Fixed Point Theorem and Character Formula

Hang Wang University of Adelaide

Index Theory and Singular Structures Institut de Math´ematiquesde 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 Geometry character index theory of 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 and Harish-Chandra’s Character Formula, ArXiv 1701.08479. Representation and character Gb: irreducible unitary representations of G (compact, Lie); For (π, V ) ∈ G,b the character of π is given by

χπ(g) = Tr[π(g): V → V ] g ∈ G.

Example 1 ∼ Consider G = SO(3) with maximal torus T = SO(2) ,→ SO(3). Let Vn ∈ SO\(3) with highest weight n, i.e.,

2n ∼ M Vn|T1 = Cj−n j=0

1 j 1 where Cj = C, on which T acts by g · z = g z, g ∈ T , z ∈ C. Then 2n X j−n 1 χVn (g) = g g ∈ T . j=0 Weyl character formula

Let G be a compact Lie group with maximal torus T . √ Let π ∈ Gb. Denote by λ ∈ −1t∗ its highest weight. Theorem (Weyl character formula) At a regular point g of T :

P w(λ+ρ) w∈W det(w)e χπ(g) = ρ −α (g). e Πα∈∆+ (1 − e )

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

+ − 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).

± ev/od ker D ↔ harmonic forms ↔ HDR (M, R). Lefschetz number, denoted by L(g):

indGD(g) =Tr(g|ker D+ ) − Tr(g|ker D− ) X i = (−1) 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

indGD(g) = Tr(g|ker D+ ) − Tr(g|ker D− ) Z g ch [σD|M g ](g) Todd(TM ⊗ C) = V TM g ch NC (g) g where NC is the complexiﬁed normal bundle of M in M. Equivariant index and representation √ Let π ∈ Gb with highest weight λ ∈ −1t∗. Choose M = G/T ¯ 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 Lλ on the homogenous space G/T .

Theorem (Atiyah-Bott) For g ∈ T reg,

ind (∂¯ + ∂¯∗ )(g) = Weyl character formula. G Lλ Lλ Example ¯ ¯∗ Let ∂n + ∂n be the Dolbeault–Dirac operator on 2 ∼ 1 S = SO(3)/T , coupled to the line bundle

2 Ln := SO(3) ×T1 Cn → S . By Borel-Weil-Bott,

¯ ¯∗ indSO(3)(∂n + ∂n) = [Vn] ∈ R(SO(3)).

By the Atiyah-Segal-Singer’s formula

2n gn g−n X ind (∂¯ + ∂¯∗)(g) = + = gj−n. SO(3) n 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 indGD recovers character formulas for discrete series representations of G. The context is K-theory:

∗ “representation, equivariant index ∈ K0(Cr G).” Discrete series (π, V ) ∈ Gb is a discrete series of G if the matrix corﬁcient cπ given by cπ(g) = hπ(g)x, xi for kxk = 1 is L2-integrable.

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

∗ 0 K0(Cr G) ' R(G) ' K (Gbd).

When G is noncompact,

0 ∗ K (Gbd) ≤ K0(Cr G) −2 where [π] corresponds [dπcπ](dπ = kcπkL2 formal degree.) Note that 1 cπ ∗ cπ = cπ. dπ Character of discrete series G: connected semisimple Lie group with discrete series. T : maximal torus, Cartan subgroup. A discrete series π ∈ Gb has a distribution valued character 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 Θ parametrised by λ, where π √ λ ∈ −1t∗ is regular; λ − ρ is an integral weight which can be lifted to a character λ−ρ (e , 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 det(w)ew(λ+ρ) w∈WK Θλ(g) = ρ −α (g). e Πα∈R+ (1 − e ) 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, 1 P ρ = 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(Cr G).

G ∗ indG : K∗ (M) → K∗(Cr G)[D] 7→ indGD

where

2 S0 S0(1 + S0)B 0 0 indGD = + 2 − . S1D 1 − S1 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))mΞ(g)−1|L(Xα)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 Fr´echet algebra under convolusion. If π ∈ Gb is a discrete series, then cπ ∈ C(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 −1 τg(f) = f(hgh )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(indGD). Calculation of τg(indGD) If G y M properly with compact M/G, then ∞ R −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,

−tD−D+ −tD+D− τg(indGD) = Trg(e ) − Trg(e )

where Z −1 Trg(T ) = Tr(hgh cT )d(hZ). G/ZG(g)

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

2 − + + − = Str(gh−1e−tD h) =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 odd-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

Z g c(x)ch [σD](g) Todd(TM ⊗ C) τg(indGD) = V TM g ch NC (g)

g where c is a cutoﬀ function on M 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 ∂¯ + ∂¯∗ on G/T which is Lλ−ρ Lλ−ρ 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 ,

τ (ind (∂¯ + ∂¯∗ )) = Harish-Chandra character formula. g G Lλ−ρ Lλ−ρ Idea of proof

[dπcπ] is the image of [Vλ−ρc ] under the Connes-Kasparov isomorphism ∗ R(K) → K0(Cr G).

∗ dim G/K ind (∂¯ + ∂¯ ) = (−1) 2 [d c ]. G Lλ−ρ Lλ−ρ π π dim G/K (−1) 2 τg[dπcπ] = Θλ(g) for g ∈ T. τ (ind (∂¯ + ∂¯∗ )) can be calculated by the main g G Lλ−ρ Lλ−ρ 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 ∈ Gb 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

Z g c(x)ch [σD|M g ](g) Todd(TM ⊗ C) V TM g 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(indGD). Fixed point formulas and charatcer formulas can be obtained for more general groups (e.g., unimodular Lie, algebraic groups over nonarchemedean ﬁelds). Could the nondiscrete spectrum of the tempered dual Gb of G be studied using index theory?