Parahoric Restriction for GSp(4)

Parahoric restriction is the parahoric analogue of Jacquet’s functor. The group GSp(4, F) of symplectic similitudes of genus two over a local number field F/? _p has five conjugacy classes of parahoric subgroups. For each we determine the parahoric restriction of the non-cuspidal irreducible smooth representations of GSp(4, F) in terms of explicit character values.     

A remark on the failure of multiplicity one for GSp(4)

We revisit a classical result of Howe and Piatetski-Shapiro on the failure of strong multiplicity one for GSp(4).     

Klingen $${mathfrak {p}}^2$$ p 2 vectors for $$mathrm{GSp}(4)$$ GSp ( 4 )

The Ramanujan Journal - We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${mathfrak {p}}^2$$ for all irreducible, admissible...     

The fundamental lemma for

The fundamental lemma is a conjectural identity between the orbital integrals on two reductive groups. The fundamental lemma is required for the stabilization of the trace formula and for various applications to automorphic forms. This paper proves the fundamental lemma for the group and its endoscopic groups.

     

The arithmetic fundamental lemma: An update

Science China Mathematics - This is an expository article on the recent progress on the arithmetic fundamental lemma conjecture, based largely on Zhang (2019). Beside stating the local conjecture,...     

A remark on the fundamental lemma of Jacquet

We conjecture a generalization of the fundamental lemma of Jacquet in the context of ${\mathit{GL}}_n$ over a quadratic extension. We provide a heuristic argument for our expectation and prove our conjecture for ${\mathit{GL}}_2$. To cite this article: O. Offen, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A uniqueness theorem for representations of GSO(6) and the strong multiplicity one theorem for generic representations of GSp(4)

Consider the θ-correspondence from GSp(4) to GSO(6). We prove that locally over a nonarchimedean fieldF, this correspondence is injective on generic representations (i.e. with Whittaker model) of GSp(4,F). We use this to show the strong multiplicity one property for irreducible, automorphic, cuspidal representations of GSp(4,A), which are generic.     

The maximal subgroups of F4(2) and its automorphism group

We completely determine the conjugacy classes of maximal subgroups of the finite simple group F₄(2) of Lie type, and of its automorphism group.     

A finiteness theorem for subgroups of SP (4,Z)

This paper proves that there are only finitely many subgroupsH of finite index in Sp(4,Z) such that coresponding quotient of the Siegel upper half space of rank two is not of general type. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. vol. 56. Algebraic Geometry-9, 1998.     

Sign (di)lemma for dimension shifting

There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the result − which we call as the sign lemma for dimension shifting − by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.     

Maximal subgroups of the Coxeter group W(H4) and quaternions

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H₄) of order 14,400. Its derived subgroup is the largest finite subgroup W(H₄)/Z₂ of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H₂) × W(H₂)] ⋊ Z₄ and W(H₃) × Z₂ possess non-crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H₄) with sets of quaternion pairs acting on the quaternionic root systems.     

Parabolic subgroups in3D4(R)

We describe parabolic subgroups in the Chevalley group of crossed type³D₄ over a commutative local ring. Previously the analogous question was considered for groups of normal and certain crossed types in RZhMat, 1976, 10A152; 1977, 10A302; 1978, 6A476; 1980, 5A437.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 106–113, 1980.     

Hybrid integral transformations (Bessel,Legendre, Bessel)

By means of the method of delta-type sequences, hybrid integral transformations of the Hankel 1st kind-Legendre-Weber and the Hankel 2nd kind-Legendre-Weber on the polar axis with two conjunction points are constructed and decomposability theorems are proved; basic identities for the integral transform of the differential operator are obtained. A logical scheme for application of the integral transformations obtained for solving the corresponding problems of mathematical physics is presented.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 770–779, June, 1991.     

Connected subgroups of the conformal group C(1, 4)

A method is given for describing maximal subalgebras of rank r, 1 r 4, of the conformal algebra AC(1, 4), which is the maximal invariance algebra of the eikonal equation. With the help of this method a classification is made up to C(1, 4)-equivalence of all maximal subalgebras L of rank 1, 2, 3, and 4 of the algebra AC(1, 4) satisfying the condition L V p₁, P₂, P₃, p₄, where V is the space of translations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7–8, pp. 870–884, July–August, 1991.     

Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.     

The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (?1, 1), $$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$ holds for z ∈ D, where Λ _U (z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U(z), and the equality occurs for some point in D if and only if $U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$ , z ∈ D, with a Möbius transformation φ of D onto itself.