L<sup>2</sup>spectral decomposition on the Heisenberg group associated to the action of U(p,q)

Godoy, T.; Saal, Linda

doi:10.2140/pjm.2000.193.327

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…By another way, it is easy to see that this subalgebra is generated by L and U where L = and U is as above (cf. [7]). The description of the unitary dual of U(p, q)H n is given in [15].…”

Section: Z T) = (Gz T) So U(p Q) Acts On L 2 (H N ) S (H N ) Andmentioning

confidence: 99%

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

Godoy¹,

Saal²

2009

MATH. SCAND.

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“…By another way, it is easy to see that this subalgebra is generated by L and U where L = and U is as above (cf. [7]). The description of the unitary dual of U(p, q)H n is given in [15].…”

Section: Z T) = (Gz T) So U(p Q) Acts On L 2 (H N ) S (H N ) Andmentioning

confidence: 99%

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

Godoy¹,

Saal²

2009

MATH. SCAND.

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“…As pointed out in [5], this construction also works to describe the space S ′ (C n ) U (p,q) , i.e., there exists a linear, continuous and surjective map, still denoted by N :…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…[5]). Let us recall some facts concerning the compact case p = n, q = 0, i.e., when U (p, q) = U (n).…”

mentioning

confidence: 99%

A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)

Godoy

Saal

2006

Colloq. Math.

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Abstract. Let S(H n ) be the space of Schwartz functions on the Heisenberg group H n . We define a spherical transform on S(H n ) associated to the action (by automorphisms) of U (p, q) on H n , p + q = n. We determine its kernel and image and obtain an inversion formula analogous to the Godement-Plancherel formula.1. Introduction. Let n ≥ 2 and let p, q be natural numbers such that p + q = n. Let H n be the Heisenberg group defined bywhereFor x = (x 1 , . . . , x n ) ∈ R n , we write x = (x ′ , x ′′ ) with x ′ ∈ R p , x ′′ ∈ R q . So, R 2n can be identified with C n via the mapIn this setting, the form − Im B(z, w) agrees with the standard symplectic form on R 2(p+q) , and the vector fieldsform a standard basis for the Lie algebra h n of H n . Thus H n can be viewed as R n × R n × R via the map (x, y, t) → (ϕ(x, y), t). From now on, we will use freely this identification. Let S(H n ) be the Schwartz space on H n and let S ′ (H n ) be the space of corresponding tempered distributions. Consider the action of U (p, q) on H n given by g · (z, t) = (gz, t) (note that since we have assumed that p, q ≥ 1, U (p, q) is noncompact). So U (p, q) acts on L 2 (H n ), S(H n ) and S ′ (H n ) in 2000 Mathematics Subject Classification: Primary 43A80; Secondary 22E25.

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“…The pair (U (p, q), H n ) is a generalized Gelfand pair and the corresponding harmonic analysis has been developed in [1], [2], [4].…”

Section: Introductionmentioning

confidence: 99%

“…Up to a constant, S λ,k is the unique tempered solution of (1.1) (see [2]). In contrast, the solution space of (1.2) in S (H n ) is one-dimensional for ν = 0, and for ν = 0 it is two-dimensional, generated by S 0 and the trivial solution 1 (see [4]).…”

Section: Introductionmentioning

confidence: 99%

A Wiener type theorem for (U(p,q) ,H_n)

Saal

2010

Colloq. Math.

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L²spectral decomposition on the Heisenberg group associated to the action of U(p,q)

Cited by 7 publications

References 8 publications

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)

A Wiener type theorem for (U(p,q) ,H_n)

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L2spectral decomposition on the Heisenberg group associated to the action of U(p,q)

Cited by 7 publications

References 8 publications

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

On the spectrum of the generalized Gelfand pair $(u(p,q), h_{n})$, $p+q=n$

A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)

A Wiener type theorem for (U(p,q) ,Hn)

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L²spectral decomposition on the Heisenberg group associated to the action of U(p,q)

A Wiener type theorem for (U(p,q) ,H_n)