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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