# Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension

Hélène Airault; Habib Ouerdiane

Banach Center Publications (2011)

- Volume: 96, Issue: 1, page 9-34
- ISSN: 0137-6934

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topHélène Airault, and Habib Ouerdiane. "Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension." Banach Center Publications 96.1 (2011): 9-34. <http://eudml.org/doc/281605>.

@article{HélèneAirault2011,

abstract = {
Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ and in L²(μ), the measure μ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on 𝓓. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator (differential operator associated to the representation) in the following elementary cases:
A) The commutative groups (ℝ,+) and (ℝ* = ℝ-0,×).
B) The multiplicative group M of 2×2 complex invertible matrices and some subgroups of M.
C) The three-dimensional Heisenberg group.
},

author = {Hélène Airault, Habib Ouerdiane},

journal = {Banach Center Publications},

keywords = {Lie group; invariant measure; Heisenberg group; Ornstein-Uhlenbeck operator},

language = {eng},

number = {1},

pages = {9-34},

title = {Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension},

url = {http://eudml.org/doc/281605},

volume = {96},

year = {2011},

}

TY - JOUR

AU - Hélène Airault

AU - Habib Ouerdiane

TI - Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension

JO - Banach Center Publications

PY - 2011

VL - 96

IS - 1

SP - 9

EP - 34

AB -
Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ and in L²(μ), the measure μ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on 𝓓. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator (differential operator associated to the representation) in the following elementary cases:
A) The commutative groups (ℝ,+) and (ℝ* = ℝ-0,×).
B) The multiplicative group M of 2×2 complex invertible matrices and some subgroups of M.
C) The three-dimensional Heisenberg group.

LA - eng

KW - Lie group; invariant measure; Heisenberg group; Ornstein-Uhlenbeck operator

UR - http://eudml.org/doc/281605

ER -

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