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 Weyl character formula Harish-Chandra character formula Geometry index theory of elliptic operators Atiyah-Segal-Singer Fixed point theorem P. Hochs, H.Wang, A Fixed Point Formula and Harish-Chandra s Character Formula, ArXiv 1701.08479.
Representation and character Ĝ: irreducible unitary representations of G (compact, Lie); For (π, V ) Ĝ, the character of π is given by Example χ π (g) = Tr[π(g) : V V ] g G. Consider G = SO(3) with maximal torus T 1 = SO(2) SO(3). Let V n ŜO(3) with highest weight n, i.e., V n T 1 = 2n C j n where C j = C, on which T 1 acts by g z = g j z, g T 1, z C. Then χ Vn (g) = 2n j=0 j=0 g j n g T 1.
Weyl character formula Let G be a compact Lie group with maximal torus T. Let π Ĝ. Denote by λ 1t its highest weight. Theorem (Weyl character formula) At a regular point g of T : χ π (g) = w W det(w)ew(λ+ρ) e ρ Π α +(1 e α ) (g). Here, W = N G (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 σ D (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 0 D D = D + on M, where (D 0 + ) = D is given by ind G D = [ker D + ] [ker D ] R(G); It is determined by the characters ind G 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 : Ω ev/od (M) Ω od/ev (M). ker D ± harmonic forms H ev/od (M, R). DR Lefschetz number, denoted by L(g): ind G D(g) =Tr(g ker D +) Tr(g ker D ) = i 0( 1) i Tr [g,i : H i (M, R) H i (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). 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 ind G D(g) = Tr(g ker D +) Tr(g ker D ) ( ch [σd M g](g) ) Todd(T M g C) = T M g ch ([ ] ) N C (g) where N C is the complexified normal bundle of M g in M.
Equivariant index and representation Let π Ĝ with highest weight λ 1t. Choose M = G/T and the line bundle L λ := G T C λ. 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, ind G ( Lλ + L λ )(g) = Weyl character formula.
Example Let n + n be the Dolbeault Dirac operator on coupled to the line bundle By Borel-Weil-Bott, S 2 = SO(3)/T 1, L n := SO(3) T 1 C n S 2. ind SO(3) ( n + n) = [V n ] R(SO(3)). By the Atiyah-Segal-Singer s formula ind SO(3) ( n + n)(g) = gn g n 2n + 1 g 1 1 g = g j n. 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 ind G D recovers character formulas for discrete series representations of G. The context is K-theory: representation, equivariant index K 0 (C r G).
Discrete series (π, V ) Ĝ is a discrete series of G if the matrix corficient c π given by c π (g) = π(g)x, x for x = 1 is L 2 -integrable. When G is compact, all Ĝ are discrete series, and K 0 (C r G) R(G) K 0 (Ĝd). When G is noncompact, K 0 (Ĝd) K 0 (C r G) where [π] corresponds [d π c π ] (d π = c π 2 L 2 Note that c π c π = 1 c π. d π formal degree.)
Character of discrete series G: connected semisimple Lie group with discrete series. T : maximal torus, Cartan subgroup. A discrete series π Ĝ has a distribution valued character Θ π (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 (g C, t C ). A discrete series is Θ π parametrised by λ, where λ 1t is regular; λ ρ is an integral weight which can be lifted to a character (e λ ρ, C λ ρ ) of T. Θ λ := Θ π 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 : Θ λ (g) = w W K det(w)e w(λ+ρ) e ρ Π α R +(1 e α (g). ) Here, T is a manximal torus, K is a maximal compact subgroup and W K = N K (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 are smoothing operators. 1 BD + = S 0 1 D + B = S 1 The equivariant index ind G D is an element of K 0 (C r G). ind G : K G (M) K (C r G) [D] ind G D where [ S 2 ind G D = 0 S 0 (1 + S 0 )B S 1 D + 1 S1 2 ] [ ] 0 0. 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. If π Ĝ is a discrete series, then c π C(G). C(G) C r (G) and the inclusion induces K 0 (C(G)) K 0 (C r G).
Character of an equivariant index Definition Let g be a semisimple element of G. The orbital integral is well defined. τ g : C(G) C τ g (f) = G/Z G (g) f(hgh 1 )d(hz) τ g continuous trace, i.e., τ g (a b) = τ g (b a) for a, b C(G), which induces τ g : K 0 (C(G)) R. Definition The g-index of D is given by τ g (ind G D).
Calculation of τ g (ind G D) If G M properly with compact M/G, then c C c (M), c 0 such that G c(g 1 x)dg = 1, x M. Proposition (Hochs-W) For g G semisimple and D Dirac type, τ g (ind G D) = Tr g (e td D + ) Tr g (e td+ D ) where Tr g (T ) = G/Z G (g) Tr(hgh 1 ct )d(hz). When G, M are compact, then c = 1 and Str(hgh 1 e td2 ) = Str(gh 1 e td2 h) =Tr(ge td D + ) Tr(ge td+ D ) =Tr(g ker D +) Tr(g ker D ). τ g (ind G D) = vol(g/z G (g))ind G 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 τ g (ind G 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 ( c(x)ch [σd ](g) ) Todd(T M g C) τ g (ind G D) = T M g ch ([ ] ) N C (g) where c is a cutoff function on M g with respect to Z G (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 We have for regular g T, L λ ρ := G T C λ ρ G/T. τ g (ind G ( Lλ ρ + L λ ρ )) = Harish-Chandra character formula.
Idea of proof [d π c π ] is the image of [V λ ρc ] under the Connes-Kasparov isomorphism R(K) K 0 (C r G). ind G ( Lλ ρ + L dim G/K λ ρ ) = ( 1) 2 [d π c π ]. dim G/K ( 1) 2 τ g [d π c π ] = Θ λ (g) for g T. τ g (ind G ( Lλ ρ + L λ 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 Θ λ Ĝ 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 ( c(x)ch [σd M g](g) ) Todd(T M g C) T M g ch ([ ] ) N C (g) can be obtained for a general locally compact group using localisation techniques. It is important to show that it factors through K 0 (C r G), i.e., equal to τ g (ind G D). 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 Ĝ of G be studied using index theory?