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Properties of the Cauchy Harish-Chandra integral for some dual pairs of Lie algebras. (Propriétés de l’intégrale de Cauchy Harish-Chandra pour certaines paires duales d’algèbres de Lie). (French) Zbl 1050.22017
The author considers a symplectic group \(Sp=Sp(W)\) and an irreducible dual pair \((G,G')\) in \(Sp\) in the sense of Howe. Let \(\mathfrak g\) (resp. \(\mathfrak g'\)) be the Lie algebra of \(G\) (resp. of \(G'\)). T. Przebinda has defined a map \( \mathbf{Chc}\), called the Cauchy Harish-Chandra integral, from the space of smooth compactly supported functions \(\mathcal D(\mathfrak g)\) on \(\mathfrak g\) to the functions defined on the set \({\mathfrak g'}^{\text{reg}}\) of semi-simple regular elements of \(\mathfrak g'\): for a subgroup of the Cartan subgroup \(H'\) of \(G'\) denote its compact part by \(T'\) and by \(A'\) its elements with positive eigenvalues. Let \(A''\) be the centralizer of \(A'\) in \(Sp\) and \(A'''\) the centralizer of \(A''\) in \(Sp\). There exists an open dense subset \(W_{A'''}\) of \(W\), such that \(A'''\setminus W_{A'''}\) is a variety. Consider a measure \(d\dot w\) on \(A'''\setminus W_{A'''}\). For a subspace \(u\) of \(\mathfrak s\mathfrak p(W)\), consider the moment map \(\tau_u: W\to u^*\) and, for \(x\in \mathbb R\), let \(\chi(x)=e^{2\pi i x}\) and \(\chi_x(w)=\chi({1\over 4 }\tau_{\mathfrak s\mathfrak p(W)}(w)(x)),w\in W\).
For \(\psi\in{\mathcal D}(\mathfrak a'')\), Przebinda has shown that \(\int_{A'''\setminus W_{A'''}} \left|\int_{\mathfrak a''}\psi(x)\chi_x(w)dx\right| d\dot w <\infty.\) We obtain a distribution on \(\mathfrak a''\): \({Chc}(\psi)=\int_{A'''\setminus W_{A'''}} \int_{\mathfrak a''}\psi(x)\chi_x(w)dx d\dot w.\) For a semi-simple regular element \(x'\in \mathfrak h'\), the injection \(i_{x'}: \mathfrak g \to \mathfrak a'', x\mapsto x+x',\) is transverse to the wave front set WF\((Chc)\) of \(Chc.\) This allows us to define the inverse image of \(Chc\) by the injection \(i_{x'}\), which we denote by \(Chc_{x'}\). For \(\phi\in \mathcal D (\mathfrak g)\), we let \(\mathbf{Ch}(\phi)\) be the function defined on \({\mathfrak g'}^{\text{reg}}\) by \(Chc(\phi)(x')=Chc_{x'}(\phi)\). In the case where \((G,G')\) is a dual pair with stable range, for every nilpotent orbit \(\mathcal{O'}\) of \(\mathfrak g'\), there exists a unique nilpotent orbit \(\mathcal O\) of \(\mathfrak g'\) which is dense in \(\tau_{\mathfrak g}\circ \tau_{\mathfrak g'}(\mathcal O)\). Denote by \(\mu(\mathcal O)\) and \(\mu(\mathcal O')\) the invariant measures on \(\mathcal O\) and \(\mathcal O'\). Then up to a multiplicative constant, on has that \(\hat \mu_{\mathcal O}(\phi)=\Sigma {1 \over{W(H')}}\int_{\mathfrak h'}\hat \mu_{\mathcal O'}(x')D_{\mathfrak g'}(x')^2 {\mathbf{Chc}}(\phi)(x')dx'\) for \(\phi\in\mathcal D(\mathfrak g)\), where the sum is made over a system of representatives of Cartan subalgebras of \(\mathfrak g\) and \(D_{\mathfrak g'}(x')=| \text{ det}(ad(x')_{\mathfrak g'/\mathfrak h'})|^{1/2}\) for \(x'\in {\mathfrak h'}^{\text{reg}}\).
In this article it is shown that for irreducible dual pairs of type I made of unitary groups of the same rank, the function \({\mathbf{Chc}}(\phi), \phi\in\mathcal D(\mathfrak g)\), which is defined on \({\mathfrak h'}^{\text{reg}}\), has the local properties of orbit integrals of \(\mathfrak g'\). This means that \({\mathbf{Chc}}(\phi)\) is a smooth \(G'\)-invariant function on \({\mathfrak h'}^{\text{ reg}}\). The author presents a family of Cartan subalgebras \((\mathfrak h_N)_{N\in \mathcal O'}\) of \(\mathfrak g'\) indexed by a set \(\mathcal O'\) which represents the set of all admissible root systems of a system of positive roots of a compact Cartan subalgebra of \(\mathfrak g'\). For every \(N\in \mathcal O'\), a function \({\mathbf{Chc}}(\phi)_N\) is constructed on \({\mathfrak h_N'}^{\text{ reg}}\). It is shown then that all the derivatives of \({\mathbf{Chc}}(\phi)_N\) are locally bounded and can be extended continuously on the closure of the connected component of the set of all the elements of \(\mathfrak h'_N\) which are not zero on any imaginary noncompact root of \(\mathfrak h'_N\). For an imaginary noncompact root, we denote by \(\langle \partial(w){\mathbf{Chc}}(\phi)_N\rangle\) the jump function which is defined by the preceding remarks. For two successive Cartan subalgebras \(\mathfrak h_N',\mathfrak h_M' \) (with \(N,M\in \mathcal O')\) for the order of Hiraï it is shown that there exists a constant \(C\) such that for every element \(w\) in the symmetric algebra of \(\mathfrak h_N'\), we have that \(\langle \partial(w){\mathbf{Chc}}(\phi)_N\rangle\)=i C\(\langle \partial(w){\mathbf{Chc}}(\phi)_M\rangle\).
Reviewer: Jean Ludwig (Metz)
22E46 Semisimple Lie groups and their representations
43A80 Analysis on other specific Lie groups
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