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The Capelli identity and unitary representations. (English) Zbl 0758.22008
Let $$\Omega=G/K$$ be an irreducible Hermitian symmetric space of tube type of rank $$n$$, $$G/P$$ its Shilov boundary where $$P=LN$$ is a maximal parabolic subgroup of $$G$$. Let $$\varphi$$ be the Jordan norm on the Lie algebra $$\mathfrak n$$ of $$N$$. If $$\overline P$$ is the opposite parabolic and $$t\in \mathbb{R}$$, consider the induced representation $$I(G)=Ind^ G_{\overline P}(\nu^ t\oplus 1)$$, where $$\nu^{-2}$$ is the positive character of $$L$$ associated with $$\varphi$$. Using a canonical isomorphism from $$\mathfrak n$$ to $$\mathfrak n^*$$ one can associate to $$\varphi$$ a differential operator $$D$$. On $$I(m)$$, $$m\in\mathbb{Z}$$, one can define the invariant Hermitian form $$(f_ 1,f_ 2)_ m=\langle D^ mf_ 1,f_ 2\rangle$$, where $$\langle , \rangle$$ is the Hermitian pairing between $$I(m)$$ and $$I(-m)$$. The author finds an explicit formula for the signature of this form for each $$K$$-type. He also obtains information on $$I(m)/\text{Ker }D^ m$$.

##### MSC:
 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 43A85 Harmonic analysis on homogeneous spaces 22D30 Induced representations for locally compact groups 11E39 Bilinear and Hermitian forms
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