scholarly journals The size of characters of exceptional lie groups

2004 ◽  
Vol 77 (2) ◽  
pp. 233-248 ◽  
Author(s):  
Kathryn E. Hare ◽  
Karen Yeats
Keyword(s):  
Lie Groups ◽  
Classical Result ◽  
Continuous Measure ◽  
Compact Lie Groups ◽  
Absolutely Continuous ◽  
Exceptional Lie Groups

AbstractPointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

scholarly journals Pointwise estimates of the size of characters of compact Lie groups

2000 ◽  
Vol 69 (1) ◽  
pp. 61-84 ◽  
Author(s):  
Kathryn E. Hare ◽  
David C. Wilson ◽  
Wai Ling Yee
Keyword(s):  
Lie Groups ◽  
Classical Result ◽  
Pointwise Estimates ◽  
Compact Lie Groups ◽  
Simple Lie Groups ◽  
Orbital Measure ◽  
Continuous Measures ◽  
Minimal Integer

AbstractPointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained with which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L2. We also prove that if k = rank G then μ 2k ∈ L1 for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be dimension of the group G.

scholarly journals Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra

2018 ◽  
Vol 33 (20) ◽  
pp. 1850118 ◽  
Author(s):  
Ivan Todorov ◽  
Michel Dubois-Violette
Keyword(s):  
Standard Model ◽  
Lie Groups ◽  
Jordan Algebra ◽  
Particle Physics ◽  
Space Time ◽  
Quantum Algebra ◽  
Compact Lie Groups ◽  
Finite Dimensional ◽  
The Standard Model ◽  
Exceptional Lie Groups

We continue the study undertaken in Ref. 16 of the exceptional Jordan algebra [Formula: see text] as (part of) the finite-dimensional quantum algebra in an almost classical space–time approach to particle physics. Along with reviewing known properties of [Formula: see text] and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel–de Siebenthal theory of maximal connected subgroups of simple compact Lie groups.

scholarly journals Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra

2016 ◽  
Vol 68 (4) ◽  
pp. 841-875 ◽  
Author(s):  
Sanjiv Kumar Gupta ◽  
Kathryn Hare
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Absolute Continuity ◽  
Classical Result ◽  
Continuous Measure ◽  
Empty Interior ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Orbital Measure ◽  
Classical Lie Algebra

AbstractLet 𝓰 be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbitalmeasures is absolutely continuous with respect to Lebesgue measure on 𝓰, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ 𝓰.In this paper 𝓰 is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1 , . . . , XL), with Xi ∊ 𝓰, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of 𝓰 and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

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).

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