An equivariant index for proper actions III: The invariant and discrete series indices

Hochs, Peter; Song, Yanli

doi:10.1016/j.difgeo.2016.07.003

Cited by 14 publications

(13 citation statements)

References 33 publications

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“…Proof The presence of the grading operator in means that

D^{2} = D_{G, K}^{2} \otimes 1 + 1 \otimes D_{N}^{2} .

Since the two terms on the right hand side commute, we therefore have

e^{- t D^{2}} = e^{- t (D_{G, K}^{2} \otimes 1 + 1 \otimes D_{N}^{2})} = e^{- t D_{G, K}^{2}} \otimes e^{- t D_{N}^{2}} .

Lemma 4.1 in states that the Riemanian density

d m

M

equals the measure

d [x, n]

G \times_{K} N

induced by the product measure

d x d n

G \times N

. Since

K

has unit volume, this implies that for all

φ \in C^{\infty} false(G false) \otimes S_{frakturp} \cap L^{2} false(G false) \otimes S_{frakturp}

and

ψ \in {normalΓ}^{\infty} false(S_{N} \otimes W |_{N} false)

such that

φ \otimes ψ

K

‐invariant, and

x \in G

and

n \in N

\begin{matrix} left (e^{- t D^{2}} φ \otimes ψ) \end{matrix}

…”

Section: Localisationmentioning

confidence: 99%

See 1 more Smart Citation

A fixed point formula and Harish‐Chandra's character formula

Hochs

Wang

2017

Proc. London Math. Soc.

Self Cite

101

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The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations.

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“…Proof The presence of the grading operator in means that

D^{2} = D_{G, K}^{2} \otimes 1 + 1 \otimes D_{N}^{2} .

Since the two terms on the right hand side commute, we therefore have

e^{- t D^{2}} = e^{- t (D_{G, K}^{2} \otimes 1 + 1 \otimes D_{N}^{2})} = e^{- t D_{G, K}^{2}} \otimes e^{- t D_{N}^{2}} .

Lemma 4.1 in states that the Riemanian density

d m

M

equals the measure

d [x, n]

G \times_{K} N

induced by the product measure

d x d n

G \times N

. Since

K

has unit volume, this implies that for all

φ \in C^{\infty} false(G false) \otimes S_{frakturp} \cap L^{2} false(G false) \otimes S_{frakturp}

and

ψ \in {normalΓ}^{\infty} false(S_{N} \otimes W |_{N} false)

such that

φ \otimes ψ

K

‐invariant, and

x \in G

and

n \in N

\begin{matrix} left (e^{- t D^{2}} φ \otimes ψ) \end{matrix}

…”

Section: Localisationmentioning

confidence: 99%

“…Let

ε

be the grading operator on

S_{p}

. By Proposition 3.1 in , there is a

{Spin}^{c}

‐Dirac operator

D_{N}

{normalΓ}^{\infty} false(S_{N} false)

such that

D

is the operator

D_{G, K} \otimes 1 + ε \otimes D_{N}

on . Here we use the metric

B_{p}

.…”

Section: Localisationmentioning

confidence: 99%

A fixed point formula and Harish‐Chandra's character formula

Hochs

Wang

2017

Proc. London Math. Soc.

Self Cite

101

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show abstract

“…Finally, other versions of G-index theory, for non-compact M/G and G, have been developed elsewhere. This includes the work of Hochs-Mathai [17], [18], Braverman [8], and Hochs-Song [21], [19], [20].…”

Section: Introductionmentioning

confidence: 99%

Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

Guo

2019

J Geom Anal

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We formulate, for any Lie group G acting isometrically on a manifold M , the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M . We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that Gequivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds.

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“…In the third part [24], we consider Spin c -Dirac operators. For semisimple Lie groups with discrete series representations, the equivariant index is then directly related to multiplicities of discrete series representations, in cases where the Riemannian metric has a certain product form.…”

Section: The Main Resultsmentioning

confidence: 99%