Integral operators on the Oshima compactification of a Riemannian symmetric space

Parthasarathy, Aprameyan; Ramacher, Pablo

doi:10.1016/j.jfa.2014.05.002

Cited by 3 publications

(16 citation statements)

References 12 publications

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“…P r o o f . The proof is similar to the proof of Theorem 7 in[17]. By Proposition 4.1,Tr ζ π ( f ) = d j=1 R s+r |y s+1 · · · y s+r | ζ (α • ϕ )(y) j j Q f (y, 0) dyis a meromorphic function in ζ with possible poles at −1, −3, .…”

mentioning

confidence: 68%

“…Proof The proof is similar to the proof of Theorem 7 in . By Proposition ,

\begin{matrix} true \\ right {Tr}_{ζ} π (f) & left = \sum_{j = 1}^{d} \sum_{ϱ} \int_{R^{s + r}} {false| y_{s + 1} \dots y_{s + r} false|}^{ζ} (α_{ϱ} \circ φ_{ϱ}) (y)^{j j} {\tilde{Q}}_{f}^{ϱ} (y, 0) d y \end{matrix}

is a meromorphic function in ζ with possible poles at

- 1, - 3, \dots

.…”

Section: Regularized Traces and Fixed Point Formulaementioning

confidence: 81%

“…Similarly, if E and F are vector bundles over M , a continuous linear map

Q : {normalC}_{normalc}^{\infty} false(M, E false) ⟶ {normalC}^{\infty} false(M, F false)

is called a totally characteristic pseudodifferential operator of order

m \in R

, if for every open subset

U \subset M

and trivializations

τ_{E} : E_{| U} ⟶ U \times R^{d_{E}}, τ_{F} : F_{| U} ⟶ U \times R^{d_{F}},

there is a

(d_{F} \times d_{E})

‐matrix of totally characteristic pseudodifferential operators

Q_{i j}

of order m such that

{(τ_{F} \circ {(Q s)}_{| U})}_{i} = \sum_{j} Q_{i j} {(τ_{E} \circ s)}_{j}, s \in {normalC}_{normalc}^{\infty} false(U, E false) .

In this case, one says that Q is of class

L_{b}^{m}

. For a more detailed exposition on totally characteristic pseudodifferential operators the reader is referred to .…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

“…Remark The proof of Proposition is modelled on the proof of [, Theorem 4] and [, Theorem 2], where the integral transform

F_{spher}

was not explicitly introduced yet, but is tacitly present.…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

“…In this case, one says that Q is of class L m b . For a more detailed exposition on totally characteristic pseudodifferential operators the reader is referred to [17].…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

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Microlocal analysis on wonderful varieties. Regularized traces and global characters

2016

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Abstract. Let G be a connected reductive complex algebraic group with split real form (G, σ). Consider a strict wonderful G-variety X equipped with its σ-equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G-vector bundle over X. In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E, and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.

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mentioning

confidence: 68%

“…Proof The proof is similar to the proof of Theorem 7 in . By Proposition ,

\begin{matrix} true \\ right {Tr}_{ζ} π (f) & left = \sum_{j = 1}^{d} \sum_{ϱ} \int_{R^{s + r}} {false| y_{s + 1} \dots y_{s + r} false|}^{ζ} (α_{ϱ} \circ φ_{ϱ}) (y)^{j j} {\tilde{Q}}_{f}^{ϱ} (y, 0) d y \end{matrix}

is a meromorphic function in ζ with possible poles at

- 1, - 3, \dots

.…”

Section: Regularized Traces and Fixed Point Formulaementioning

confidence: 81%

“…Similarly, if E and F are vector bundles over M , a continuous linear map

Q : {normalC}_{normalc}^{\infty} false(M, E false) ⟶ {normalC}^{\infty} false(M, F false)

is called a totally characteristic pseudodifferential operator of order

m \in R

, if for every open subset

U \subset M

and trivializations

τ_{E} : E_{| U} ⟶ U \times R^{d_{E}}, τ_{F} : F_{| U} ⟶ U \times R^{d_{F}},

there is a

(d_{F} \times d_{E})

‐matrix of totally characteristic pseudodifferential operators

Q_{i j}

of order m such that

{(τ_{F} \circ {(Q s)}_{| U})}_{i} = \sum_{j} Q_{i j} {(τ_{E} \circ s)}_{j}, s \in {normalC}_{normalc}^{\infty} false(U, E false) .

In this case, one says that Q is of class

L_{b}^{m}

. For a more detailed exposition on totally characteristic pseudodifferential operators the reader is referred to .…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

“…Remark The proof of Proposition is modelled on the proof of [, Theorem 4] and [, Theorem 2], where the integral transform

F_{spher}

was not explicitly introduced yet, but is tacitly present.…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

“…In this case, one says that Q is of class L m b . For a more detailed exposition on totally characteristic pseudodifferential operators the reader is referred to [17].…”

Section: Microlocal Analysis Of Integral Operators On Wonderful Variementioning

confidence: 99%

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Microlocal analysis on wonderful varieties. Regularized traces and global characters

2016

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Equivariant heat asymptotics on spaces of automorphic forms

Paniagua-Taboada

Ramacher

2015

Trans. Amer. Math. Soc.

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Let G be a connected, real, semisimple Lie group with finite center, and K a maximal compact subgroup of G. In this paper, we derive K-equivariant asymptotics for heat traces with remainder estimates on compact Riemannian manifolds carrying a transitive and isometric G-action. In particular, we compute the leading coefficient in the Minakshishundaram-Pleijel expansion of the heat trace for Bochner-Laplace operators on homogeneous vector bundles over compact locally symmetric spaces of arbitrary rank.

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Compactification of semi-simple Lie groups

Albin,

Dimakis,

Melrose

et al. 2019

Preprint

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We discuss the 'hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary face fibering over two copies of the corresponding flag variety with fiber modeled on the (compactification of the) reductive part. On the hdcompactification Harish-Chandra's Schwartz space is identified with a space of conormal functions of rapid-logarithmic decay relative to square-integrable functions.

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Integral operators on the Oshima compactification of a Riemannian symmetric space

Cited by 3 publications

References 12 publications

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Equivariant heat asymptotics on spaces of automorphic forms

Compactification of semi-simple Lie groups

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