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A local trace formula for the generalized Shalika model. (English) Zbl 1426.22011

Let \(F\) be a \(p-\)adic field, let \(A=\operatorname{Mat}_n(D)\) be a central simple algebra where \(D/F\) is a division algebra. Let \(G=\mathrm{GL}_2(A)\) and \(H\) be its subgroups consisting of elements \(\begin{pmatrix}a&b\\&a\end{pmatrix}\) with \(a\in A^*\) and \(b\in A\). Let \(\omega\) be a character of \(H\). Given an irreducible admissible representation \(\pi\) of \(G\), a general Shalika model of \(\pi\) is an embedding of \(\pi\) in \(\operatorname{Ind}_H^G \omega\).
The paper proves a formula in the case of essentially square integrable representations \(\pi\) for the multiplicity \(m(\pi,\omega)\) of the Shalika model, which is the dimension of \(\operatorname{Hom}_H(\pi,\omega)\). The multiplicity formula is \[ m(\pi,\omega)=\sum_T |\operatorname{Norm}_{H_0}(T)/T|^{-1}\int_{Z_G\backslash T}D^H(t)c_\pi(t)\omega^{-1}(t)\,dt. \] Here \(H_0\subset H\) consisting of \(\begin{pmatrix}a&\\&a\end{pmatrix}\), \(T\) runs through the elliptic maximal tori in \(H_0\), \(D^H\) is the Weyl discriminant and \(c_\pi\) is a character of \(\pi\).
As an immediate consequence of the multiplicity formula, it is shown that for discrete series representations, the multiplicity \(m(\pi,\omega)\) is preserved under the Jacquet-Langlands correspondence.
The multiplicity formula is similar to what Waldspurger established for the case of Gross-Prasad models, as well as those in prior works of Beuzart-Plessis. The proof is based on the local trace formula.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:

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