for R-struc on alg var G(R).
Reason: conj transpose is real Galois action on GL(N, C).
A τ -invt sesq form on (g, K )-module V is pairing h, iτ with
hZ · v , wi = hv , −τ (σ0 (Z )) · wi,
hk · v , wi = hv , τ (k −1 ) · wi
(Z ∈ g; k ∈ K ; v , w ∈ V ).
Prop. τ -invt sesq form on V ! (g, K )-map T : V → V h,τ :
hv , wiT = (Tv )(w).
Form is Hermitian ⇐⇒ T h = T .
Assume from now on V is irreducible.
V ' V h,τ ⇐⇒ ∃τ -invt sesq ⇐⇒ ∃τ -invt Herm
τ -invt Herm form on V unique up to real scalar mult.
T → T h ! real form of cplx line Homg,K (V , V h,τ ).
Deciding existence of τ -invt Hermitian form amounts to
[ easy.
computing the involution V 7→ V h,τ on G(R);
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
Hermitian forms and unitary reps
Unitary
representations
Vogan
G(R) real reductive, τ inv aut preserving K . . .
π rep of G(R) on complete loc cvx Vπ , Vπh Hermitian dual
space. Hermitian dual reps are
π h (g) = π(g −1 )h ,
π h,τ (g) = π(τ (g −1 )h
Def. A τ -invariant form is continuous Hermitian pairing
h, iτπ : Vπ × Vπ → C,
hπ(g)v , wiτπ = hv , π(τ (g −1 ))wiτπ .
Equivalently: T ∈ HomG(R) (Vπ , Vπh,τ ), T = T h .

6
Because infl equiv easier than topol equiv, Vπ ' Vπh,τ =⇒
existence of a continuous map Vπ → Vπh . So invt forms may not
exist on topological reps even if they exist on (g, K )-modules.
Thm. (Harish-Chandra) Passage to K -finite vectors
[ onto
defines bijection from the unitary dual G(R)
u
equivalence classes of irreducible (g, K ) modules
admitting a pos def invt Hermitian form.
[ .
Despite warning, get perfect alg param of G(R)
u
SL(2, R)
b
Picture of G
Formulas for
signatures
What are we planning to do?
Unitary
representations
Vogan
Know: irr (g, K )-module π unitary ⇐⇒ (a) π admits invt
herm form, and (b) form is positive definite.
Know: π has invt herm form ⇐⇒ π ' π h : fixed by herm
[ Easy to list these π.
dual involution of G(R).
If π ' π h , want to know: is corr form h, iπ pos def?
Method: compute the signature of h, iπ . Meaning?
SIGNATURE: (π, Vπ ) (g, K )-module with form h, iπ .
X
Vπ =
Eδ ⊗ Vπδ ,
Vπδ =def HomK (Eδ , Vπ )
b
(δ,Eδ )∈K
h, iπ =
X
h, iδ ⊗ h, iδπ
δ
Def. Signature of h, iπ at δ is sig (pπ (δ), qπ (δ), zπ (δ)) of
h, iδπ on mult space Vπδ . Signature of h, iπ is (pπ , qπ , zπ ),
b → N.
three functions K
Note pπ + qπ + zπ = mπ , mult fn for reps of K in π.
SL(2, R)
b
Picture of G
Formulas for
signatures
What are we planning to do (continued)?
b → N3 .
signature: (pπ , qπ , zπ ) : K
Invt form h, iπ on Vπ
Form semidefinite ⇐⇒ one of pπ , qπ is zero.
b is infinite, so
What’s it mean to compute pπ ? Domain K
tabulating won’t work.
Answer: write pπ as finite linear combination
X
b → Z)
pπ =
aπi mi
(ai ∈ Z, mi : K
i∈I
with {mi | i ∈ I} lin ind set of “standard” functions.
COMPUTE means compute the finitely many integers aπi .
Checking pπ = 0 means checking finitely many ints are all
zero.
P
If qπ = i bπi mi , then realex writes signature as
X
(aπi + sbπi )mi .
i
Definite means either all aπi = 0 or all bπi = 0.
Unitary
representations
Vogan
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
The “standard multiplicity functions.”
b → Z. (
Need basis {mi } of mult fns mi : K
Obvious choice: delta functions mδ (µ) =
Vogan
1
0
(µ = δ)
.
(µ 6= δ)
Difficulty: need inf many mδ to write mult for inf-diml π.
Pretty obvious choice: multiplicity functions for standard
representations. Case of SL(2, R):
(
1 j≡
0 j≡
6 (
1 j = ±(n + 1), ±(n + 3), · · ·
0 otherwise
mI(ν,) (j) =
m(L)DS± (n) (j) =
(mod 2)
(mod 2).
Difficulty: not linearly independent: mI(ν,) − mI(ν 0 ,) = 0,
mLDS+ (0) + mLDS− (0) − mI(ν,odd) = 0.
Right choice: mult fns for I = {irr temp, real inf char}.
Thm. The mult fns {mi | i ∈ I} are basis for signature and
mult fns of finite-length (g, K )-mods. πi has unique lowest
b.
K -type δi with mult=1; gives bijection I ↔ K
SL(2, R)
b
Picture of G
Formulas for
signatures
Unitary
representations
Character formulas
Can decompose Verma module into irreducibles
X
V (λ) =
mµ,λ L(µ)
(mµ,λ ∈ N)
µ≤λ
µ≤λ
Can decompose standard HC module into irreducibles
X
I(x) =
my ,x J(y )
(my ,x ∈ N)
y ≤x
or write a formal character for an irreducible
X
My ,x I(y )
(My ,x ∈ Z)
J(x) =
y ≤x
Matrices m and M upper triang, ones on diag, mutual
inverses. Entries are KL polynomials eval at 1:
My ,x = ±Py ,x (1)
SL(2, R)
b
Picture of G
or write a formal character for an irreducible
X
L(λ) =
Mµ,λ V (µ)
(Mµ,λ ∈ Z)
my ,x = Qy ,x (1),
Vogan
(Qy ,x , Py ,x ∈ N[q]).
Formulas for
signatures
Unitary
representations
Character formulas for SL(2, R)
Vogan
Rewrite formulas from Jeff’s talk in general G notation. . .
Had I(ν, ) J(ν, ); and disc ser I± (n) = DS± (n) (n ≥ 1)
I− (n)|SO(2) = −n − 1, −n − 3, −n − 5 · · ·
Discrete series reps are irr: I± (n) = J± (n)
Decompose principal series (m pos int)
I(m, (−1)m+1 ) = J(m, (−1)m+1 ) + J+ (m) + J− (m).
Character formula
J(m, (−1)m+1 ) = I(m, (−1)m+1 ) − I+ (m) − I− (m).
I(m, (−1)m+1 )
1
0
0
I+ (m)
−1
1
0
b
Picture of G
Formulas for
signatures
I+ (n)|SO(2) = n + 1, n + 3, n + 5 · · ·
±Px,y
J(m, (−1)m+1 )
J+ (m)
J− (m)
SL(2, R)
I− (m)
−1
0
1