On the $C^{\infty }$ wave-front set  of traces of CR   functions  on maximally real submanifolds

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

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Precise propagation of singularities for a hyperbolic system with characteristics of variable multiplicity

Nagoya Mathematical Journal ◽

10.1017/s0027763000000350 ◽

1986 ◽

Vol 101 ◽

pp. 111-130 ◽

Cited By ~ 1

Author(s):

Chisato Iwasaki ◽

Yoshinori Morimoto

Keyword(s):

Cauchy Problem ◽

Wave Front ◽

Hyperbolic System ◽

Wave Front Set ◽

Variable Multiplicity ◽

The Cauchy Problem ◽

Propagation Of Singularities ◽

Admissible Trajectory

In this paper we consider the Cauchy problem for a hyperbolic system with characteristics of variable multiplicity and construct a certain solution whose wave front set propagates precisely along the so-called “broken null bicharacteristic flow”, in other words, along the admissible trajectory if we use the terminology of [6].

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The wave front set of the solution of a simple initial-boundary value problem with glancing rays. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410000030x ◽

1977 ◽

Vol 81 (1) ◽

pp. 97-120 ◽

Cited By ~ 19

Author(s):

F. G. Friedlander ◽

R. B. Melrose

Keyword(s):

Boundary Value Problem ◽

Wave Front ◽

Boundary Value ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Principal Result ◽

Wave Front Set ◽

Precise Meaning ◽

Image Position ◽

Initial Boundary Value

This paper is a sequel to an earlier paper in these Proceedings by one of us ((5); this will be referred to as [I]). The question considered there was that of determining the wave front set of the solution of the boundary value problemwhere x∈+, y∈n, and n > 1; the precise meaning of the boundary condition at x = 0 is explained in section 1 below. The principal result of [I] can be expressed concisely by saying that singularities do not propagate along the boundary; a detailed statement is given in Theorem 1·9 of the present paper.

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The Gabor wave front set

Monatshefte für Mathematik ◽

10.1007/s00605-013-0592-0 ◽

2013 ◽

Vol 173 (4) ◽

pp. 625-655 ◽

Cited By ~ 20

Author(s):

Luigi Rodino ◽

Patrik Wahlberg

Keyword(s):

Wave Front ◽

Wave Front Set ◽

Gabor Wave Front Set

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Motivic Wave Front Sets

International Mathematics Research Notices ◽

10.1093/imrn/rnz196 ◽

2019 ◽

Author(s):

Michel Raibaut

Keyword(s):

Tensor Product ◽

Wave Front ◽

Lie Groups ◽

Wave Front Set ◽

Singular Support ◽

Valued Field ◽

Wave Front Sets ◽

Multiplicative Subgroups ◽

Products Of Distributions ◽

Lambda Wave

Abstract The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions.

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Wave Front Sets with respect to the Iterates of an Operator with Constant Coefficients

Abstract and Applied Analysis ◽

10.1155/2014/438716 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17 ◽

Cited By ~ 5

Author(s):

C. Boiti ◽

D. Jornet ◽

J. Juan-Huguet

Keyword(s):

Differential Operator ◽

Wave Front ◽

Partial Differential Operator ◽

Regularity Theorem ◽

Wave Front Set ◽

Open Set ◽

Wave Front Sets ◽

Constant Coefficients ◽

New Type ◽

Classical Distribution

We introduce the wave front setWF*P(u)with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distributionu∈𝒟′(Ω)in an open set Ω in the setting of ultradifferentiable classes of Braun, Meise, and Taylor. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.

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