scholarly journals TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

2021 ◽  
pp. 1-62
Author(s):  
Zongbin Chen
Keyword(s):  
Fundamental Domain ◽  
Recurrence Relations ◽  
Rational Points ◽  
Orbital Integrals ◽  
Orbital Integral ◽  
Comparison Of Results ◽  
Fundamental Domains ◽  
Springer Fiber ◽  
Springer Fibers

Abstract We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

scholarly journals Components of Affine Springer Fibers

2018 ◽  
Vol 2020 (6) ◽  
pp. 1882-1919
Author(s):  
Cheng-Chiang Tsai
Keyword(s):  
Weyl Group ◽  
Characteristic Zero ◽  
Reductive Group ◽  
Semisimple Element ◽  
Orbital Integrals ◽  
Number Of Components ◽  
Springer Fiber ◽  
Springer Fibers

Abstract Let G be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma _0\in (\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t\gamma _0$. Using methods from p-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.

Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets

2018 ◽  
Vol 9 (2) ◽  
pp. 93-107
Author(s):  
Heidi Burgiel ◽  
Vignon Oussa
Keyword(s):  
Fundamental Domain ◽  
Full Rank ◽  
Unit Cube ◽  
Gabor Frames ◽  
Invertible Matrix ◽  
Orthonormal Bases ◽  
Fundamental Domains ◽  
Uncountable Family ◽  
Indicator Functions

AbstractThe main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in{\mathbb{R}^{d}}admitting common connected fundamental domains of the type{N[0,1)^{d}}, whereNis an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type{N[0,1)^{d}}. We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type{N[0,1)^{2}}, whereNis an invertible matrix.

scholarly journals Fundamental Domains for Rhombic Lattices with Dihedral Symmetry of Order 8

10.37236/7802 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Joseph Ray Clarence G. Damasco ◽  
Dirk Frettlöh ◽  
Manuel Joseph C. Loquias
Keyword(s):  
Symmetry Group ◽  
Fundamental Domain ◽  
Point Group ◽  
Symmetry Groups ◽  
Fundamental Domains ◽  
Open Case ◽  
Point Groups ◽  
Index 2 ◽  
Rhombic Lattice ◽  
Planar Lattices

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in a preprint by the authors on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.  

scholarly journals Number systems and tilings over Laurent series

2009 ◽  
Vol 147 (1) ◽  
pp. 9-29 ◽  
Author(s):  
TOBIAS BECK ◽  
HORST BRUNOTTE ◽  
KLAUS SCHEICHER ◽  
JÖRG M. THUSWALDNER
Keyword(s):  
Finite Field ◽  
Fundamental Domain ◽  
Laurent Series ◽  
Residue Class ◽  
Number Systems ◽  
Fundamental Domains ◽  
Ring Of Polynomials ◽  
Residue Class Ring ◽  
Class Ring

AbstractLet be a field and [x, y] the ring of polynomials in two variables over . Let f ∈ [x, y] and consider the residue class ring R := [x, y]/f[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element of R admits a digit representation of the form d0 + d1x + ⋅ ⋅ ⋅ + dℓxℓ with digits di ∈ [y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base” x. In particular, we define and characterize digit representations for the ring S := ((x−1, y−1))/f((x−1, y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers of x in their “x-ary” representation. The translates of the fundamental domain induce a tiling of S. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose =q to be a finite field, these unions become finite.

scholarly journals LA VARIANTE INFINITÉSIMALE DE LA FORMULE DES TRACES DE JACQUET-RALLIS POUR LES GROUPES LINÉAIRES

2016 ◽  
Vol 17 (4) ◽  
pp. 735-783 ◽  
Author(s):  
Michał Zydor
Keyword(s):  
Trace Formula ◽  
Haar Measure ◽  
Fourier Transforms ◽  
Zeta Functions ◽  
Adjoint Action ◽  
Linear Groups ◽  
Orbital Integrals ◽  
Absolute Value ◽  
General Linear Groups ◽  
Orbital Integral

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated à la Arthur multiplied by the absolute value of the determinant to the power $s\in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s,\cdot )$ indexed by the invariants of the adjoint action of $\text{GL}_{n}(\text{F})$ on $\mathfrak{gl}_{n+1}(\text{F})$ as well as a «spectral side» consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s,\cdot )$ are invariant and depend only on the choice of the Haar measure on $\text{GL}_{n}(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s,\cdot )$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s,\cdot )$ in terms of relative orbital integrals regularised by means of zeta functions.

scholarly journals Fundamental domains for genus-zero and genus-one congruence subgroups

2010 ◽  
Vol 13 ◽  
pp. 222-245
Author(s):  
C. J. Cummins
Keyword(s):  
Fundamental Domain ◽  
Congruence Subgroup ◽  
Discrete Subgroup ◽  
Magma Source ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Fundamental Domains ◽  
Finite Set ◽  
Data Files ◽  
Genus One

AbstractIn this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups ofPSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integerf, which determines a maximal discrete subgroup Γ0(f)+ofSL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+is in a subgroupG; and the index ofGin Γ0(f)+. The output consists of: a fundamental domain forG, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators ofG. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

Elliptic and hypoelliptic orbital integrals

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Space ◽  
Heat Kernel ◽  
Orbital Integrals ◽  
Orbital Integral ◽  
Finite Dimensional

This chapter constructs semisimple orbital integrals associated with the heat kernel for the hypoelliptic Laplacian ℒbX. By making b → 0, the chapter shows that the corresponding supertrace coincides with the orbital integral associated with the standard elliptic heat kernel. Throughout this chapter, the same assumptions as in chapters 2 and 3 will be made, and this chapter uses corresponding notation. Also if V = V₊ ⊕ V₋ is a finite dimensional Z₂-graded vector space, if τ‎ = ±1 is the involution of V that defines the Z₂-grading, if A ∈ End(V), the chapter defines its supertrace Tr″[A] by Tr″[A] = Tr[τ‎A].

Unipotent Orbital Integrals of Hecke Functions for GL(n)

1994 ◽  
Vol 46 (2) ◽  
pp. 308-323
Author(s):  
Rebecca A. Herb
Keyword(s):  
Linear Functional ◽  
Elementary Proof ◽  
Hecke Algebra ◽  
Conjugacy Classes ◽  
Spherical Functions ◽  
Orbital Integrals ◽  
Orbital Integral ◽  
Linearly Independent ◽  
Complete Set

AbstractLet G = GL(n, F) where F is a p-adic field, and let 𝓗(G) denote the Hecke algebra of spherical functions on G. Let u1,..., up denote a complete set of representatives for the unipotent conjugacy classes in G. For each 1 ≤ i ≤ p, let μi be the linear functional on such that μi(f) is the orbital integral of f over the orbit of ui. Waldspurger proved that the μi, 1 ≤ i ≤ p, are linearly independent. In this paper we give an elementary proof of Waldspurger's theorem which provides concrete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspurger's theorem and discuss the consequences for SL(n, F).

scholarly journals GEOMETRY OF KOTTWITZ–VIEHMANN VARIETIES

2019 ◽  
pp. 1-65
Author(s):  
Jingren Chi
Keyword(s):  
Conjugacy Class ◽  
Dual Group ◽  
Geometric Properties ◽  
Orbital Integrals ◽  
Irreducible Components ◽  
Springer Fibers

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

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