Let \(F\) be a non-archimedean local field of characteristic 0 and residual field of characteristic \(\neq 2\). Let \(V_1\) be a vector space over \(F\) with \(\dim_F V_1=2\). If \(E\) is a quadratic extension, \(E=F(\sqrt{\xi})\), \(a= x+y\sqrt{\xi}\), \(\overline{a}= x-y\sqrt{\xi}\) put, for \(A\in M_2(E)\), \(q(A)= -\det A=- xy+z^2+ t\xi^2\). Denote \(V_2= \{A\in M_2(E)\), \({}^t\overline{A}=A\}\), thus \(V_2=\left\{\left( \begin{smallmatrix} y &\overline{b}\\ b &-x\end{smallmatrix} \right),\;x,y\text{ in }F,\;b\text{ in }E\right\}\), \(\dim_F V_2=4\) \(V_0=\left\{\left( \begin{smallmatrix} 0 &z-t\sqrt{\xi}\\ z+t\sqrt{\xi} &0\end{smallmatrix} \right),\;z,t\text{ in }F\right\}\) and \(H =\left\{\left( \begin{smallmatrix} y &0\\ 0&-x\end{smallmatrix} \right),\;x,y\text{ in }F\right\}\).
The dimensions of \(V_2\), \(V_0\) and \(H\) are even. Let \(H_1= O(V_2)\), \(H_0= O(V_0)\) and \(J= O(H)\) be the corresponding orthogonal groups. Let \(\omega_\chi^\infty\) be the oscillator representation of \(\text{SL}_2(F)\) corresponding to the additive character \(\chi\) of \(F\). The author studies the theta correspondences for the \((\text{SL}_2(F), O(V_2))\), \((\text{SL}_2(F), O(V_0))\) and \((\text{SL}_2(F), O(H))\) \((\dim_F V_2=4\), \(\dim_F V_0=2\), \(\dim_F H=2)\).
In Section 1, Theorem 1.1 the fundamental results of Labesse-Langlands are stated, and in Lemma 2 the results of Casselman on the theta correspondence attached to \(\chi\) and the pairs \((\text{SL}_2(F), O(V_2))\), \((\text{SL}_2(F), O(V_0))\) and \((\text{SL}_2(F), O(H))\) are recalled. Lemma 1.4 states the results of Shalika-Tanaka on the theta correspondence attached to \(\chi\) and \((\text{SL}_2(F), O(H))\). Using Prasad’s result on the existence of a unique invariant trilinear form on \(\pi_1\times \pi_2\times \pi_3\), where \(\pi_1, \pi_2, \pi_3\) are infinite-dimensional irreducible smooth representations of \(\text{GL}_2(F)\) the author obtains the main result of this paper.
Theorem 2.3: If \(\pi\) is an irreducible smooth representation of \(\text{SL}_2(F)\), whose \(L\)-packet is not that of a Weil representation, then \(\pi\) occurs in the theta correspondence.
There exists the base change correspondence between the set of irreducible admissible representations of \(\text{GL}_2(F)\) and a set of irreducible admissible representations of \(\text{GL}_2(E)\), \(E\) quadratic extension of \(F\) [Langlands]. There is a theta correspondence between the representations of \(\text{GL}_2(F)\) and the representations of \(\text{GL}_2(E)\) when we decompose the Weil representation [Howe] [M. Cognet, Bull. Soc. Math. Fr. 113, 403-457 (1985; Zbl 0609.12016)].