1. Character Formulas, Dario Beraldo, UC Berkeley
Weyl character formula for compact Lie groups. Let G be a compact
Lie group and fix a maximal torus T ⊂ G, with Lie algebra t. Inside it∗ we
have some lattices:
roots of g ⊂ Hom(T, S 1 ) ⊂ weights of g
Here we consider Hom(T, S 1 ) ⊂ it∗ as follows: there is an isomorphism (L :=
ker exp|t )
∼
=
exp : t/L → T.
Defining
Λ :=
L
2πi
∗
= {φ ∈ it∗ | φ(L) ⊂ 2πiZ},
we see that Λ corresponds to Hom(T, S 1 ) via exponentiation. The key point
is that, for φ ∈ Λ, the following function on T is well-defined: eφ : exp(X) 7→
eφ(X) .
It is well-known that irreducible reps of G are in bijection with Λ+ (the
intersection of Λ with the positive Weyl chamber), where λ ∈ Λ+ maps to the
irrep (πλ , Vλ ) with highest weight λ. The Weyl character formula for X ∈ t
computes that trace of πλ (eX ):
P
(w·(λ+ρ))(X)
w (w)e
,
χλ (exp(X)) := tr(πλ (exp X)) = ρ(X)
e
Πα>0 (1 − e−α(X) )
P
P
where ρ = 12 α>0 α = `i=1 wi is the Weyl vector. The denominator of this
formula is independent of λ and is called the Weyl denominator.
Remark: in general w · (λ + ρ) and ρ don’t belong to Λ, so that ew · (λ + ρ)
and eρ are not well defined, but their ratio is (as w(ρ) − ρ is always in the root
lattice).
Kirillov character formula for compact Lie groups. Now we will explain
Kirillov’s orbit method philosophy. Let G any Lie group (not necessarily
compact). We know that G acts on its Lie algebra g by the adjoint action;
dualizing we get an action of G on g∗ , the coadjoint action, denoted by K.
g · φ = K(g)φ = φ ◦ Ad(g −1 ). (φ ∈ g∗ , g ∈ G).
It’s also worthwhile to write down the action K∗ of g on its dual:
K∗ (X)φ = φ ◦ (−ad(X))
(φ ∈ g∗ , X ∈ g).
Now, g∗ is partitioned into coadjoint orbits and Kirillov’s philosophy says:
the set of coadjoint orbits in g∗ is in bijection with the set of unitary irreducible representations of G.
1
2
The way of attaching to Ω (a coadjoint orbit) a unitary irrep πΩ may go
under the name of quantization. Indeed, the key property of coadjoint orbits
is the following:
Proposition 1.1. Each coadjoint orbit is a homogeneous symplectic manifold.
Then the orbit Ω, being symplectic, may be regarded as the classical phase
space of a physical system with symmetry G. Quantizing, one gets a quantum
phase space, namely a unitary representation πΩ of G on an Hilbert space.
The fact that Ω is homogeneous should correspond to the irreducibility of πΩ .
Unfortunately, this doesn’t work for all Lie groups, for example the above
is false for a compact Lie group, as we will see. However this does work if G
is 1-connected with nilpotent Lie algebra.
Using the symplectic form, denoted σ, we can define a measure on Ω,
namely the volume form σ d /d!, and Kirillov character formula says that
χπΩ = µˆΩ .
Again, the character is the trace of the representation evaluated on the exponential of a Lie algebra element.
Before looking at an example, let me explain Proposition 1.1. Picking a
point φ ∈ Ω, we get the following isomorphism
G/Stabφ → Ω
g 7→ K(g)φ.
Since Stabφ = {g | K(g)φ = φ} is a closed group of G, each orbit is Ghomogeneous.
Now we wish to define a form σ (that will turn out to be symplectic). By
homogeneity, we just need to define a skew-symmetric bilinear form on Tφ Ω,
invariant under Stabφ . Clearly,
Tφ Ω = g/sφ
where
sφ = Lie(Stabφ ) = {X ∈ g|K∗ (X)φ = 0}.
Then
σ :g×g→R
(X, Y ) 7→ φ([X, Y ]).
By the very definition of K∗ , one sees that ker σ = sφ , so that σ descends
to a non-degenerate form on Tφ Ω. Furthermore, one can easily show that σ
is closed using the Jacobi identity. This completes the proof of the proposition.
3
For an illustration of the orbit philosophy, let’s consider the Heisenberg
group of 3 × 3 upper triangular matices with 1’s on the diagonal:


1 a c
H := {gabc :=  0 1 b  |a, b, c ∈ R}.
0 0 1
It is simply connected and nilpotent.
First, let me describe the irreducible representations of H: they are either
infinite-dimensional, denoted πλ with λ ∈ R∗ , or one-dimensional, πµ,ν with
(µ, ν) ∈ R2 .
Here are the definitions:
πλ (gabc ) : L2 (R, dx) → L2 (R, dx)
with
(πλ (gabc )f )(x) = e2πλi(bx+c) f (x + a)
(this rep is obviously unitary and it is irreducible by Wiener’s theorem).
πµ,ν (gabc ) = e2πi(aµ+bν) .
Accepting this, let’s compute the coadjoint orbits. Using the non-degenerate
pairing
hA, Bi = tr(AB),
we identify g∗ = M at(3 × 3)/g⊥ , which is M at(3 × 3) modulo the entries on
and above the main diagonal. Then cadjoint orbits are equivalent to adjoint
ones, and we compute






∗ ∗ ∗
∗
∗
∗
∗ ∗ ∗
−1
∗
∗ 
=  x + bz
K(gabc )  x ∗ ∗  = gabc  x ∗ ∗  gabc
z y ∗
z
y − az ∗
z y ∗
Thus we see that the coordinate z is fixed on the orbit; let z = λ. If λ 6= 0, by
changing a, b we notice that the orbit is the entire plane {z = λ} in g∗ ∼
= R3 .
On the contrary, if λ = 0, orbits are points {(µ, ν, 0)}.
This suggests the correspondence of Ω := Ωλ = {z = λ} and πλ ; let’s
verify the character formula in this case.
We have a measure on these orbits, given by the symplectic volume (I omit
the short calculation),
dx ∧ dy
µΩ := σ =
λ
and need to verify
χπΩ = µˆΩ ,
in distributional sense (since we are dealing with an infinite-dimensional rep),
namely (χπΩ , φ) = (µ̂Ω , φ) = (φ̂, µΩ ).
Given a test function φ ∈ S(G) (G is the Heisenberg group) we define the
operator (it’s πλ weighted by φ)
Z
πλ (φ) = (πλ (g), φ) =
πλ (gabc )φ(a, b, c)dadbdc.
G
4
Now,
Z
φ(a, b, c)e2πiλ(bx+c) f (x + a)dadbdc
(πλ (φ)(f ))(x) =
R3
Z
φ(y − x, b, c)e2πiλ(bx+c) f (y)dydbdc
Z Z
=
φ̂(y − x, λx, c)e2πiλc f (y)dcdy
R R
Z Z
2πiλc
=
φ̂(y − x, λx, c)e
dc f (y)dy
R
R
Z
=
K(x, y)f (y)dy,
=
R3
R
where K(x, y) =
R
2πiλc
φ̂(y
−
x,
λx,
c)e
.
dc
R
Since φ̂ is Schwartz, this operator is trace class and the trace is obtained
integrating the kernel K along the diagonal:
Z Z
Z
φ̂(0, λx, c)e2πiλc dcdx =
K(x, x)dx =
trπλ (φ) =
R
R
1
λ
Z Z
2πiλc
φ̂(0, u, c)e
R
R
R
Z
dcdu =
φ(0, 0, c)
R
Thus
χπΩ =
e2πiλc
dc,
λ
e2πiλc
δ(a)δ(b).
λ
On the other hand,
dx ∧ dy
φ(a, b, c)e
dadbdc
(µˆΩ , φ) = (µΩ , φ̂) =
λ
R3
Ω
Z λ Z
dxdy
φ(a, b, c)e2πi(ax+by+cλ)
dadbdc
=
λ
R3
Ωλ
Z
Z
2πicz
Z
e
2πiax
2πiby
e
dx
e
dy
dadbdc
= φ(a, b, c)
λ
Z
e2πicz
= φ(a, b, c)δ(a)δ(b)
dadbdc,
λ
in accordance with Kirillov’s formula.
Z
Z
2πi(ax+by+cz)
For compact groups, Kirillov’s character formula needs to be modified in
three ways. First, there are too many coadjoint orbits and too few irreducible
representations of G. Namely, using the Killing form on g, we know that
coadjoint orbits are identified with coadjoint orbits. Now, adjoint orbits are
flag varieties, G/Γ, for Γ conjugated to a subgroup containing T , and it is
well-known that each orbit intersects the positive Weyl chamber in exactly one
point. So, coadjoint orbits are parametrized by the positive Weyl chamber,
while unirreps are parametrized by the discrete set Λ+ .
5
Another problem is the more complicated (compared to nilpotent groups)
nature of the map exp : g → G, that forces to introduce a distorsion factor
p(X) =
sinh(ad X/2)
ad X/2
in the character formula. A third problem is a ρ-shift.
As an example, let’s look at SU (2). Coadjoint orbits are flag varieties,
i.e. spheres, which are centered at 0 ∈ su(2). So let Ωr be the sphere of
radius r, and, after passing to spherical cordinates, σ = r sin φdθ ∧ dφ (this is
a cumbersome computation). Let Z = diag(iz, −iz) ∈ su2 . Then
Z
sin rz
e2πi<Z,F > dσ(F ) =
z
Ωr
Dividing by
p(Z) =
sin z
sinh(ad Z/2)
=
ad Z/2
z
we get
Z
1 sin rz
sin rz
1
e2πi<Z,F > dσ(F ) =
=
.
p(Z)
p(Z) z
sin z
Now notice that, if r = n + 1, the RHS is exactly the character of the
representation of SU (2) corresponing to the weight n. We see here the ρ-shift,
as ρ ∼
= 1 in this case.
To summarize, the Kirillov character formula (for G compact):
Theorem 1.2 (Kirillov). For λ ∈ Λ+ and X ∈ t, we have
Z
1
χλ (exp(X)) =
e2πi<X,F > dµ(F )
p(X) Ωλ+ρ
Loop groups. Both formulas admit generalizations to the case of loop groups.
We have a central extension of Lg
˜ ∼
Lg
= Lg ⊕ RC
and we also define
ˆ = Lg ⊕ RC ⊕ RD
Lg
with
[D, X ⊗ z n ] = nX ⊗ z n , [D, C] = 0.
We have an invariant bilinear form defined by
< D, C >= 1,
< X⊗z m , Y ⊗z n >=< X, Y > δm,−n , the other products are zero.
ˆ acting on Lg
ˆ ∗ , but we’ll
In principle we want to calculate the action of LG
˜ ∗ . Suitably interpreted, this action has an intrinsic
just do LG acting on Lg
geometric meaning. We have the sequence
˜ → Lg → 0
0 → RC → Lg
and its dual
˜ ∗ → Rδ → 0
0 → (Lg)∗ → Lg
6
where δ is an element determined by δ : C 7→ 1. Then
˜ ∗ := {φ : Lg
˜ → R|φ(C) = `} = lδ + (Lg)∗ .
(Lg)
`
Moreover,
(Lg)∗ = Ω1 (S 1 ; g∗ )
R
via the pairing (X ⊗ f, Y ⊗ ω) =< X, Y > f ω. Using the Killing form, we
finally obtain
(Lg)∗ = Ω1 (S 1 ; g∗ ) = Ω1 (S 1 ; g),
˜ ∗ = lδ + Ω1 (S 1 ; g), which is isomorphic to the space of connections
so that (Lg)
`
on the trivial principle G bundle over S 1 .
˜ ∗ , interpreted as a space of
Let’s now describe the coadjoint action on (Lg)
`
connections.
Let x, y ∈ Lg, ω ∈ Ω1 (g), α ∈ R:
< K∗ (x)(`δ + ω), y + αC > = − < `δ + ω, [x, y + αC] >
= − < `δ + ω, [x, y] >
= −` < dx, y > − < ω, [x, y] >
= −` < dx, y > + < [x, ω], y >
=< −`dx + [x, ω], y + αC >
so we find that the coadjoint action corresponds to the infinitesimal gauge
action
x · (`δ + ω) = −`dx + [x, ω] = [x, `d + ω]
So LG acts on (Lg)∗` by γ · (d + ω) = γ(d + ω)γ −1 , and we see
(Lg)∗` /LG ∼
= A/LG = G//G
ˆ I. Frenkel has extended Kirillov
1.1. Kirillov character formula for LG.
character formula to loop groups. Let λ be a dominant weight of LG. Then
Z
1
b/a(H,H)
χλ (exp(bD + Y )) =
e
e2πi<bD+Y,F > dµa/b (F ).
p(bD + Y )
Ωλ+ρ̂
Hereρ̂ is the Kac-Weyl vector to be defined below, p is a function depending
only on G and µa/b is a particular measure on the orbit Ωλ+ρ̂ (that I won’t
describe). We also need to know what H and a are.
This formula is defined only when Im(b) < 0; indeed only in this case
the exponential inside the integral is rapidly descreasing, making the integral
converge.
ˆ are
Recall that the roots of LG
G
{Rg + ZC} (Z − {0})C,
where Rg are the roots of G.
The simple roots are α̂1 := α1 , . . . , α̂` := α` and α̂0 := C − θ. In other
words, the simple roots for L̂G are the simple roots of G, plus α̂0 . Here θ
7
is the top root of G. The fundamental weights, denoted {ωˆ0 , . . . , ω̂` }, must
satisfy the following defining relations:
2(ω̂i , α̂j )
= δij .
(α̂j , α̂j )
The generalized Weyl vector is defined to be
ρ̂ =
`
X
ω̂i
i=0
and it can be computed explicitely: ρ̂ =
G). Now, we can uniquely write
D
2
+ ρ (here ρ is the Weyl vector of
λ + ρ̂ = aD + H
for some a > 0 and H ∈ t ⊗ C.