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 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 � G : irreducible unitary representations of G (compact, Lie); For ( π, V ) ∈ � G, the character of π is given by χ π ( g ) = Tr[ π ( g ) : V → V ] g ∈ G. Example Consider G = SO (3) with maximal torus T 1 ∼ = SO (2) ֒ → SO (3) . Let V n ∈ � SO (3) with highest weight n , i.e., 2 n � V n | T 1 ∼ C j − n = j =0 where C j = C , on which T 1 acts by g · z = g j z, g ∈ T 1 , z ∈ C . Then 2 n � g j − n g ∈ T 1 . χ V n ( g ) = j =0

Weyl character formula Let G be a compact Lie group with maximal torus T . G . Denote by λ ∈ √− 1 t ∗ its highest weight. Let π ∈ � Theorem (Weyl character formula) At a regular point g of T : � w ∈ W det( w ) e w ( λ + ρ ) χ π ( g ) = 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 α ∈ ∆ + α. 2

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 ) . ∂ ∗ on a complex manifold. Dolbeault operator ¯ ∂ + ¯

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 − on M , where ( D + ) ∗ = D − is given by D = D + 0 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 − ) � ( − 1) i Tr [ g ∗ ,i : H i ( M, R ) → H i ( M, R )] . = i ≥ 0 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 − ) � � � Todd( TM g ⊗ C ) ch [ σ D | M g ]( g ) ��� N C � � = ch ( g ) TM g where N C is the complexified normal bundle of M g in M .

Equivariant index and representation G with highest weight λ ∈ √− 1 t ∗ . Choose M = G/T Let π ∈ � 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 ∂ L λ + ¯ ¯ ∂ ∗ L λ on the homogenous space G/T . Theorem (Atiyah-Bott) For g ∈ T reg , ind G (¯ ∂ L λ + ¯ ∂ ∗ L λ )( g ) = Weyl character formula .

Example Let ¯ ∂ n + ¯ ∂ ∗ n be the Dolbeault–Dirac operator on S 2 ∼ = SO (3) / T 1 , coupled to the line bundle L n := SO (3) × T 1 C n → S 2 . By Borel-Weil-Bott, ind SO (3) (¯ ∂ n + ¯ ∂ ∗ n ) = [ V n ] ∈ R ( SO (3)) . By the Atiyah-Segal-Singer’s formula 2 n � 1 − g − 1 + g − n g n ind SO (3) (¯ ∂ n + ¯ g j − n . ∂ ∗ n )( g ) = 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 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 ) ∈ � G 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 � G are discrete series, and K 0 ( C ∗ r G ) ≃ R ( G ) ≃ K 0 ( � G d ) . When G is noncompact, K 0 ( � G d ) ≤ K 0 ( C ∗ r G ) where [ π ] corresponds [ d π c π ] ( d π = � c π � − 2 L 2 formal degree.) Note that c π ∗ c π = 1 c π . d π

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 � f ∈ C ∞ Θ π ( f ) := Tr( π ( f )) = Tr f ( g ) π ( g ) dg c ( 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 ∈ √− 1 t ∗ 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 : � w ∈ W K det( w ) e w ( λ + ρ ) Θ λ ( g ) = ( g ) . e ρ Π α ∈ R + (1 − e − α ) 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 α ∈ R + α. 2

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 + = S 0 1 − D + B = S 1 are smoothing operators. 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 � � 0 � S 0 (1 + S 0 ) B 0 0 ind G D = − . S 1 D + 1 − S 2 0 1 1

Harish-Chandra Schwartz algebra The Harish-Chandra Schwartz space , denoted by C ( G ), consists of f ∈ C ∞ ( G ) where (1 + σ ( g )) m Ξ( g ) − 1 | L ( X α ) R ( Y β ) f ( g ) | < ∞ sup 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 ∈ � G 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 τ g : C ( G ) → C � f ( hgh − 1 ) d ( hZ ) τ g ( f ) = G/Z G ( g ) is well defined. 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 ∞ G c ( g − 1 x )d g = 1 , ∀ x ∈ M. c ( M ) , c ≥ 0 such that 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( hgh − 1 cT ) d ( hZ ) . Tr g ( T ) = G/Z G ( g ) When G, M are compact, then c = 1 and Str( hgh − 1 e − tD 2 ) = Str( gh − 1 e − tD 2 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 � � � Todd( TM g ⊗ C ) c ( x )ch [ σ D ]( g ) ��� N C � � τ g (ind G D ) = ch ( g ) TM 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 ∈ √− 1 t ∗ . Corollary (P. Hochs-W) Choose an elliptic operator ¯ ∂ L λ − ρ + ¯ ∂ ∗ L λ − ρ on G/T which is the Dolbeault operator on G/T coupled with the homomorphic line bundle L λ − ρ := G × T C λ − ρ → G/T. We have for regular g ∈ T , τ g (ind G (¯ ∂ L λ − ρ + ¯ ∂ ∗ L λ − ρ )) = Harish-Chandra character formula .