K -types of tempered representations Peter Hochs University of - PowerPoint PPT Presentation

K -types of tempered representations Peter Hochs University of Adelaide Conference Index theory and singular structures Mathematics Institute of Toulouse Paul Sabatier University 29 May 2017 Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 0 / 24

Joint work with Yanli Song (Dartmouth College/Washington University St. Louis) Shilin Yu (Chinese University of Hong Kong) ‘A geometric realisation of tempered representations restricted to maximal compact subgroups’ (ArXiv:1705.02088) Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 1 / 24

Geometric realisation 1 Multiplicity formula 2 Example: G = SL(2 , R ) 3 Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 1 / 24

I Geometric realisation Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 1 / 24

Tempered representations Throughout this talk G will be a connected, linear, real reductive Lie group with compact centre. Let K < G be maximal compact. Let π : G → U( H ) be a unitary irreducible representation. A vector x ∈ H is K -finite if π ( K ) x spans a finite-dimensional subspace of H . The representation π is tempered if for all K -finite vectors x , y ∈ H , the function g �→ ( x , π ( g ) y ) H is in L 2+ ε ( G ) for all ε > 0. Tempered representations are important because of the Plancherel formula � ⊕ π ⊗ π ∗ d µ ( π ) L 2 ( G ) = ˆ G temp they are used in Langlands’ classification of all admissible representations. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 2 / 24

Example: unitary irreducible representations of SL(2 , R ) A representation is in the discrete series if its matrix coefficients are in L 2 ( G ). These exist for SL( n , R ) only if n = 2. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 3 / 24

K -types From now on, let π be a tempered representation of G . Idea: the restriction π | K contains a lot of useful information about π . Analogy: restricting an irreducible representation of K to a maximal torus. Goals: give a geometric realisation of π | K determine the multiplicities m δ in � π | K = m δ · δ. δ ∈ ˆ K The δ ∈ ˆ K for which m δ � = 0 are the K -types of π . We will do this using index theory of Dirac operators. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 4 / 24

Clifford modules Let M be a complete Riemannian manifold, acted on isometrically by (any compact Lie group) K . Let E → M be a Hermitian K -vector bundle with Z 2 -grading E = E + ⊕ E − . Suppose we have a Clifford action c : TM → End( E ) landing in the odd endomorphisms, such that for all v ∈ TM , c ( v ) 2 = −� v � 2 Id E . Then E is a Clifford module . Example Let J be a K -invariant almost complex structure on M , and L → M a K -line bundle. Use E = � J TM ⊗ L and c ( v ) = ( v ∧ · ) − ( v ∗ � · ) . Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 5 / 24

Dirac operators Let E → M be a Clifford module. Let ∇ be a Hermitian connection on E such that for all vector fields v and w , [ ∇ v , c ( w )] = c ( ∇ LC v w ) . Definition The Dirac operator associated to ∇ is the composition D : Γ ∞ ( E ) ∇ = Γ ∞ ( TM ⊗ E ) c → Γ ∞ ( T ∗ M ⊗ E ) ∼ → Γ ∞ ( E ) . − − If M is compact, we have its equivariant index index K ( E ) = index K ( D ) = [ker D + ] − [ker D − ] with D ± the restriction of D to Γ ∞ ( E ± ). Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 6 / 24

Taming maps If M is noncompact, we can still define an equivariant index. Let ψ : M → k be an equivariant map. It induces a vector field � v ψ ( m ) = d � exp( − t ψ ( m )) m . � dt � t =0 Definition The map ψ is taming if the set of zeroes of v ψ is compact. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 7 / 24

Deformed Dirac operators Let ψ : M → k be a taming map. Consider the deformed Dirac operator D ψ := D − ic ( v ψ ) . (Such deformations were first used by Tian and Zhang.) Theorem (Braverman, 2002) For a function f with suitable growth behaviour, and for all δ ∈ ˆ K, the multiplicity m ± δ of δ in ker( D f ψ ) ∩ L 2 ( E ± ) is finite . And m + δ − m − δ is independent of ∇ and f . Definition The equivariant index of ( E , ψ ) is � ( m + δ − m − δ ) δ = [ker( D f ψ ) ∩ L 2 ( E + )] − [ker( D f ψ ) ∩ L 2 ( E − )] index K ( E , ψ ) = δ ∈ ˆ K Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 8 / 24

Example: the discrete series Let T < K be a maximal torus. Suppose π is a discrete series representation. Let λ ∈ i t ∗ be its Harish-Chandra parameter. Consider the taming map → k ∗ ∼ = Ad ∗ ( G ) λ ⊂ g ∗ restriction ψ : G / T ∼ − − − − − − = k . Theorem (Paradan, 2003) For a G-invariant complex structure J on G / T and a line bundle L → G / T, π | K = ( − 1) dim( G / K ) / 2 index K ( � J T ( G / T ) ⊗ L , ψ ) . Schmid had already realised the whole representation π in the L 2 -kernel of a Dirac operator on G / T in 1976. Paradan’s result is useful because it implies a multiplicity formula for the K -types of π it can be generalised to arbitrary tempered representations Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 9 / 24

General tempered representations Theorem (H-Song-Yu, 2017) Let π be any tempered representation. Then � π | K = ± index K ( J T ( G / H ) ⊗ L , ψ ) Here the sign ± is explicit H < G is a Cartan subgroup J is a K -invariant almost complex structure on G / H L → G / H is a line bundle under a regularity assumption on π , → k ∗ ∼ = Ad ∗ ( G ) λ ⊂ g ∗ restriction ψ : G / H ∼ − − − − − − = k for some λ ∈ h ∗ without this assumption, ψ is defined differently. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 10 / 24

II Multiplicity formula Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 10 / 24

Multiplcities of K -types Let π be a tempered representation of G . Problem: find m δ in � π | K = m δ · δ. δ ∈ ˆ K These completely determine π | K explicitly, which says a lot about π . For discrete series representations, there is Blattner’s formula (proved by Hecht and Schmid in 1975). For general tempered representations, there is an algorithm in the ATLAS software. (Vogan, du Cloux and many others.) But: neither gives much insight into the general behaviour of multiplicities, e.g. when they are zero. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 11 / 24

Spin c -moment maps Suppose a Clifford module E → M is the spinor bundle of a Spin c -structure, with determinant line bundle L det → M . Example If E = � J TM ⊗ L for a K -invariant almost complex structure J and a K -line bundle L → M , then L det = � dim( M ) / 2 TM ⊗ L ⊗ 2 . J Let ∇ L det be a K -invariant connection on L det that preserves the metric. Definition The Spin c -moment map associated to ∇ L det is the map µ : M → k ∗ determined by 2 i � µ, X � = L X − ∇ L det X M for X ∈ k . Here L X is the Lie derivative, and X M is the vector field induced by X . Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 12 / 24

Indices on reduced spaces Let µ : M → k ∗ be a Spin c -moment map. Definition For ξ ∈ k ∗ , the reduced space at ξ is M ξ := µ − 1 ( ξ ) / K ξ . If ξ is a regular value of µ , then ξ is an orbifold, and inherits a Spin c -structure from M . If M ξ is compact, let index( M ξ ) ∈ Z be the orbifold index of the Spin c -Dirac operator on M ξ . If ξ is a singular value, this can still be defined. This is nontrivial, and was done by Paradan and Vergne. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 13 / 24

Quantisation commutes with reduction (1) Theorem (Paradan-Vergne, 2014) Suppose M is compact, and K acts with abelian stabilisers. Then the equivariant index of the Spin c -Dirac operator on E is � index K ( E ) = index( M λ δ + ρ ) · δ, δ ∈ ˆ K where λ δ is the highest weight of δ and ρ is half the sum of the positive roots. Paradan and Vergne proved a more complicated statement without assuming abelian stabilisers. This followed results for compact K¨ ahler manifolds by Guillemin and Sternberg in 1982 and compact symplectic manifolds by Meinrenken and Sjamaar in 1998. (With later proofs by Tian and Zhang and by Paradan.) Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 14 / 24

Quantisation commutes with reduction (2) Theorem (H-Song, 2015) Suppose the Spin c -moment map µ is proper and taming, and K acts with abelian stabilisers. Then the equivariant index of the Spin c -Dirac operator on E deformed by µ is � index K ( E , µ ) = index( M λ δ + ρ ) · δ, δ ∈ ˆ K where λ δ is the highest weight of δ and ρ is half the sum of the positive roots. Again, the original result is slightly more complicated, without the assumptions of abelian stabilisers or taming moment map. For symplectic manifolds, the analogous statement was conjectured by Vergne in 2006 and proved by Ma and Zhang in 2014, and later by Paradan. We will need the Spin c -version. Peter Hochs (Adelaide) K -types of tempered representations 29-5-2017 15 / 24