scholarly journals Equivariant holomorphic maps into the Siegel disc and the metaplectic representation

1997 ◽  
Vol 62 (2) ◽  
pp. 160-174 ◽  
Author(s):  
Jean-Louis Clerc
Keyword(s):  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Holomorphic Maps ◽  
Harmonic Polynomials ◽  
Metaplectic Representation ◽  
Siegel Disc

AbstractWe restrict the metaplectic representation to subgroupsGof the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup ofG.

scholarly journals Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups

2021 ◽  
Vol 25 (34) ◽  
pp. 975-993
Author(s):  
Dubravka Ban ◽  
Joseph Hundley
Keyword(s):  
Maximal Compact Subgroup ◽  
Series Representation ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Principal Series ◽  
Principal Series Representation ◽  
Content Type ◽  
Series Representations ◽  
Principal Series Representations

In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G 0 G_0 -invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G 0 G_0 -representations by duality.

scholarly journals Residual automorphic representations of Sp4

1992 ◽  
Vol 127 ◽  
pp. 15-47 ◽  
Author(s):  
Takao Watanabe
Keyword(s):  
Hilbert Space ◽  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Borel Subgroup ◽  
Compact Subgroup ◽  
Algebraic Number Field ◽  
Orthogonal Decomposition ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Adele Group

Let G = Sp4 be the symplectic group of degree two defined over an algebraic number field F and K the standard maximal compact subgroup of the adele group G (A). By the general theory of Eisenstein series ([14]), one knows that the Hilbert space L2(G(F)\G(A)) has an orthogonal decomposition of the formL2(G(F)\G(A)) = L2(G) ⊕ L2(B) ⊕ L2(P1) ⊕ L2(P1),where B is a Borel subgroup and Pi are standard maximal parabolic subgroups in G for i = 1,2. The purpose of this paper is to study the space L2d(B) associated to discrete spectrurns in L2(B).

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

The relativistic SL (6, C ) strong vertex and kinematic form factors

1968 ◽  
Vol 305 (1480) ◽  
pp. 27-53
Keyword(s):  
Form Factor ◽  
Closed Form ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Form Factors ◽  
Large Momentum ◽  
Group Representations ◽  
Technical Problems ◽  
Infinite Dimensional ◽  
Dimensional Group

The SL (6, C ) invariant baryon-baryon-meson vertex, involving three infinite multiplets, is expanded in terms of representations of the maximal compact subgroup SU (6) p for particles of arbitrary momentum. The terms in the expansion which involve the lowest-dimensional SU (6) multiplets are evaluated in closed form. The assumption of SL (6, C ) invariance leads to a breaking of SU (6) by spurions in a well-defined way. Each type of spurion coupling is accompanied by a function of momentum which vanishes rapidly for large momentum transfer and can be interpreted as a kinematic form factor. All calculations are performed with the ‘generalized tensor’ formalism, and a number of mathematical results are obtained which greatly simplify the technical problems of dealing with infinite-dimensional group representations.

scholarly journals Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Runlin Zhang
Keyword(s):  
Invariant Theory ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Linear Algebraic Group ◽  
Natural Question ◽  
Geometric Invariant ◽  
Geometry Of Numbers ◽  
Push Forward ◽  
The Right ◽  
Refined Version

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> be a semisimple linear algebraic group defined over rational numbers, <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{K} $\end{document}</tex-math></inline-formula> be a maximal compact subgroup of its real points and <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> be an arithmetic lattice. One can associate a probability measure <inline-formula><tex-math id="M4">\begin{document}$ \mu_{ \mathrm{H}} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> for each subgroup <inline-formula><tex-math id="M6">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M7">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> defined over <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Q} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. As G acts on <inline-formula><tex-math id="M9">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> from the right, we can push forward this measure by elements from <inline-formula><tex-math id="M10">\begin{document}$ \mathrm{G} $\end{document}</tex-math></inline-formula>. By pushing down these measures to <inline-formula><tex-math id="M11">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula>, we call them homogeneous. It is a natural question to ask what are the possible weak-<inline-formula><tex-math id="M12">\begin{document}$ * $\end{document}</tex-math></inline-formula> limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of <inline-formula><tex-math id="M13">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general <inline-formula><tex-math id="M15">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on <inline-formula><tex-math id="M16">\begin{document}$ {\text{SL}}_n $\end{document}</tex-math></inline-formula> proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. <b>193</b> words.</p>

scholarly journals Spherical functions on orthogonal groups

1996 ◽  
Vol 141 ◽  
pp. 157-182 ◽  
Author(s):  
Yasuhiro Kajima
Keyword(s):  
Algebraic Group ◽  
Root System ◽  
Explicit Formula ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Spherical Functions ◽  
Orthogonal Groups ◽  
Reductive Algebraic Group

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

scholarly journals Stable base change for spherical functions

1987 ◽  
Vol 106 ◽  
pp. 121-142 ◽  
Author(s):  
Yuval Z. Flicker
Keyword(s):  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Base Change ◽  
Conjugacy Classes ◽  
Natural Homomorphism ◽  
Convolution Algebra ◽  
Orbital Integral ◽  
Twisted Conjugacy ◽  
Complex Valued

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.

Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

2018 ◽  
Vol 61 (3) ◽  
pp. 553-571
Author(s):  
Camelia Karimianpour
Keyword(s):  
Linear Group ◽  
Local Field ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Principal Series ◽  
Special Linear Group ◽  
Covering Group ◽  
Branching Rules ◽  
Series Representations ◽  
Covering Groups

AbstractLet G be the n-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of G to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

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