Fixed Point Theorem and Character Formula

Hang WangUniversity of Adelaide

Index Theory and Singular StructuresInstitut de Mathematiques de Toulouse

29 May, 2017

OutlineAim: Study representation theory of Lie groups from the pointof view of geometry, motivated by the developement of

K-theory and representation;

Harmonic analysis on Lie groups.

Representation theory Geometry

character index theoryof representations of elliptic operators

Weyl character formula Atiyah-Segal-Singer

Harish-Chandra character formula Fixed point theorem

P. Hochs, H.Wang, A Fixed Point Formula andHarish-Chandra’s Character Formula, ArXiv 1701.08479.

Representation and characterG: irreducible unitary representations of G (compact, Lie);

For (π, V ) ∈ G, the character of π is given by

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⊕j=0

Cj−n

where Cj = C, on which T1 acts by g · z = gjz, g ∈ T1, z ∈ C.Then

χVn(g) =

2n∑j=0

gj−n g ∈ T1.

Weyl character formula

Let G be a compact Lie group with maximal torus T .

Let π ∈ G. Denote by λ ∈√−1t∗ its highest weight.

Theorem (Weyl character formula)

At a regular point g of T :

χπ(g) =

∑w∈W det(w)ew(λ+ρ)

eρΠα∈∆+(1− e−α)(g).

Here, W = NG(T )/T is the Weyl group, ∆+ is the set ofpositive roots and ρ = 1

2

∑α∈∆+ α.

Elliptic operatorsM : closed manifold.

Definition

A differential operator D on a manifold M is elliptic if itsprincipal symbol σD(x, ξ) is invertible whenever ξ 6= 0.

Dirac type operators ⊂ elliptic operators.

Example

de Rham operator on a closed oriented even dimensionalmanifold 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 representationring of G (identified as rings of characters).

Definition

The equivariant index of a G-invariant elliptic operator

D =

[0 D−

D+ 0

]on M , where (D+)∗ = D− is given by

indGD = [kerD+]− [kerD−] ∈ R(G);

It is determined by the characters

indGD(g) := Tr(g|kerD+)− Tr(g|kerD−) ∀g ∈ G.

Example. Lefschetz number

Consider the de Rham operator on a closed oriented evendimensional manifold M :

D± = d+ d∗ : Ωev/od(M)→ Ωod/ev(M).

kerD± ↔ harmonic forms ↔ Hev/odDR (M,R).

Lefschetz number, denoted by L(g):

indGD(g) =Tr(g|kerD+)− Tr(g|kerD−)

=∑i≥0

(−1)iTr [g∗,i : Hi(M,R)→ Hi(M,R)] .

Theorem (Lefschetz)

If L(g) 6= 0, then g has a fixed-point in M.

Fixed point formula

M : compact manifold.

g ∈ Isom(M).

Mg: fixed-point submanifold of M .

Theorem (Atiyah-Segal-Singer)

Let D : C∞(M,E)→ C∞(M,E) be an elliptic operator on M .Then

indGD(g) = Tr(g|kerD+)− Tr(g|kerD−)

=

∫TMg

ch([σD|Mg ](g)

)Todd(TMg ⊗ C)

ch([∧

NC](g))

where NC is the complexified normal bundle of Mg in M .

Equivariant index and representation

Let π ∈ G with highest weight λ ∈√−1t∗. Choose M = G/T

and the line bundle Lλ := G×T Cλ. Let ∂ be the Dolbeaultoperator on M .

Theorem (Borel-Weil-Bott)

The character of an irreducible representation π of G is equal tothe equivariant index of the twisted Dolbeault operator

∂Lλ + ∂∗Lλ

on the homogenous space G/T .

Theorem (Atiyah-Bott)

For g ∈ T reg,

indG(∂Lλ + ∂∗Lλ)(g) = Weyl character formula.

ExampleLet ∂n + ∂∗n be the Dolbeault–Dirac operator on

S2 ∼= SO(3)/T1,

coupled to the line bundle

Ln := SO(3)×T1 Cn → S2.

By Borel-Weil-Bott,

indSO(3)(∂n + ∂∗n) = [Vn] ∈ R(SO(3)).

By the Atiyah-Segal-Singer’s formula

indSO(3)(∂n + ∂∗n)(g) =gn

1− g−1+

g−n

1− g=

2n∑j=0

gj−n.

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 indGD recoverscharacter formulas for discrete series representations of G.

The context is K-theory:

“representation, equivariant index ∈ K0(C∗rG).”

Discrete series(π, V ) ∈ G is a discrete series of G if the matrix corficient cπgiven by

cπ(g) = 〈π(g)x, x〉 for ‖x‖ = 1

is L2-integrable.

When G is compact, all G are discrete series, and

K0(C∗rG) ' R(G) ' K0(Gd).

When G is noncompact,

K0(Gd) ≤ K0(C∗rG)

where [π] corresponds [dπcπ] (dπ = ‖cπ‖−2L2 formal degree.)

Note that

cπ ∗ cπ =1

dπcπ.

Character of discrete seriesG: connected semisimple Lie group with discrete series.T : maximal torus, Cartan subgroup.

A discrete series π ∈ G has a distribution valued character

Θπ(f) := Tr(π(f)) = Tr

∫Gf(g)π(g)dg f ∈ C∞c (G).

Theorem (Harish-Chandra)

Let ρ be half sum of positive roots of (gC, tC). A discrete seriesis 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 anopen dense subset of G.

Harish-Chandra character formula

Theorem (Harish-Chandra Character formula)

For every regular point g of T :

Θλ(g) =

∑w∈WK

det(w)ew(λ+ρ)

eρΠα∈R+(1− e−α)(g).

Here,

T is a manximal torus,

K is a maximal compact subgroup and WK = NK(T )/T isthe compact Weyl group,

R+ is the set of positive roots,

ρ = 12

∑α∈R+ α.

Equivariant Index. Noncompact CaseLet G be a connected seminsimple Lie group acting on Mproperly 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 indGD is an element of K0(C∗rG).

indG : KG∗ (M)→ K∗(C

∗rG) [D] 7→ indGD

where

indGD =

[S2

0 S0(1 + S0)BS1D

+ 1− S21

]−[0 00 1

].

Harish-Chandra Schwartz algebraThe Harish-Chandra Schwartz space, denoted by C(G), consistsof f ∈ C∞(G) where

supg∈G,α,β

(1 + σ(g))mΞ(g)−1|L(Xα)R(Y β)f(g)| <∞

∀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 Frechet algebra under convolusion.

If π ∈ G is a discrete series, then cπ ∈ C(G).

C(G) ⊂ C∗r (G) and the inclusion induces

K0(C(G)) ' K0(C∗rG).

Character of an equivariant index

Definition

Let g be a semisimple element of G. The orbital integral

τg : C(G)→ C

τg(f) =

∫G/ZG(g)

f(hgh−1)d(hZ)

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(indGD).

Calculation of τg(indGD)If GyM properly with compact M/G, then∃c ∈ C∞c (M), c ≥ 0 such that

∫G c(g

−1x)dg = 1,∀x ∈M.

Proposition (Hochs-W)

For g ∈ G semisimple and D Dirac type,

τg(indGD) = Trg(e−tD−D+

)− Trg(e−tD+D−

)

where

Trg(T ) =

∫G/ZG(g)

Tr(hgh−1cT )d(hZ).

When G,M are compact, then c = 1 and Str(hgh−1e−tD2)

= Str(gh−1e−tD2h) =Tr(ge−tD

−D+)− Tr(ge−tD

+D−)

=Tr(g|kerD+)− Tr(g|kerD−).

⇒ τg(indGD) = vol(G/ZG(g))indGD(g).

Fixed point theorem

Theorem (Hochs-W)

Let G be a connected semisimple group acting on M properlyisometrically with compact quotient. Let g ∈ G be semisimple.Ifg is not contained in a compact subgroup of G, or if G/K isodd-dimensional, then

τg(indGD) = 0

for a G-invariant elliptic operator D.If G/K is even-dimensional and g is contained in compactsubgroups of G, then

τg(indGD) =

∫TMg

c(x)ch([σD](g)

)Todd(TMg ⊗ C)

ch([∧

NC](g))

where c is a cutoff function on Mg with respect to ZG(g)-action.

Geometric realisation

Let G be a connected semisimple Lie group with compactCartan subgroup T. Let π be a discrete series withHarish-Chandra parameter λ ∈

√−1t∗.

Corollary (P. Hochs-W)

Choose an elliptic operator ∂Lλ−ρ + ∂∗Lλ−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 + ∂∗Lλ−ρ)) = Harish-Chandra character formula.

Idea of proof

[dπcπ] is the image of [Vλ−ρc ] under the Connes-Kasparovisomorphism

R(K)→ K0(C∗rG).

indG(∂Lλ−ρ + ∂∗Lλ−ρ) = (−1)dimG/K

2 [dπcπ].

(−1)dimG/K

2 τg[dπcπ] = Θλ(g) for g ∈ T.τg(indG(∂Lλ−ρ + ∂∗Lλ−ρ)) can be calculated by the main

theorem and be reduced to a sum over finite set (G/T )g.

Summary

We obtain a fixed point theorem generalizingAtiyah-Segal-Singer index theorem for a semisimple Liegroup G acting properly on a manifold M with compactquotient;

Given a discrete series Θλ ∈ G with Harish-Chandraparameter λ ∈

√−1t∗, the fixed point formula for the

Dolbeault operator on M = G/T twisted by the linebundle determined by λ recovers the Harish-Chandra’scharacter formula.

This generalizes Atiyah-Bott’s geometric method towardsthe Wyel character formula for compact groups.

Outlook

The expression∫TMg

c(x)ch([σD|Mg ](g)

)Todd(TMg ⊗ C)

ch([∧

NC](g))

can be obtained for a general locally compact group usinglocalisation techniques.

It is important to show that it factors through K0(C∗rG),

i.e., equal to τg(indGD).

Fixed point formulas and charatcer formulas can beobtained for more general groups (e.g., unimodular Lie,algebraic groups over nonarchemedean fields).

Could the nondiscrete spectrum of the tempered dual G ofG be studied using index theory?