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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)
MSC:
 22E46 Semisimple Lie groups and their representations 43A80 Analysis on other specific Lie groups
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