Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group

Adler, Jeffrey D.; DeBacker, Stephen

doi:10.1307/mmj/1028575734

Cited by 48 publications

(88 citation statements)

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“…The terms "unipotent" and "nilpotent" are sometimes given other definitions. See §2.5 of [2] for a discussion.…”

Section: Some Notation and Results In A General Settingmentioning

confidence: 99%

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A generalization of a result of Kazhdan and Lusztig

Adler

DeBacker

2003

Proc. Amer. Math. Soc.

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“…The terms "unipotent" and "nilpotent" are sometimes given other definitions. See §2.5 of [2] for a discussion.…”

Section: Some Notation and Results In A General Settingmentioning

confidence: 99%

“…Let G(E) + F denote the pro-unipotent radical of G(E) F . As in [2], we define lattices g(E) F and g(E) + F in g(E); for any x ∈ B(G, E), we define g(E) x and g(E) + x to be g(E) F and g(E)…”

Section: Introductionmentioning

confidence: 99%

A generalization of a result of Kazhdan and Lusztig

Adler

DeBacker

2003

Proc. Amer. Math. Soc.

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“…The second is proved in the "⊂" part of the proof of Lemma 4.3.2 of [16] for r = 0, and in the proof of Lemma 3.7.6 of [3] for r > 0.…”

Section: Filtrations and Descentmentioning

confidence: 99%

Good Product Expansions for Tame Elements of p-Adic Groups

Adler

Spice

2010

International Mathematics Research Papers

101

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“…It is proven in [3] that g r (resp. G r ) is a G-domain: a G-invariant, open and closed subset of g (resp.…”

Section: Moy-prasad Filtrationsmentioning

confidence: 99%

The local character expansion near a tame, semisimple element

Adler¹,

Korman²

2007

ajm

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Abstract. Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected, nor that the underlying field has characteristic zero. IntroductionLet G denote the group of k-points of a reductive k-group G, where k is a nonarchimedean local field. To simplify the present discussion, assume for now that G is connected and that k has characteristic zero. Let (π, V ) denote an irreducible admissible representation of G. Let dg denote a fixed Haar measure on G, and let C ∞ c (G) denote the space of complex-valued, locally constant, compactly supported functions on G. The distribution character Θ π of π is the map C ∞ c (G) → C given by Θ π (f ) := tr π(f ), where π(f ) is the (finite-rank) operator on V given by π(f )v := G f (g)π(g)v dg. From Howe [12] and Harish-Chandra [9], the distribution Θ π is represented by a locally constant function on the set of regular semisimple elements in G. We will denote this function also by Θ π .For any semisimple γ ∈ G, the local character expansion about γ (see [11] and [10]) is the identityvalid for all regular semisimple Y in the Lie algebra m of the centralizer of γ such that Y is close enough to 0. Here, the sum is over the set of nilpotent orbits O in m; µ O is the function that represents the distribution that is the Fourier transform of the orbital integral µ O associated to O; c O = c O,γ (π) ∈ C; and e is the exponential map, or some suitable substitute. This is a qualitative result, in the sense that it gives no indication of how close Y must be to 0 in order for the identity to be valid. Many questions in harmonic analysis on G require more quantitative versions of such qualitative results. As an example of a quantitative result, DeBacker [6] has determined (under some hypotheses on G) a neighborhood of validity for the local character expansion near the identity, thus verifying a conjecture of Hales, Moy, and Prasad (see [15]).1991 Mathematics Subject Classification. 22E35, 22E50, 20G25.

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Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group

Cited by 48 publications

References 4 publications

A generalization of a result of Kazhdan and Lusztig

A generalization of a result of Kazhdan and Lusztig

Good Product Expansions for Tame Elements of p-Adic Groups

The local character expansion near a tame, semisimple element

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