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

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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)

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1

Geometric realisation

2

Multiplicity formula

3

Example: G = SL(2, R)

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I Geometric realisation

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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 L2+ε(G) for all ε > 0. Tempered representations are important because

• f the Plancherel formula

L2(G) = ⊕

ˆ Gtemp

π ⊗ π∗ dµ(π) they are used in Langlands’ classification of all admissible representations.

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Example: unitary irreducible representations of SL(2, R)

A representation is in the discrete series if its matrix coefficients are in L2(G). These exist for SL(n, R) only if n = 2.

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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 mδ in π|K =

• δ∈ ˆ

K

mδ · δ. The δ ∈ ˆ K for which mδ = 0 are the K-types of π. We will do this using index theory of Dirac operators.

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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 Z2-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 = −v2 IdE . 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∗ ·).

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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∞(T ∗M ⊗ E) ∼ = Γ∞(TM ⊗ E) c − → Γ∞(E). If M is compact, we have its equivariant index indexK(E) = indexK(D) = [ker D+] − [ker D−] with D± the restriction of D to Γ∞(E ±).

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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 vψ(m) = d dt

• t=0

exp(−tψ(m))m.

Definition

The map ψ is taming if the set of zeroes of vψ is compact.

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Deformed Dirac operators

Let ψ: M → k be a taming map. Consider the deformed Dirac operator Dψ := D − ic(vψ). (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(Df ψ) ∩ L2(E ±) is finite. And m+

δ − m− δ is independent of ∇ and f .

Definition

The equivariant index of (E, ψ) is indexK(E, ψ) =

• δ∈ ˆ

K

(m+

δ −m− δ )δ = [ker(Df ψ)∩L2(E +)]−[ker(Df ψ)∩L2(E −)]

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Example: the discrete series

Let T < K be a maximal torus. Suppose π is a discrete series

• representation. Let λ ∈ it∗ be its Harish-Chandra parameter.

Consider the taming map ψ: G/T ∼ = Ad∗(G)λ ⊂ g∗ restriction − − − − − − → k∗ ∼ = 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 indexK(

JT(G/T) ⊗ L, ψ).

Schmid had already realised the whole representation π in the L2-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

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General tempered representations

Theorem (H-Song-Yu, 2017)

Let π be any tempered representation. Then π|K = ± indexK(

• JT(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, ψ: G/H ∼ = Ad∗(G)λ ⊂ g∗ restriction − − − − − − → k∗ ∼ = k for some λ ∈ h∗ without this assumption, ψ is defined differently.

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II Multiplicity formula

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Multiplcities of K-types

Let π be a tempered representation of G. Problem: find mδ in π|K =

• δ∈ ˆ

K

mδ · δ. 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.

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Spinc-moment maps

Suppose a Clifford module E → M is the spinor bundle of a Spinc-structure, with determinant line bundle Ldet → M.

Example

If E =

J TM ⊗ L for a K-invariant almost complex structure J and a

K-line bundle L → M, then Ldet = dim(M)/2

J

TM ⊗ L⊗2. Let ∇Ldet be a K-invariant connection on Ldet that preserves the metric.

Definition

The Spinc-moment map associated to ∇Ldet is the map µ: M → k∗ determined by 2iµ, X = LX − ∇Ldet

X M

for X ∈ k. Here LX is the Lie derivative, and X M is the vector field induced by X.

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Indices on reduced spaces

Let µ: M → k∗ be a Spinc-moment map.

Definition

For ξ ∈ k∗, the reduced space at ξ is Mξ := µ−1(ξ)/Kξ. If ξ is a regular value of µ, then ξ is an orbifold, and inherits a Spinc-structure from M. If Mξ is compact, let index(Mξ) ∈ Z be the orbifold index of the Spinc-Dirac operator on Mξ. If ξ is a singular value, this can still be defined. This is nontrivial, and was done by Paradan and Vergne.

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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 Spinc-Dirac operator on E is indexK(E) =

• δ∈ ˆ

K

index(Mλδ+ρ) · δ, where λδ 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.)

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Quantisation commutes with reduction (2)

Theorem (H-Song, 2015)

Suppose the Spinc-moment map µ is proper and taming, and K acts with abelian stabilisers. Then the equivariant index of the Spinc-Dirac operator

• n E deformed by µ is

indexK(E, µ) =

• δ∈ ˆ

K

index(Mλδ+ρ) · δ, where λδ 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 Spinc-version.

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A multiplicity formula

We had two results: π|K = ± indexK(

• JT(G/H) ⊗ L, ψ)

indexK(E, µ) =

• δ∈ ˆ

K

index(Mλδ+ρ) · δ

Theorem (H-Song-Yu, 2017)

We have π|K = ±

• δ∈ ˆ

K

index((G/H)λδ+ρ) · δ. (There is some work involved in showing that we can get µ = ψ.)

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Consequences

We have π|K =

• δ∈ ˆ

K

mδ · δ with mδ = ± index((G/H)λδ+ρ). So if mδ = 0 then λδ + ρ ∈ µ(G/H) if (G/H)λδ+ρ is a point, then mδ ∈ {0, 1}. (There is an explicit criterion for this to be 1.) The index on a point can be zero because this is an orbifold index: the dimension of the part if a one-dimensional space invariant under a representation of a finite group.

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Earlier results

The relation mδ = ± index((G/H)λδ+ρ) was proved for the discrete series by Paradan in 2003, using a similar approach for tempered representations satisfying a regularity assumption by Duflo and Vergne in 2011, using a very different approach (Kirillov’s character formula) that is hard to generalise to the non-regular case, and involves heavy representation theoretic machinery.

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III Example: G = SL(2, R)

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Tempered representations of SL(2, R)

For SL(2, R), the tempered representations are: the discrete series D±

n for n = 1, 2, 3, . . .

the principal series P±

iν, where ν ≥ 0 for + and ν > 0 for −

the limits of discrete series D±

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The discrete series (1)

Consider the discrete series representation D+

n for n = 1, 2, 3, . . ..

Now µ−1(0, 0, l) is empty if l < n a point if l = n a circle if l > n We get index((G/T)l) = 1 if l = n + s for s > 0 odd. So D+

n |K = Cn+1 ⊕ Cn+3 ⊕ Cn+5 ⊕ · · ·

Here Cl is the representation of K = SO(2) in C with weight l ∈ Z.

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The discrete series (2)

For D−

n , we similarly get

D+

n |K = C−n−1 ⊕ C−n−3 ⊕ C−n−5 ⊕ · · ·

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The principal series (1)

Consider the principal series representation P+

iν for ν ≥ 0.

Now µ−1(0, 0, l) is always a circle. We get index((G/T)l) = 1 if l is even. So P+

iν|K = · · · ⊕ C−2 ⊕ C0 ⊕ C2 ⊕ · · ·

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The principal series (2)

Consider the principal series representation P−

iν for ν > 0.

Again, µ−1(0, 0, l) is always a cir- cle. We get index((G/T)l) = 1 if l is

• dd, because a different line bun-

dle L is used. So P−

iν|K = · · · C−3 ⊕ ⊕C−1 ⊕ C1 ⊕ C3 ⊕ · · ·

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Limits of discrete series

Consider the limit of discrete series representation D+

0 . The the map µ is

now a shifted and deformed version of the the projection map from a coadjoint orbit. Similarly to the discrete series case, we obtain D+

0 |K = C1 ⊕ C3 ⊕ C5 ⊕ · · ·

and D−

0 |K = C−1 ⊕ C−3 ⊕ C−5 ⊕ · · ·

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Thank you

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