Fixed Point Theorem and Character Formula Hang Wang University of - - PowerPoint PPT Presentation

Fixed Point Theorem and Character Formula

Hang Wang University of Adelaide

Index Theory and Singular Structures Institut de Math´ ematiques de Toulouse

29 May, 2017

Outline

Aim: Study representation theory of Lie groups from the point

• f view of geometry, motivated by the developement of

K-theory and representation; Harmonic analysis on Lie groups. Representation theory Geometry character index theory

• f representations
• f 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

• G: 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 of positive roots and ρ = 1

2

• 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 = 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 D = D− D+

• n M, where (D+)∗ = D− is given by

indGD = [ker D+] − [ker D−] ∈ R(G); It is determined by the characters indGD(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). ker D± ↔ harmonic forms ↔ Hev/od

DR

(M, R). Lefschetz number, denoted by L(g): indGD(g) =Tr(g|ker D+) − Tr(g|ker D−) =

• i≥0

(−1)iTr [g∗,i : Hi(M, R) → Hi(M, R)] .

Theorem (Lefschetz)

If L(g) = 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|ker D+) − Tr(g|ker D−) =

• 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 Dolbeault

• perator 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 + ¯ ∂∗

Lλ

• n the homogenous space G/T.

Theorem (Atiyah-Bott)

For g ∈ T reg, indG(¯ ∂Lλ + ¯ ∂∗

Lλ)(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 → 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 recovers character formulas for discrete series representations of G. The context is K-theory: “representation, equivariant index ∈ K0(C∗

r G).”

Discrete series

(π, V ) ∈ G is a discrete series of G if the matrix corficient cπ given 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∗

r G) ≃ R(G) ≃ K0(

Gd). When G is noncompact, K0( Gd) ≤ K0(C∗

r G)

where [π] corresponds [dπcπ] (dπ = cπ−2

L2 formal degree.)

Note that cπ ∗ cπ = 1 dπ cπ.

Character of discrete series

G: 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

• G

f(g)π(g)dg f ∈ C∞

c (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

• pen 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 is the compact Weyl group, R+ is the set of positive roots, ρ = 1

2

• 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 indGD is an element of K0(C∗

r G).

indG : KG

∗ (M) → K∗(C∗ r G)

[D] → indGD where indGD = S2 S0(1 + S0)B S1D+ 1 − S2

1

• −

1

• .

Harish-Chandra Schwartz algebra

The Harish-Chandra Schwartz space, denoted by C(G), consists

• f f ∈ C∞(G) where

sup

g∈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 Fr´ echet 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∗

r G).

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 G M 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|ker D+) − Tr(g|ker D−). ⇒ τg(indGD) = vol(G/ZG(g))indGD(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

• dd-dimensional, then

τg(indGD) = 0 for a G-invariant elliptic operator D. If G/K is even-dimensional and g is contained in compact subgroups 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 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 + ¯ ∂∗

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-Kasparov isomorphism R(K) → K0(C∗

r G).

indG(¯ ∂Lλ−ρ + ¯ ∂∗

Lλ−ρ) = (−1)

dim G/K 2

[dπcπ]. (−1)

dim G/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 generalizing Atiyah-Segal-Singer index theorem for a semisimple Lie group G acting properly on a manifold M with compact quotient; Given a discrete series Θλ ∈ G with Harish-Chandra 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

• TMg

c(x)ch

• [σD|Mg](g)
• Todd(TMg ⊗ C)

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(C∗

r G),

i.e., equal to τg(indGD).

Fixed point formulas and charatcer formulas can be

• btained for more general groups (e.g., unimodular Lie,

algebraic groups over nonarchemedean fields). Could the nondiscrete spectrum of the tempered dual G of G be studied using index theory?