Simon Gindikin, Bernhard Kroetz
We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.
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A convexity theorem for certain G-orbits in a complexified Riemannian symmetric space G_C/K_C is proved. Applications to analytically continued spherical functions will be given.
Distinction of the Steinberg representation and the dual group of a symmetric space
研究Steinberg表示相对于对称子群的区分问题，通过超图上的调和函数存在性联系，并验证Steinberg表示在p-adic域上的区分性与Langlands参数与对偶群的关系。
We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic deco…
We prove a remarkable generalization of a convexity theorem for semisimple symmetric spaces G/H established earlier in 1986 by the second named author. The latter result generalized Kostant's non-linear convexity theorem for the Iwasawa decomposition of a real semisimple Lie group. The present generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type relative to H considered in 1986.
For a symmetric pair $(G,H)$ of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of $G$ and one for $H$ and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.
It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in the ``Iwasawa domain'' using complex analysis. But we can see that we need no complex analysis to prove it. In this paper, we generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry of two associated symmetric subgroups, we also give a direct proof of the known fact that the Akhiezer-Gindikin domain is included in all cycle spaces.
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …
For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$ (G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F})\times U(n-m;\mathbb{F}))), \qquad \mathbb{F}=…
We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.
We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GL_n(q) acting on the tensor product of the symmetric algebra and the exterior algebra of the natural GL_n(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GL_n(q) in the tensor product of the symmetric algebra and the exterior algebra of V, twisted by an arbitrary determinant power.
We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.
Distinction of the Steinberg representation with respect to a symmetric pair
本文将 Steinberg 表示实现为调和上链空间，研究其关于对称对固定子群的区分空间维数，给出可达上界并应用于一般线性群与特殊正交子群完成分类。
We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associated to symmetric diffusion semigroups to obtain a characterization of inhomogeneous abstract Besov spaces on the abstract Hilbert spaces. Then we apply our results for the abstract Besov spaces defined through the Poisson and Gauss-Weierstrass semigroups.
Let $X=G/K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\in G^\mbb{C}$ be the associated compression semi-group. Let $Γ$ be a discrete subgroup of $G$. We give a sufficient condition for $Γ\backslash S$ to be a Stein manifold. Moreover, we show that in general $Γ\backslash S$ is not Stein, which disproves a conjecture by Achab, Betten and Krötz.
Stein's method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by i-cycles, or equivalently for the character ratio of a Plancherel distributed representation on an i-cycle. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations o…
One of the purposes of this paper is to prove that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L^{2,1}(G)\ast L^{2,1}({G})\subseteq L^{2,\infty}({G}). Let {K} be a maximal compact subgroup of {G} and X=G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator {\cal M}_2f(z) = \sup_{z\in B} \frac{1}{|B|} \int_{B}|f(z')|\,dz' is bounded from L^{2,1}(X) to L^{2,\infty}(X) and from L^p(X) to L^p(X) in the sharp range of exponents p\in(2,\infty]. The supremum in the definition of {\cal M}_2f(z) is taken over al…
The main result of this paper is a ``real form'' of Kirwan's convexity theorem, which in the abelian case was proved earlier by Duistermaat. We apply our result to flag varieties of real semisimple groups and obtain eigenvalue inequalities, which generalize inequalities found by Weyl, Ky Fan, Kostant, Klyachko, and many others.
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