Fixed Point Theorem and Character Formula
Hang Wang
University of Adelaide
Index Theory and Singular Structures
Institut de Mathématiques de Toulouse
29 May, 2017
Outline
Aim: Study representation theory of Lie groups from the point
of view of geometry, motivated by the developement of
K-theory and representation;
Harmonic analysis on Lie groups.
Representation theory
character
of representations
Geometry
index theory
of elliptic operators
Weyl character formula
Atiyah-Segal-Singer
Harish-Chandra character formula
Fixed point theorem
P. Hochs, H.Wang, A Fixed Point Formula and
Harish-Chandra’s Character Formula, ArXiv 1701.08479.
Representation and character
b irreducible unitary representations of G (compact, Lie);
G:
b the character of π is given by
For (π, V ) ∈ G,
χπ (g) = Tr[π(g) : V → V ]
g ∈ G.
Example
Consider G = SO(3) with maximal torus T1 ∼
= SO(2) ,→ SO(3).
\
Let Vn ∈ SO(3) with highest weight n, i.e.,
Vn |T1
∼
=
2n
M
Cj−n
j=0
where Cj = C, on which T1 acts by g · z = g j z, g ∈ T1 , z ∈ C.
Then
2n
X
χVn (g) =
g j−n
g ∈ T1 .
j=0
Weyl character formula
Let G be a compact Lie group with maximal torus T .
√
b Denote by λ ∈ −1t∗ its highest weight.
Let π ∈ G.
Theorem (Weyl character formula)
At a regular point g of T :
P
det(w)ew(λ+ρ)
χπ (g) = w∈W
(g).
eρ Πα∈∆+ (1 − e−α )
Here, W = NG (T )/T is P
the Weyl group, ∆+ is the set of
positive roots and ρ = 12 α∈∆+ α.
Elliptic operators
M : closed manifold.
Definition
A differential operator D on a manifold M is elliptic if its
principal symbol σD (x, ξ) is invertible whenever ξ 6= 0.
{Dirac type operators} ⊂ {elliptic operators}.
Example
de Rham operator on a closed oriented even dimensional
manifold M :
D± = d + d∗ : Ω∗ (M ) → Ω∗ (M ).
Dolbeault operator ∂¯ + ∂¯∗ on a complex manifold.
Equivariant Index
G: compact Lie group acting on compact M by isometries.
R(G) := {[V ] − [W ] : V, W fin. dim. rep. of G} representation
ring of G (identified as rings of characters).
Definition
The equivariant
index of a G-invariant elliptic operator
0 D−
D=
on M , where (D+ )∗ = D− is given by
D+ 0
indG D = [ker D+ ] − [ker D− ] ∈ R(G);
It is determined by the characters
indG D(g) := Tr(g|ker D+ ) − Tr(g|ker D− )
∀g ∈ G.
Example. Lefschetz number
Consider the de Rham operator on a closed oriented even
dimensional manifold M :
D± = d + d∗ : Ωev/od (M ) → Ωod/ev (M ).
ev/od
ker D± ↔ harmonic forms ↔ HDR (M, R).
Lefschetz number, denoted by L(g):
indG D(g) =Tr(g|ker D+ ) − Tr(g|ker D− )
X
=
(−1)i Tr [g∗,i : Hi (M, R) → Hi (M, R)] .
i≥0
Theorem (Lefschetz)
If L(g) 6= 0, then g has a fixed-point in M.
Fixed point formula
M : compact manifold.
g ∈ Isom(M ).
M g : fixed-point submanifold of M .
Theorem (Atiyah-Segal-Singer)
Let D : C ∞ (M, E) → C ∞ (M, E) be an elliptic operator on M .
Then
indG D(g) = Tr(g|ker D+ ) − Tr(g|ker D− )
Z
ch [σD |M g ](g) Todd(T M g ⊗ C)
V =
ch
NC (g)
T Mg
where NC is the complexified normal bundle of M g in M .
Equivariant index and representation
√
b with highest weight λ ∈ −1t∗ . Choose M = G/T
Let π ∈ G
and the line bundle Lλ := G ×T Cλ . Let ∂¯ be the Dolbeault
operator on M .
Theorem (Borel-Weil-Bott)
The character of an irreducible representation π of G is equal to
the equivariant index of the twisted Dolbeault operator
∂¯Lλ + ∂¯L∗ λ
on the homogenous space G/T .
Theorem (Atiyah-Bott)
For g ∈ T reg ,
indG (∂¯Lλ + ∂¯L∗ λ )(g) = Weyl character formula.
Example
Let ∂¯n + ∂¯n∗ be the Dolbeault–Dirac operator on
S2 ∼
= SO(3)/T1 ,
coupled to the line bundle
Ln := SO(3) ×T1 Cn → S 2 .
By Borel-Weil-Bott,
indSO(3) (∂¯n + ∂¯n∗ ) = [Vn ] ∈ R(SO(3)).
By the Atiyah-Segal-Singer’s formula
2n
indSO(3) (∂¯n + ∂¯n∗ )(g) =
X
g −n
gn
+
=
g j−n .
1 − g −1 1 − g
j=0
Overview of main results
Let G be a compact group acting on compact M by isometries.
From index theory,
G-inv elliptic operator D → equivariant index → character
Geometry plays a role in representation by
R(G) → special D and M → character formula
When G is noncompact Lie group, we
Construct index theory and calculate fixed point formulas;
Choose M and D so that the character of indG D recovers
character formulas for discrete series representations of G.
The context is K-theory:
“representation, equivariant index ∈ K0 (Cr∗ G).”
Discrete series
b is a discrete series of G if the matrix corficient cπ
(π, V ) ∈ G
given by
cπ (g) = hπ(g)x, xi for kxk = 1
is L2 -integrable.
b are discrete series, and
When G is compact, all G
b d ).
K0 (Cr∗ G) ' R(G) ' K 0 (G
When G is noncompact,
b d ) ≤ K0 (C ∗ G)
K 0 (G
r
where [π] corresponds [dπ cπ ] (dπ = kcπ k−2
formal degree.)
L2
Note that
1
cπ ∗ cπ =
cπ .
dπ
Character of discrete series
G: connected semisimple Lie group with discrete series.
T : maximal torus, Cartan subgroup.
b has a distribution valued character
A discrete series π ∈ G
Z
Θπ (f ) := Tr(π(f )) = Tr
f (g)π(g)dg
f ∈ Cc∞ (G).
G
Theorem (Harish-Chandra)
Let ρ be half sum of positive roots of (gC , tC ). A discrete series
is Θπ parametrised by λ, where
√
λ ∈ −1t∗ is regular;
λ − ρ is an integral weight which can be lifted to a character
(eλ−ρ , Cλ−ρ ) of T .
Θλ := Θπ is a locally integrable function which is analytic on an
open dense subset of G.
Harish-Chandra character formula
Theorem (Harish-Chandra Character formula)
For every regular point g of T :
P
Θλ (g) =
w(λ+ρ)
w∈WK det(w)e
(g).
eρ Πα∈R+ (1 − e−α )
Here,
T is a manximal torus,
K is a maximal compact subgroup and WK = NK (T )/T is
the compact Weyl group,
R+ is the set of positive roots,
P
ρ = 12 α∈R+ α.
Equivariant Index. Noncompact Case
Let G be a connected seminsimple Lie group acting on M
properly and cocompactly.
Let D be a G-invariant elliptic operator D.
Let B be a parametrix where
1 − BD+ = S0
1 − D + B = S1
are smoothing operators.
The equivariant index indG D is an element of K0 (Cr∗ G).
indG : K∗G (M ) → K∗ (Cr∗ G)
[D] 7→ indG D
where
S02
S0 (1 + S0 )B
0 0
indG D =
−
.
S1 D +
1 − S12
0 1
Harish-Chandra Schwartz algebra
The Harish-Chandra Schwartz space, denoted by C(G), consists
of f ∈ C ∞ (G) where
sup (1 + σ(g))m Ξ(g)−1 |L(X α )R(Y β )f (g)| < ∞
g∈G,α,β
∀m ≥ 0, X, Y ∈ U (g).
L and R denote the left and right derivatives;
σ(g) = d(eK, gK) in G/K (K maximal compact);
Ξ is the matrix coefficient of some unitary representation.
Properties:
C(G) is a Fréchet algebra under convolusion.
b is a discrete series, then cπ ∈ C(G).
If π ∈ G
C(G) ⊂ Cr∗ (G) and the inclusion induces
K0 (C(G)) ' K0 (Cr∗ G).
Character of an equivariant index
Definition
Let g be a semisimple element of G. The orbital integral
τg : C(G) → C
Z
τg (f ) =
f (hgh−1 )d(hZ)
G/ZG (g)
is well defined.
τg continuous trace, i.e., τg (a ∗ b) = τg (b ∗ a) for a, b ∈ C(G),
which induces
τg : K0 (C(G)) → R.
Definition
The g-index of D is given by τg (indG D).
Calculation of τg (indG D)
If G y M properly with compact
then
R M/G,
∞
−1
∃c ∈ Cc (M ), c ≥ 0 such that G c(g x)dg = 1, ∀x ∈ M.
Proposition (Hochs-W)
For g ∈ G semisimple and D Dirac type,
τg (indG D) = Trg (e−tD
where
Z
− D+
) − Trg (e−tD
+ D−
)
Tr(hgh−1 cT )d(hZ).
Trg (T ) =
G/ZG (g)
2
When G, M are compact, then c = 1 and Str(hgh−1 e−tD )
2
= Str(gh−1 e−tD h) =Tr(ge−tD
− D+
) − Tr(ge−tD
+ D−
=Tr(g|ker D+ ) − Tr(g|ker D− ).
⇒ τg (indG D) = vol(G/ZG (g))indG D(g).
)
Fixed point theorem
Theorem (Hochs-W)
Let G be a connected semisimple group acting on M properly
isometrically with compact quotient. Let g ∈ G be semisimple.If
g is not contained in a compact subgroup of G, or if G/K is
odd-dimensional, then
τg (indG D) = 0
for a G-invariant elliptic operator D.
If G/K is even-dimensional and g is contained in compact
subgroups of G, then
Z
c(x)ch [σD ](g) Todd(T M g ⊗ C)
V τg (indG D) =
ch
NC (g)
T Mg
where c is a cutoff function on M g with respect to ZG (g)-action.
Geometric realisation
Let G be a connected semisimple Lie group with compact
Cartan subgroup T. Let π be a √
discrete series with
Harish-Chandra parameter λ ∈ −1t∗ .
Corollary (P. Hochs-W)
Choose an elliptic operator ∂¯Lλ−ρ + ∂¯L∗ λ−ρ on G/T which is
the Dolbeault operator on G/T coupled with
the homomorphic line bundle
Lλ−ρ := G ×T Cλ−ρ → G/T.
We have for regular g ∈ T ,
τg (indG (∂¯Lλ−ρ + ∂¯L∗ λ−ρ )) = Harish-Chandra character formula.
Idea of proof
[dπ cπ ] is the image of [Vλ−ρc ] under the Connes-Kasparov
isomorphism
R(K) → K0 (Cr∗ G).
dim G/K
indG (∂¯Lλ−ρ + ∂¯L∗ λ−ρ ) = (−1) 2 [dπ cπ ].
dim G/K
2
τg [dπ cπ ] = Θλ (g) for g ∈ T.
τg (indG (∂¯Lλ−ρ + ∂¯L∗ λ−ρ )) can be calculated by the main
theorem and be reduced to a sum over finite set (G/T )g .
(−1)
Summary
We obtain a fixed point theorem generalizing
Atiyah-Segal-Singer index theorem for a semisimple Lie
group G acting properly on a manifold M with compact
quotient;
b with Harish-Chandra
Given a discrete√series Θλ ∈ G
∗
parameter λ ∈ −1t , the fixed point formula for the
Dolbeault operator on M = G/T twisted by the line
bundle determined by λ recovers the Harish-Chandra’s
character formula.
This generalizes Atiyah-Bott’s geometric method towards
the Wyel character formula for compact groups.
Outlook
The expression
Z
T Mg
c(x)ch [σD |M g ](g) Todd(T M g ⊗ C)
V ch
NC (g)
can be obtained for a general locally compact group using
localisation techniques.
It is important to show that it factors through K0 (Cr∗ G),
i.e., equal to τg (indG D).
Fixed point formulas and charatcer formulas can be
obtained for more general groups (e.g., unimodular Lie,
algebraic groups over nonarchemedean fields).
b of
Could the nondiscrete spectrum of the tempered dual G
G be studied using index theory?