scholarly journals Isometry theorem for the Segal–Bargmann transform on a noncompact symmetric space of the complex type

2008 ◽  
Vol 254 (6) ◽  
pp. 1575-1600 ◽  
Author(s):  
Brian C. Hall ◽  
Jeffrey J. Mitchell
Keyword(s):  
Symmetric Space ◽  
Complex Type ◽  
Bargmann Transform ◽  
Noncompact Symmetric Space

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 Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

1980 ◽  
Vol 78 ◽  
pp. 1-44 ◽  
Author(s):  
Ramesh Gangolli ◽  
Garth Warner
Keyword(s):  
Symmetric Space ◽  
Zeta Function ◽  
Symmetric Spaces ◽  
Zeta Functions ◽  
Discrete Subgroup ◽  
Uniform Lattice ◽  
Rank One ◽  
Noncompact Symmetric Space

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).

scholarly journals PATH INTEGRALS AND PERTURBATIVE EXPANSIONS FOR NONCOMPACT SYMMETRIC SPACES

1992 ◽  
Vol 07 (27) ◽  
pp. 6873-6885
Author(s):  
NOAH LINDEN ◽  
MALCOLM J. PEFIRY
Keyword(s):  
Perturbation Theory ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Path Integrals ◽  
Perturbation Expansion ◽  
Quantum Mechanical ◽  
Soluble Model ◽  
Exactly Soluble Model ◽  
Quantum Mechanical Systems ◽  
Noncompact Symmetric Space

We show how to construct path integrals for quantum-mechanical systems where the space of configurations is a general noncompact symmetric space. Associated with this path integral is a perturbation theory which respects the global structure of the system. This perturbation expansion is evaluated for a simple example and leads to a new exactly soluble model. This work is a step towards the construction of a strong coupling perturbation theory for quantum gravity.

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