Lectures on Gaussian Integral Operators and Classical Groups

A q-analogue of the weighted Bergman space on the disk and associated second q-Bargmann transform

“…The first one is the well-known Segal-Bargmann transform that made the quantum mechanical configuration space L 2 (R; dx) unitary isomorphic to the phase space (Bargmann-Fock space) consisting of all e −|z| 2 dxdy-square integrable holomorphic functions on the complex plane (for details see [4,7,21,18] and the references therein). He also considered the generating function of the classical Laguerre polynomials L (α) n as kernel function and pointed out that the corresponding integral transform gives a concrete isometric realization of the Hilbert space L 2,α (R + ) = L 2 R + ;…”

Self-adjoint elements in the pseudo-unitary group U(p,p)

“…U (p, q) also contains certain subgroups that are of particular interest to researchers in modern math and physics. One of them is the maximal compact subgroup U (p) ⊕ U (q) ( [11]). As we shall see, this subgroup is precisely the set of unitary elements inside the pseudo-unitary group.…”

“…This paper aims to describe the set U s (p, q) of self-adjoint (Hermitian) elements in U (p, q) for the case p = q. By the Cartan decomposition theorem ( [11], Theorem 3.4), every matrix M ∈ U (p, q) is of the form M = U S, where U ∈ U (p) ⊕ U (q) and S is self-adjoint. It is easy to check that S ∈ U s (p, q), and hence there is the following factorization:…”

The Fourier Transform on the Group $$\mathrm {GL}_2({\mathbb R })$$ GL 2 ( R ) and the Action of the

“…Notice that canonical overgroups exist for all 10 series of real classical groups. 3 Moreover overgroups exist for all 52 series of classical semisimple symmetric spaces G/H , see [8,15], see also [18], Addendum D.6. So the problem makes sense for all classical symmetric spaces.…”