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 ) 2 G;b the character of π is given by χπ(g) = Tr[π(g): V ! V ] g 2 G: Example 1 ∼ Consider G = SO(3) with maximal torus T = SO(2) ,! SO(3): Let Vn 2 SO\(3) with highest weight n, i.e., 2n ∼ M VnjT1 = Cj−n j=0 1 j 1 where Cj = C, on which T acts by g · z = g z; g 2 T ; z 2 C: Then 2n X j−n 1 χVn (g) = g g 2 T : j=0 Weyl character formula Let G be a compact Lie group with maximal torus T . p Let π 2 Gb. Denote by λ 2 −1t∗ its highest weight. Theorem (Weyl character formula) At a regular point g of T : P w(λ+ρ) w2W det(w)e χπ(g) = ρ −α (g): e Πα2∆+ (1 − e ) + Here, W = NG(T )=T is the Weyl group, ∆ is the set of 1 P positive roots and ρ = 2 α2∆+ 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: fDirac type operatorsg ⊂ felliptic operatorsg: 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) := f[V ] − [W ]: V; W fin. dim. rep. of Gg 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 ] 2 R(G); It is determined by the characters indGD(g) := Tr(gjker D+ ) − Tr(gjker D− ) 8g 2 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(gjker D+ ) − Tr(gjker 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 2 Isom(M): M g: fixed-point submanifold of M. Theorem (Atiyah-Segal-Singer) Let D : C1(M; E) ! C1(M; E) be an elliptic operator on M. Then indGD(g) = Tr(gjker D+ ) − Tr(gjker D− ) Z g ch [σDjM 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 p Let π 2 Gb with highest weight λ 2 −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 2 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] 2 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 2 K0(Cr G):" Discrete series (π; V ) 2 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 2 Gb has a distribution valued character Z 1 Θπ(f) := Tr(π(f)) = Tr f(g)π(g)dg f 2 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 p λ 2 −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) w2WK Θλ(g) = ρ −α (g): e Πα2R+ (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 α2R+ 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 2 C1(G) where sup (1 + σ(g))mΞ(g)−1jL(Xα)R(Y β)f(g)j < 1 g2G,α,β 8m ≥ 0; X; Y 2 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 2 Gb is a discrete series, then cπ 2 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 2 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 1 R −1 9c 2 Cc (M); c ≥ 0 such that G c(g x)dg = 1; 8x 2 M: Proposition (Hochs-W) For g 2 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(gjker D+ ) − Tr(gjker 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 2 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 ap discrete series with Harish-Chandra parameter λ 2 −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 2 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 2 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.