scholarly journals CONVOLUTIONS OF GENERIC ORBITAL MEASURES IN COMPACT SYMMETRIC SPACES

2009 ◽  
Vol 79 (3) ◽  
pp. 513-522 ◽  
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Invariant Measure ◽  
Symmetric Space ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Nonempty Interior ◽  
Absolutely Continuous ◽  
Double Cosets ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces ◽  
Dense Set

AbstractWe prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

scholarly journals CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 470-485 ◽  
Author(s):  
BOUDJEMÂA ANCHOUCHE ◽  
SANJIV KUMAR GUPTA
Keyword(s):  
Positive Integer ◽  
Symmetric Space ◽  
Lie Algebra ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Double Coset ◽  
Cartan Decomposition ◽  
Absolutely Continuous ◽  
Orbital Measure ◽  
Noncompact Symmetric Space

AbstractLet G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

scholarly journals Reflections and symmetries in compact symmetric spaces

1988 ◽  
Vol 38 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Bang-Yen Chen ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces

Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

scholarly journals SMOOTHNESS OF CONVOLUTION PRODUCTS OF ORBITAL MEASURES ON RANK ONE COMPACT SYMMETRIC SPACES

2016 ◽  
Vol 94 (1) ◽  
pp. 131-143
Author(s):  
KATHRYN E. HARE ◽  
JIMMY HE
Keyword(s):  
Density Function ◽  
Symmetric Spaces ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Double Cosets ◽  
Rank One ◽  
Convolution Products ◽  
Compact Symmetric Spaces

We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning that their density function is in $L^{2}$). We characterise the pairs whose convolution product is either absolutely continuous or in $L^{2}$ in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $\text{SU}(2)/\text{SO}(2)$, then the convolution of any two regular orbital measures is in $L^{2}$, while in $\text{SU}(2)/\text{SO}(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

scholarly journals Integral geometry under cut loci in compact symmetric spaces

1995 ◽  
Vol 137 ◽  
pp. 33-53 ◽  
Author(s):  
Hiroyuki Tasaki
Keyword(s):  
Invariant Measure ◽  
Lie Group ◽  
Integral Geometry ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Integral Invariant ◽  
Identity Component ◽  
Integral Invariants ◽  
Compact Symmetric Spaces ◽  
Group Of Isometries

The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

scholarly journals Homotopy of compact symmetric spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 221-228 ◽  
Author(s):  
John M. Burns
Keyword(s):  
Symmetric Spaces ◽  
Unified Approach ◽  
Topological Characterization ◽  
New Approach ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Compact Symmetric Spaces

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

Singular measures for classP-circle homeomorphisms with several break points

2012 ◽  
Vol 34 (2) ◽  
pp. 423-456 ◽  
Author(s):  
ABDELHAMID ADOUANI ◽  
HABIB MARZOUGUI
Keyword(s):  
Invariant Measure ◽  
Rotation Number ◽  
Haar Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Points ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Break Points ◽  
Irrational Rotation Number

AbstractLetfbe a classP-homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Letfhave irrational rotation number andDfbe absolutely continuous on every continuity interval ofDf. We prove that if the product of thef-jumps along any subset of break points is distinct from 1 then the invariant measureμfis singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation numberρ(f) is irrational of bounded type then (a) if the product of thef-jumps on some orbit is distinct from 1 then the invariant measureμfis singular with respect to the Haar measurem, and (b) if the product of thef-jumps on each orbit is equal to 1 andD2f∈Lp(S1) for somep>1 thenμfis equivalent to the Haar measure.

scholarly journals Helgason spheres of compact symmetric spaces and immersions of finite type

2001 ◽  
Vol 63 (2) ◽  
pp. 243-255
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Finite Type ◽  
Symmetric Spaces ◽  
Unit Vector ◽  
Geodesic Sphere ◽  
Compact Type ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Unit Speed

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

scholarly journals On some recent progress in the classification of $$(P\hbox { and }Q)$$-polynomial association schemes

2019 ◽  
Author(s):  
Alexander L. Gavrilyuk ◽  
Jack H. Koolen
Keyword(s):  
Symmetric Spaces ◽  
Recent Progress ◽  
Association Schemes ◽  
Compact Symmetric Spaces ◽  
Expository Paper

AbstractThe problem of classification of $$(P\hbox { and }Q)$$(PandQ)-polynomial association schemes, as a finite analogue of E. Cartan’s classification of compact symmetric spaces, was posed in the monograph “Association schemes” by E. Bannai and T. Ito in the early 1980s. In this expository paper, we report on some recent results towards its solution.

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