Equal rank local theta correspondence as a strong Morita equivalence. (English) Zbl 07896245
Summary: Let \((G, H)\) be one of the equal rank reductive dual pairs \((Mp_{2n}, O_{2n+1})\) or \((U_n, U_n)\) over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say \(\widehat{G}_\theta\) and \(\widehat{H}_\theta\), of the tempered duals of \(G\) and \(H\). We prove that this bijection arises from an equivalence between the categories of representations of two \(C^\ast\)-algebras whose spectra are \(\widehat{G}_\theta\) and \(\widehat{H}_\theta\). This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees.
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
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