%&amsppt \input amstex \documentstyle{amsppt}\magnification=1200 %\nologo \NoBlackBoxes \magnification\magstep1 \pagewidth{32.622pc} \pageheight{44.8pc} \define\oz{\overline{\omega}} \define\Oz{\overline{\Omega}} \define\om{\overline{\omega\omega}} \define\Om{\overline{\Omega\Omega}} \define\ind{\operatorname{ind}} \define\Ind{\operatorname{Ind}} \define\pro{\operatorname{pro}} \define\ca{\Cal C_{c}^{\infty}} \topmatter \title Exceptional $\Theta$-correspondences I \endtitle \author Kay Magaard and Gordan Savin \endauthor \address Department of Mathematics, Wayne State University, Detroit, MI 48202 \endaddress \email kaym\char'100math.wayne.edu \endemail \address Department of Mathematics, University of Utah, Salt Lake City, UT 84112\endaddress \email savin\char'100math.utah.edu\endemail \subjclass Primary 22E35 and 50, Secondary 11F70 \endsubjclass \thanks Research at MSRI is supported in part by NSF grant DMS-9022140\endthanks \abstract Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of $E_7$ to the dual pair $G_2 \times Sp(6)$ gives the non-endoscopic Langlands lift of irreducible representations of $G_2$ to $Sp(6)$. \endabstract \endtopmatter \document \head Introduction \endhead Let $F$ be a $p$-adic field and $G$ be a split simple group of type $D_n$ or $E_n$, ($n=6,7,8$) over $F$. If the type is $D_n$ then $G$ is $SO(2n)$. In \cite{KS} the minimal representation $V$ of $G$ was introduced and in \cite{S} some further properties of it were studied. It is an analogue of the Weil representation of $Sp(2n)$ and in some ways it is better behaved then the Weil representation. Therefore it makes sense to ask if one can use it to obtain dual pair correspondences. We have two types of dual pairs in $G$. In $SO(2n+2)$ we have a dual pair $$SO(2n-1)\times SO(3).$$ If $G$ is exceptional then we have a dual pair $$ G_2(F)\times H_D $$ with $H_D$ of type $A_2$, $C_3$ or $F_4$, respectively. Let $A\times B$ be a dual pair in $G$ and $X$ an irreducible representation of $A$. We say that an irreducible representation $Y$ of $B$ is a $\Theta$-lift of $X$ if $X\otimes Y$ is a quotient of $V$ (see \cite{H1}). Let $\Theta(X)$ be the set of all such $Y$. In this paper we compute Jacquet functors $V_U$ of $V$ where $U$ is a unipotent radical of a maximal parabolic subgroup of $SO(3)$ or $H_D$, with $U$ commutative, or a unipotent radical of a maximal parabolic subgroup of $G_2$ such that $U$ is a Heisenberg group. The results are reminiscent of those of Kudla \cite{Ku} and Rallis \cite{Ra} in the classical case. We then use the knowledge of Jacquet functors to obtain dual pair correspondences. For example, we compute $\Theta$-lifts of tempered spherical representations of $SO(3)$, $A_2$, $C_3$ and $G_2$. The lifts from $A_2$ to $G_2$ and from $G_2$ to $C_3$ obtained by restricting the minimal representation of $E_6$ and $E_7$ respectively, are Langlands correspondences. In a sequel to this paper we plan to compute Jacquet functors $V_U$ for unipotent radicals of remaining maximal parabolic subgroups. Once this is done, it will be possible to compute lifts of most Langlands quotients and check the conjectural answer for the exceptional correspondences as given by B. Gross \cite{G2}. Local computations are, in a way, a preparation for global correspondences (i.e. correspondences of automorphic forms). So it is worth mentioning that in a forthcoming work, D. Ginzburg, S. Rallis and D. Soudry are studying a global variant of exceptional correspondences. Another possible global application has recently been initiated by B. Gross in connection to a realization of a $G_2$-motive \cite{G1}. The paper is organized as follows. In Section 1 we recall the definition and few properties of the minimal representation from \cite{S}. In Section 2 we study the dual pair $SO(3)\times SO(2n-1)$. It is the simpliest case and as such it is a good introduction to exceptional dual pairs which form a more interesting part of this work. In Sections 3 and 4 we study the dual pair $G_2\times A_2$. We compute $\Theta$-lifts of spherical tempered representations of $A_2$. In Section 5 we study the dual pair $G_2\times C_3$. We compute $\Theta$-lifts of spherical tempered representations of $C_3$. In Section 6 we recall some facts about simple Jordan algebras or rank $3$. Exceptional groups can be described nicely in terms of Jordan algebras. We use this description in Section 7 to compute $V_U$ where $U$ is the unipotent radical of the "Heisenberg" maximal parabolic subgroup of $G_2$ in all three cases. We finish the paper by computing $\Theta$-lifts of spherical tempered representations of $G_2$ in all three cases. \head 1. Minimal representation \endhead Let $G$ be a simple, split, group of type $A_{2n-1}$, $D_n$ or $E_n$. Let $^LG$ be the complex $L$-group of $G$ (see \cite{B}). It is well known that conjugacy classes of unipotent elements in $^LG$ correspond to conjugacy classes of homomorphisms $$ \varphi : SL_2(\bold C) \rightarrow\, ^LG. $$ Assume now that $\varphi$ corresponds to the subregular unipotent orbit. Let $q$ be the order of the residual field of $F$. Let $$ s = \varphi \left( \smallmatrix q^{\frac12} & 0 \\ 0 & q^{-\frac12} \endmatrix \right). $$ Then $V$ is the spherical representation of $G$ with the Satake parameter $s$ (see \cite{Ca}). Let $\frak g$ be the Lie algebra of $G$. Let $<\cdot , \cdot >$ be the Killing form on $\frak g$. Fix $\psi : F \to \Bbb C^{\times}$, a non-trivial character. Let $\ca(\frak g)$ denote the space of locally constant, compactly supported functions on $\frak g$. Define the Fourier transform on $\ca(\frak g)$ by $$ \hat f(y)=\int_{\frak g} f(x) \psi() dx . $$ Let $\Cal O_{\min}$ be the unique minimal nilpotent $G$-orbit in $\frak g$ and $\mu_{\Cal O_{\min}}$ a $G$-invariant measure on $\Cal O_{\min}$ normalized as in \cite{MW}. Let $\frak g_{\Bbb Z}$ be the $\Bbb Z$-span of a Chevaley basis of $\frak g$. Let $\Cal R$ be the ring of integers of $F$ and $\varpi$ the uniformizing element. Let $$ \frak g _n = \frak g_{\Bbb Z}\otimes_{\Bbb Z} \varpi^n \Cal R. $$ In \cite{S} we have shown that there exists $m>0$ such that $$ tr\int_{\frak g}f(x)\pi(\exp x) dx = \int \hat f \mu_{\Cal O_{\min}} $$ for any $f \in \ca(\frak g)$ supported on $\frak g_m$ and such that $\hat f(0)=0$. This implies "smallness" of the $N$-spectrum of $V$, where $N$ is a unipotent radical of a maximal parabolic subgroup of $G$. More precisely, let $\Delta$ be the Dynkin diagram of $G$. Mark the diagram $\Delta$ as follows. Attach $0$ to the unique branch vertex (or to the middle vertex of $\Delta$ if the type of $G$ is $A_{2n-1}$) and $2$ to all other. This marking corresponds to the subregular nilpotent orbit \cite{D}. Let $P=MN$ be a maximal parabolic subgroup of $G$. Let $\Delta_M$ be the Dynkin diagram of $M$. Assume that we are in the following favorable situation: \roster \item $N$ is a commutative group. \item The marking of $\Delta$ corresponding to the subregular nilpotent orbit of $G$ restricts to the marking of $\Delta_M$ corresponding to the subregular nilpotent orbit of $M$. \endroster The possible cases are given by the following table: $$ \matrix G & M & N \\ D_{n+1} & D_{n} & F^{2n} \\ E_6 & D_5 & F^{16} \\ E_7 & E_6 & F^{27} \endmatrix $$ Here $F^{2n}$ is the standard representation of $SO(2n)$, $F^{16}$ is a spin-representation of $Spin_{10}$ and $F^{27}$ is isomorphic to the exceptional Jordan algebra. We say that a point in $N$ is singular if it is a highest weight vector for a Borel subgroup of $M$. Let $\omega$ and $\overline{\omega}$ be the sets of singular vectors in $N$ and $\overline{N}$. Note that $\omega$ is the smallest non-trivial $M$-orbit in $N$. If $G=SO(2n)$ then $\omega$ is the null-cone of the invariant quadratic form (with $0$ excluded). \proclaim{Theorem 1.1} The minimal representation $V$ of $G$ has a $P$-invariant filtration $$ 0 \to {\Cal C}_{c}^{\infty}(\oz) \to V \to V_N \to 0. $$ Here $\Cal C_{c}^{\infty}(\oz)$ denotes the space of locally constant, compactly supported functions on $\oz$ and $V_N$ is the space of $N$-coinvariants of $V$ (Jacquet functor). \roster \item Let $f\in \Cal C_{c}^{\infty}(\oz)$. The action of $P$ is given by $$\ \pi(n)f(x) = \psi()f(x), \qquad n\in N $$ and $$ \pi(m)f(x) = |\det(m)|^{\frac{s}{d}}f(m^{-1}x), \qquad m\in M. $$ \item $$ V_N \cong V_M \otimes |\det|^{\frac{t}{d}} + |\det|^{\frac{s}{d}} $$ where $V_M$ is the minimal representation of $M$ (center acting trivially). \endroster In the above formulas $<\cdot , \cdot >$ is the $F$-valued pairing between $N$ and $\overline N$ induced by the Killing form on $\frak g$, $\det$ is determinant of the representation of $M$ on $N$, $d$ is the dimension of $N$ and the values of $s$ and $t$ are given in the following table. $$ \matrix G & s & t \\ D_{n+1} & n-1 & 1 \\ E_6 & 4 & 2 \\ E_7 & 6 & 3 \endmatrix $$ \endproclaim \demo{Proof} This is just Theorem 6.5 in \cite{S} if $G$ is $E_7$. The other two cases also satisfy conditions of Proposition 4.1 in \cite{S}. Hence the proof carries over with no changes. The proof given in \cite{S}, however, is valid only when the residual characterisitic is odd. It remains to discuss the case when the residual characteristic is $2$. Let $x\in \overline N$, and define $\psi_x(n)=\psi()$. Let $V_{N,\psi_x}$ be the quotient of $V$ by the space spanned by the elements $\{\pi(n)v-\psi_x(n)v | n\in N, v \in V\}$. The key point in the proof of Theorem 6.5 in \cite{S} is to show that $$ V_{N,\psi_x}=0 $$ for $x\neq 0$ and not in $\overline \omega$, i.e. the $N$-spectrum of $V$ is concentrated on the closure of the smallest $M$-orbit in $\overline N$. This follows from the work of Moeglin and Waldspurger \cite{MW}, but only if the residual characteristic of $F$ is odd. Let $k$ be a number field and $\Bbb A$ its ring of adeles. Ginzburg, Rallis and Soudry \cite{GR} have constructed a square integrable automorphic form $\Pi=\otimes_{v}\Pi_{v}$ on $G_{\Bbb A}$ isomorphic to $V$ at every finite place. But arguing exactly as Howe (\cite{H2} Lemma 2.4), one shows that if the $N$-spectrum is concentrated on the closure of $\overline \omega$ at one place $v$, it has to be concentrated on the closure of $\overline \omega$ at all places. This completes the proof of the theorem. \enddemo \head 2. Dual pair $SO(2n-1) \times SO(3)$ \endhead Let $G=SO(2n+2)$. Let $e,h,f$ be the standard basis for $sl(2)$, the Lie algebra of $SO(3)=PGL_2(F)$. Let $P=MN$ be the maximal parabolic of $G$ as in Section 1. Let $\frak g$ be the Lie algebra of $G$ and let $$ \frak g = \overline{\frak n} + \frak m + \frak n $$ be the corresponding decomposition of $\frak g$. The embedding $SO(3) \subseteq SO(2n+2)$ corresponds to $sl(2)\subseteq \frak g$ such that \roster \item $\overline{\frak n}=\{ x\in \frak g~ |~ [h,x]=-2x\}. $ \item $\frak m=\{ x\in \frak g~ |~ [h,x]=0\}. $ \item ${\frak n}=\{ x\in \frak g~ |~ [h,x]=2x\}. $ \endroster Note that $e\in \frak n$, $f \in \overline{\frak n}$ and $SO(2n-1)$ is the centralizer of $e$ in $M$. Let $Q=LU =P\cap SO(3)$ be a Borel subgroup of $SO(3)$. \proclaim{Proposition 2.1} Let $\pmatrix a & \\ & b \endpmatrix \in L \subset PGL_2(F)=SO(3)$. \roster \item $$ \pi(\pmatrix a & \\ & b \endpmatrix)f(x) = |\frac{a}{b}|^{n-1} f(\frac{a}{b} x), \qquad f\in \ca (\oz). $$ \item The eigenvalues of $\pi(\pmatrix a & \\ & b \endpmatrix)$ on $V_N$ are $|\frac{a}{b}|$ and $|\frac{a}{b}|^{n-1}$. \endroster \endproclaim \demo{Proof} This is just Theorem 1.1. \enddemo Let $\chi$ be a multiplicative character of $F$. Let $\rho\chi$ denote the character of $L$ defined by $$ \rho\chi(\pmatrix a & \\ & b \endpmatrix) = |\frac{a}{b}|^{\frac{1}{2}}\chi(\frac{a}{b}). $$ Let $\tau_{\chi}= \Ind_Q^{SO(3)}\rho\chi$. If $\chi$ is unitary than $\tau_{\chi}$ is an irreducible tempered spherical representation of $SO(3)$. Let $\sigma \in \Theta(\tau_{\chi})$. By Frobenius reciprocity $$ Hom_{SO(2n-1)\times SO(3)}(V, \sigma\otimes \tau_{\chi})= Hom_{SO(2n-1) \times L}(V_U, \sigma\otimes \rho\chi). $$ Therefore, $\sigma\in \Theta(\tau_{\chi})$ is a quotient of $V$ if and only if $\sigma\otimes \rho\chi$ is a quotient of $V_U$. Since $$ 0 \to {\Cal C}_{c}^{\infty}(\oz)_U \to V_U \to V_N \to 0 $$ we need to understand $\ca(\oz)_U$. Let $NN$ and $\overline{NN}$ be the complements of $\overline U$ and $U$ in $N$ and $\overline N$. Let $\overline \omega$, as before, be the set of singular vectors in $\overline N$. Put $$ \om =\overline{\omega} \cap \overline{NN}. $$ \proclaim{Lemma 2.2} \endproclaim $$ \ca(\overline\omega)_U=\ca(\om). $$ \demo{Proof} Let us recall few known facts about Jacquet functors. Let $E$ be a $U$-module. Then $E_U= E / E(U)$ where $E(U)$ can be defined either as the space spanned by the elements $\{\pi(u)v - v | u\in U, v\in E\}$ or the space of all $v$ such that $$ \int_{U_c}\pi(u)v du = 0 $$ for some open compact subgroup $U_c \subset U$ depending on $v$ (2.33 \cite{BZ}). Obviously, $\ca(\om)$ is a quotient of $\ca(\overline\omega)$ and by Theorem 1.1 (1), $U$ acts trivially on $\ca(\om)$. Let $f\in \ca(\overline\omega)$ such that $f|_{\om}=0$. To prove the lemma, we need to find an open compact subgroup $U_c$ such that $$ \int_{U_c}\psi()f(x)du=0 $$ for all $x\in \overline\omega$. Let $x$ be such that $f(x)\neq 0$. Since $x$ is not in $\overline{NN}$, there exists an open compact neighborhood $\Cal O$ of $x$ and an open compact subgroup $U_{\Cal O}$ such that $\psi()$ is a non-trivial character of $U_{\Cal O}$ for any $y\in \Cal O$. Since the support of $f$ is compact, a finite collection of $\Cal O$ covers the support of $f$. The union of the corresponding $U_{\Cal O}$ is the desired $U_c$. The lemma follows. \enddemo We can, therefore, summarize the situation with the following proposition. \proclaim{Proposition 2.3} $V_U$ has a filtration with two quotients: $$ \ca(\om), \, \text { and } V_N. $$ As $SO(2n-1)\times L$-modules: \roster \item the action on $\ca(\om)$ is geometric, twisted with $ |\frac{a}{b}|^{n-1} $ for $ \pmatrix a & \\ & b \endpmatrix \in L$. \item $$V_N\cong V_M \otimes |\frac{a}{b}| + 1\otimes |\frac{a}{b}|^{n-1} $$ where $V_M$ is the minimal representation of $M$ (center acting trivially). \endroster \endproclaim Note that $\overline{NN} = F^{2n-1}$ and $\om$ is the null-cone of the $SO(2n-1)$-invariant quadratic form (with $0$ excluded). Let $\Cal C^{\infty}(\om)$ be the space of locally constant functions on $\om$. We can define degenerate principal series representations $\sigma_{\chi}$ by $$ \sigma_{\chi}=\{ f\in \Cal C^{\infty}(\om) \mid f(cx)=\chi(c) |c|^{\frac{3}{2}-n}f(x)\}. $$ Analogously, $\sigma_{\chi}$ can be defined as a quotient of $\ca(\om)$ consisting of $\overline f$ such that $$ \overline f(cx)=\chi(c) |c|^{\frac{3}{2}-n}\overline f(x). $$ If $\chi$ is unramified and unitary than $\sigma_{\chi}$ is an irreducible unitarizable spherical representation by a result of Tadi\'c \cite{T2}, Theorem 9.2. We are now ready to state and prove the main result of this section. \proclaim{Proposition 2.4} Let $\chi$ be an unramified, unitary multiplicative character. Then $$ \Theta(\tau_{\chi})= \{\sigma_{\chi}\}. $$ \endproclaim \demo{Proof} Let $\sigma \in \Theta(\tau_{\chi})$. Then $\sigma \times \rho\chi$ is a quotient of $V_U$. Since $\rho\chi$ is different from $|\cdot|^{n-1}$ and $|\cdot|$ if $\chi$ is unitary, it follows that $\sigma\times \rho \chi$ is quotient of $\ca(\om)$. Therefore $\sigma$ must be isomorphic to $\sigma_{\chi}$. To finish the theorem, we have to show that $\sigma_{\chi} \in \Theta(\tau_{\chi})$. By Frobenius reciprocity, it suffices to show that $\sigma_{\chi} \times \rho\chi$ is a quotient of $V_U$. We know that it is a quotient of $\ca(\om)$. We need the following lemma. \proclaim{Lemma 2.5} Let $H$ be a group and $$ 0\to V_1 \to V_2 \to V_3 \to 0 $$ a sequence of $H$-modules. Let $W$ be a $H$-invariant quotient of $V_1$. Assume that there exists a $H$-invariant linear map $T$ on $V_2$ reducing to scalar multiplications by $\lambda$ and $\mu$ on $W$ and $V_3$. If $\lambda \neq \mu$ then $W$ is a quotient of $V_3$. \endproclaim \demo{Proof} Let $V_0$ be such that $$ 0\to V_0 \to V_1 \to W \to 0. $$ Then $$ 0\to W \to V_2/V_0 \to V_3 \to 0 $$ Since the eigenvalues of $T$ are different, the last sequence splits and the lemma follows. \enddemo Let $\varpi$ be the uniformizing element of $F$. Put $$ T = \pi \pmatrix \varpi & \\ & 1 \endpmatrix. $$ If $\chi$ is unitary, $T$ satisfies the conditions of Lemma, hence $\sigma_{\chi}\times \rho\chi$ is a quotient of $V_U$ and the theorem is proved. \enddemo \head 3. Group $E_6$ \endhead Let $J$ be the 27-dimensional exceptional Jordan algebra and $(x,y,z)$ be the Dickson trilinear form on $J$ (see \cite{J2}). Let $G_S$ be the isometry group of the Dickson form. It is a split, simply-connected groups of type $E_6$ \cite{A1}. Let $\frak g_S$ be the Lie algebra of $G_S$. Then $$ \frak g_S = \bar \frak n + \frak m_S + \frak n $$ where $[\frak m_S, \frak m_S]= so(10)$ and $\bar \frak n$, $\frak n$ are two spin-representations. Let $P_S=M_S N$ be the corresponding maximal parabolic subgroup in $G_S$. Then $[M_S,M_S]=Spin_{10}$ and $Spin_{10}$ can be defined as the subgroup of $G_S$ stabilizing the $10$-dimensional subspace $J_{10}$ of $J$ consisting of matrices in $J$ such that the coefficients in the third row and the third column are zero and fixing the vector $$ d= \pmatrix 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \endpmatrix . $$ In particular, $Q_{d}(x,y)=(x,y,d)$ is the invariant quadratic form on $J_{10}$ and $N$ (one of the spin-modules) can be identified with the space in $J$ consisting of matrices $$ \pmatrix 0 & 0 & y \\ 0 & 0 & z \\ \overline{y} & \overline{z} & 0 \endpmatrix . $$ The group $G_S$ has a unique smallest, non-trivial orbit $\Omega$ on $J$. It is the orbit of $d$ and its dimension is $17$. On the other hand, $M_S$ has two non-trivial orbits on $N$ and the smaller (call it $\omega$) can be identified with $\Omega \cap N$ \cite{A1}. \head 4. Dual pair $G_2(F) \times PGL_3(F)$ \endhead Let $D$ be a nondegenerate subalgebra of the Octonions and let $J_D$ denote the subalgebra of $J$ obtained by restricting the off-diagonal entries of the matrices of $J$ to $D$. When $\dim(D)= 1$, i.e. $D = <1_{\Bbb O}> = F$ then $J_F$ is the algebra of symmetric $3\times 3$ matrices over $F$ and $C_{G_S}(J_F)= G_2(F)$, the split group of type $G_2$. The centralizer of $G_2(F)$ in $G_S$ is $SL_3(F)$ \cite{A1}. Let $G$, $G_S\subset G$, be a reductive group of type $E_6$ appearing as a Levi factor of a maximal parabolic subgroup of a split, simply-connected group of type $E_7$. Then $C_G(G_2(F))=GL_3(F)$. The action of $G_S$ on $J$ can be extended to $G$. Indeed, the inclusion $GL_3 \subset G$ induces an isomorphism $GL_3/[GL_3,GL_3]\cong G/[G,G]$. Since $[G,G]=G_S$, it suffices to extend the action to $GL_3 \subset SL_3$. This is done by the formula $$ gxg^t $$ where $x\in J$, $g\in GL_3(F)$ and $g^t$ is the transpose of $g$. Let $\frak g$ be the Lie algebra of $G$. Then, as in Section 3, $$ \frak g = \bar \frak n + \frak m + \frak n $$ with $\frak m_S \subset \frak m$. Let $P=MN$ be the corresponding maximal parabolic subgroup of $G$. Then $Q=LU=P\cap GL_3(F)$ is a maximal parabolic subgroup of $GL_3$ with $L=GL_2(F)\times GL_1(F)$. One can write $$ gl(3) = \bar \frak u + \frak l + \frak u $$ where $\frak l \subset \frak m$, $\overline{\frak u} \subset \overline{\frak n}$ and $\frak u \subset \frak n$. \proclaim{Proposition 4.1} Let $V$ be the minimal representation of $G$, the center acting trivially. Identify $\overline{N}$ with the set of pairs of Octonions $(y,z)$. Let $f\in \ca(\overline\omega)$. Then: \roster \item $$ \pi(g)f((y,z)) = f((g^{-1}y, g^{-1}z)), \qquad g\in G_2(F). $$ \item $$ \pi(l_2 \times l_1)f((y,z)) = \frac{|\det l_2|^2}{|l_1|^4} f(l_1^{-1}(y,z)l_2), \qquad l_2\times l_1 \in GL_2(F)\times GL_1(F) $$ \endroster where $\det$ denotes the usual determinant of $2\times 2$ matrices. \endproclaim \demo{Proof} This is just Theorem 1.1. \enddemo Let $Y$ be an irreducible representations of $GL_3(F)$ and $\tilde Y$ an irreducible representation of $G_2(F)$ such that $\tilde Y\in \Theta(Y)$. Assume that $Y$ is a submodule of $\Ind_{Q}^{GL_3}X$ for some irreducible representation $X$ of $L$. By Frobenius reciprocity $$ Hom_{G_2\times GL_3}(V, \tilde Y\otimes \Ind X)= Hom_{G_2 \times L}(V_U, \tilde Y\otimes X). $$ Therefore, $\tilde Y\otimes Y$ is a quotient of $V$ only if $\tilde Y\otimes X$ is a quotient of $V_U$. Since $$ 0 \to {\Cal C}_{c}^{\infty}(\oz)_U \to V_U \to V_N \to 0 $$ we need to understand $\ca(\oz)_U$. Let $NN$ and $\overline{NN}$ be the orthogonal complements of $\overline{U}$ and $U$ in $N$ and $\overline{N}$. Since $U$ can be identified with the space in $J$ consisting of matrices $$ \pmatrix 0 & 0 & y \\ 0 & 0 & z \\ y & z & 0 \endpmatrix $$ where $y,z \in F$, it follows that $$ NN = \{ \pmatrix 0 & 0 & y \\ 0 & 0 & z \\ \overline{y} & \overline{z} & 0 \endpmatrix |~ \overline{y} =-y \text { and } \overline{z} =-z \}. $$ Let $\om =\oz \cap \overline{NN}$. As in Lemma 2.2 $ \ca(\oz)_U = \ca(\om ), $ and we have to understand the structure of $G_2(F) \times L$ orbits on $\om$ which is, of course, the same as the structure of orbits on $\omega\omega=\omega \cap NN$. \proclaim{Proposition 4.2} \roster \item $\om = \{(y,z)\neq (0,0)| \overline{y} =-y, \overline{z} =-z \text { and } y^2=z^2=yz=0 \}$. \item Let $AA$ and $BB$ be the subsets of $\om$ consisting all pairs $(y,z)$ such that the space $Fy+Fz$ has dimension $2$ and $1$ respectively. Then $AA$ and $BB$ are $G_2(F)\times GL_2(F)$-orbits. \endroster \endproclaim \demo{Proof} Let us prove the corresponding statement for $\omega\omega$. Let $$ n= \pmatrix 0 & 0 & y \\ 0 & 0 & z \\ -y & -z & 0 \endpmatrix \in \omega\omega . $$ Since $n$ is a traceless matrix in $J$ it will be singular iff $n^2=0$. But this is equivalent to $y^2 =z^2= yz = 0$. The first part of the proposition is proved. We go on to observe that $G_2$ has three orbits on the set of spaces of traceless Octonions with the property that the Octonion multiplication is trivial. These are characterized by their dimension; the possible choices being $0,1,2$. The stabilizers of the nontrivial spaces are the maximal parabolics of $G_2$. Let $x$ be a traceless Octonion such that $x^2=0$. Let $P_1$ be the maximal parabolic subgroup of $G_2(F)$ stabilizing the line $Fx$. The Levi factor of $P_1$ is "spanned" by a long root. Consider $$ B=\{ (ax,0)|a\in F \text { and } a\neq 0\}. $$ Let $QQ$ be the maximal parabolic subgroup of $GL_2(F)$ stabilizing $B$. Then $P_1 \times QQ$ acts transitively on $B$ and $$ BB= (G_2(F)\times GL_2(F)) \times_{(P_1 \times QQ)} B. $$ Let $y$ and $z$ be two traceless and linearly independent Octonions such that $y^2=z^2=yz=0$. Let $P_2$ be the maximal parabolic subgroup of $G_2(F)$ stabilizing the space $Fy+Fz$. The Levi factor of $P_2$ is "spanned" by a short root. Consider $$ A=\{ (ay+bz,cy+zd)|a,b,c,d\in F \text { and } ad-bc\neq 0\}. $$ Then $P_2 \times GL(2)$ acts transitively on $A$ and since $G_2(F)$ acts transitively on the set of all two-dimensional spaces of traceless Octonions with trivial multiplication, $$ AA = G_2(F) \times_{P_2} A. $$ The proposition is proved. \enddemo We can now summarize the structure of $V_U$ as a $G_2(F) \times GL_2(F)$-module in the following theorem (compare \cite{Ku}). \proclaim{Theorem 4.3} $V_U$ has a filtration with three quotients: $$ \ca(AA),\,\, \ca(BB),\, \text { and } V_N. $$ Moreover \roster \item $$ \ca(AA) = \ind_{P_2}^{G_2}(\ca(A))\otimes |\det|^2 $$ \item $$ \ca(BB)=\ind_{P_1\times QQ}^{G_2\times GL_2}(\ca(B))\otimes |\det|^2 $$ \item $$ V_N\cong V_M \otimes |\det| + 1\otimes |\det|^2 $$ \endroster as $G_2(F) \times GL_2(F)$-modules. \endproclaim We are now ready to state and prove the main theorem. \proclaim{Theorem 4.4} Let $\Phi : SL_3(\Bbb C) \to G_2(\Bbb C)$ be the standard inclusion of the L-groups of $PGL_3(F)$ and $G_2(F)$. Let $Y$ be a tempered spherical representation of $PGL_3(F)$. Let $s\in SL_3(\Bbb C)$ be its Satake parameter. Let $\tilde Y$ be the tempered spherical representation of $G_2(F)$ whose Satake parameter is $\Phi(s)$. Then $$ \Theta (Y) = \{\tilde Y\}. $$ \endproclaim \demo{Proof} Write $Y=\Ind_{Q}^{GL_3}X$ (note that there are up to three different choices for $X$). Then the restriction of $X$ to $GL_2(F)$ (a factor of $L$) is the irreducible spherical representation whose parameter is $(\chi |\cdot |^{1/2}, \mu |\cdot |^{1/2})$ where $\chi$ and $\mu$ are unitary characters. Let $\tilde {X}$ be the irreducible spherical representation of $GL_2(F)$ whose parameter is $(\chi^{-1} |\cdot |^{3/2}, \mu^{-1} |\cdot |^{3/2})$. Note that $\ca(A)$ is the regular representation of $GL(2)$. After taking into account the twist with $|\det |^2$ it follows that there is a unique quotient of $\ca(AA)$ isomorphic to $\tilde X\otimes X$ as a $GL_2(F)\times GL_2(F)$-module. Here the first copy of $GL_2(F)$ is the Levi factor of $P_2$ and the second is the factor of $L$. Therefore, $$ \Ind_{P_2}^{G_2}(\tilde X)\otimes X $$ is a quotient of $\ca(AA)$. Write $ \tilde Y =\Ind_{P_2}^{G_2}\tilde X$. Let $U_2$ be the unipotent radical of $P_2$. It is Heisenberg group. Let $Z$ be the center of $U_2$. The action of $GL_2(F)$ (Levi factor of $P_2$) on $Z$ is $\det$ and on $U_2 /Z$ it is isomorphic to $S^3(F^2)\otimes \det^{-1}$. It follows that $\rho_{U_2}=|\det|^3$. Hence $\tilde Y$ is tempered. It is irreducible by a result of Keys \cite{Ke}. We have to show that $\tilde Y\otimes X$ is a quotient of $V_U$ and that $X$ does not appear as a quotient composition factors of $V_U$ different from $\ca(AA)$. We again use Lemma 2.5, but in this case $T$ will be in the Bernstein center (see \cite{BD}) of $GL_2(F)\subseteq L$. Recall that the component of the Bernstein center corresponding to representations generated by their Iwahori-fixed vectors is isomorphic to $$ \Bbb C[x,x^{-1},y,y^{-1}]^{W} $$ where $W=\{1,w\}$, $w(x)=y$ and $w(y)=x$ is the Weyl group of $GL_2$. Let $I$ be the Iwahori subgroup of $GL_2(F)$. Let $\varpi$ be the uniformizing element in $F$. Then any unramified character $\chi$ of $F^*$ is determined by its value on $\varpi$. If $E$ is a subquotient of an induced representation with the parameter $(\chi_1, \chi_2)$ then $$ (x+y) = \chi_1(\varpi) + \chi_2(\varpi) \text { and } xy = \chi_1(\varpi)\chi_2(\varpi) $$ on $E$. Let $$ T_1=q^{3/2}(x^{-1}+y^{-1})-(xy)^{-1}, $$ where $q=|\varpi|^{-1}$. On $X$ $T_1$ acts as the scalar $$ q^2(\chi(\varpi)^{-1} +\mu(\varpi)^{-1})-q\chi(\varpi)^{-1}\mu(\varpi)^{-1}, $$ which if real, is strictly less then $2q^2$. On the other hand, $\ca(B)$ is the regular representation of $GL_1$, so $\ca(B)$ as a $GL_2(F)$-module consists of induced representations whose inducing parameters are $(|\cdot|^{3/2}, \chi)$. It has an Iwahori-fixed vector only when $\chi$ is an unramified character. On such induced representations $T_1$ acts as $$ q^3=q^{3/2}(q^{3/2}+\chi(\varpi)^{-1})-q^{3/2}\chi(\varpi)^{-1}. $$ Hence $T_1$ acts on $\ca(B)^{I}$ as the scalar $q^3$, which is always bigger then $2q^2$. Therefore $X$ can not be a quotient of $\ca(B)$. Let $$ T_2 =xy. $$ Then $T_2$ acts on $X$ as $|\varpi|\chi(\varpi)\mu(\varpi)$ which is different from $|\varpi|^{2}$ and $|\varpi|^4$, the eigenvalues of $T_2$ on $V^I_N$. Therefore $X$ can not be a quotient of $V_N$. Finally, Lemma 2.5 applied to $T_1$ and $T_2$ shows that $\tilde Y\otimes X$ is a quotient of $V_U$. To finish the proof we have to check that the correspondence $Y\to \tilde Y$ is Langlands. Since $\tilde Y =\Ind_{P_2}^{G_2}\tilde X$ and $P_2$ is spanned by a short root, the Satake parameter of $\tilde Y$ sits in a Levi factor of a parabolic subgroup of $G_2(\Bbb C)$ spanned by a long root. Since $SL_3(\Bbb C)$ is spanned by long roots of $G_2$, $\tilde Y$ must be a lift of a representation of $PGL_3(F)$ induced from $Q$: $Y$ or $Y^*$. Note that replacing the pair $(\chi,\mu)$ by $(\chi^{-1},\mu^{-1})$ does not change $\tilde Y$ but replaces $Y$ by $Y^*$. The theorem is proved. \enddemo \head 5. Dual pair $G_2(F) \times PGSp_6(F)$ \endhead Let $G$ be the simply connected group of type $E_7$. Let $P=MN$ be the maximal parabolic subgroup of $G$ such that $M$ is the reductive group of type $E_6$ discussed in the last section. The nilpotent radical $N$ is commutative and isomorphic to $J$ as an $M$-module. Let $G_2(F) \times GL_3(F)$ be the dual pair in $M$, described in Section 4. The centralizer of $G_2(F)$ in $G$ is $Sp(6)$. This can be easily seen on the level of Lie algebras. Let $\frak g$ be the Lie algebra of $G$. Then $$ \frak g = \bar \frak n + \frak m + \frak n $$ and write $$ C_{\frak g}(G_2(F))= \bar \frak u + \frak l + \frak u $$ where $\frak l \subset \frak m$, $\overline{\frak u} \subset \overline{\frak n}$ and $\frak u \subset \frak n$. Then $\frak l = gl(3)$ and $\frak u \subset \frak n$ corresponds to the inclusion $J_F \subset J$ since $$ J^{G_2(F)} = J_F.$$ Therefore $C_{\frak g}(G_2(F))= sp(6)$ whose Siegel parabolic subalgebra is $\frak l + \frak u$. Note also that we have a distinguished $SL_2$, $$ SL_2(F) \subset Sp_6(F) \subset G $$ such that if $(e,h,f)$ is the standard basis of $sl(2)$ (the Lie algebra of $SL_2$) then $h \in \frak m$, $f \in \overline{\frak n}$ and $ e\in \frak n$. Moreover $e$ corresponds to the unit element of $J$ under the identification of $\frak n$ and $J$. Let $G_A$ be the split adjoint group of type $E_7$, and let $P_A=M_A N$ be the corresponding maximal parabolic subgroup. Then $M_A$ can be defined as the group of isogenies of the Dickson form. We have a dual pair $G_2(F) \times PGSp_6(F) \subset G_A$. Note that $PGSp_6(F)$ is generated by the distinguished $PGL_2(F)\subset PGSp_6(F)$ and the image of $Sp_6(F)$. Let $Q=GL_3(F)U$ be the Siegel parabolic subgroup of $Sp_6(F)$ and $Q_A=LU$ be the corresponding maximal parabolic of $PGSp_6(F)$. Let $V$ be the minimal representation of $G_A$. We want to compute $V_U$. Since $$ 0 \to {\Cal C}_{c}^{\infty}(\oz)_U \to V_U \to V_N \to 0 $$ we need to understand $\ca(\oz)_U$. Let $NN$ and $\overline{NN}$ be the orthogonal complements of $\overline{U}$ and $U$ in $N$ and $\overline{N}$. Since $U$ can be identified with $J_F$, it follows that $$ NN = \{ \pmatrix 0 & x & y \\ \overline{x} & 0 & z \\ \overline{y} & \overline{z} & 0 \endpmatrix |~ \overline{x}=-x, \overline{y} =-y \text { and } \overline{z} =-z \}. $$ Let $\om =\oz \cap \overline{NN}$. As in Lemma 2.2 $\ca(\oz)_U = \ca(\om )$ and we have: \proclaim{Proposition 5.1} Identify $\overline{NN}$ with the set of triples of traceless Octonions $(x,y,z)$. Let $f\in \ca(\om)$. The action of $G_2(F)\times L$ is given by \roster \item $$ \pi(g)f((x,y,z)) = f((g^{-1}x, g^{-1}y, g^{-1}z)), \qquad g\in G_2(F). $$ \item $$ \pi\pmatrix a & \\ & b \\ \endpmatrix f((x,y,z)) = |\frac{a}{b}|^6 f(\frac{a}{b}(x,y,z)), \qquad \pmatrix a & \\ & b \\ \endpmatrix\in PGL_2(F)\subset PGSp_6(F). $$ \item $$ \pi(g)f((x,y,z)) = |\det g|^4 f(\det g(x,y,z)\,^tg^{-1}), \qquad g \in GL_3(F). $$ \endroster \endproclaim \demo{Proof} It follows from Theorem 1.1. Note, however, that in this case the formulas are describing the action on $\ca(\om)$. \enddemo To keep our notation simple we work in $G$. We have to understand the structure of $G_2(F) \times GL_3(F)$ orbits on $\om$ which is, of course, the same as the structure of orbits on $\omega\omega=\omega \cap NN$. \proclaim{Proposition 5.2} \roster \item $\om = \{(x,y,z)|\overline{x}=-x, \overline{y} =-y, \overline{z} =-z \text { and }x^2= y^2=z^2=xy= xz=yz=0 \}$. \item Let $AA$ and $BB$ be the subsets of $\om$ consisting of all triples $(x,y,z)$ such that the space $Fx+Fy+Fz$ has dimension $2$ and $1$ respectively. Then $AA$ and $BB$ are $G_2(F)\times GL_3(F)$-orbits and $$\om = AA \cup BB. $$ \endroster \endproclaim \demo{Proof} Let us prove the corresponding statement for $\omega\omega$. Let $$ n= \pmatrix 0 & x & y \\ -x & 0 & z \\ -y & -z & 0 \endpmatrix \in \omega\omega . $$ Since $n$ is a traceless matrix in $J$ it will be singular iff $n^2=0$. But this is equivalent to $x^2=y^2= z^2=xy=xz=yz = 0$. The first part of the proposition is proved. Again, recall that $G_2$ has three orbits on the set of spaces of traceless Octonions with the property that the Octonion multiplication is trivial. These are characterized by their dimension; the possible choices being $0,1,2$. The stabilizers of the nontrivial spaces are the maximal parabolics of $G_2$. It follows that $x$, $y$ and $z$ are linearly dependent. hence $\om = AA \cup BB$. It remains to show that $AA$ and $BB$ are single orbits. The proof is analogous to the proof of Proposition 4.2. Let $x$ be a traceless Octonion such that $x^2=0$. Let $P_1$ be the maximal parabolic subgroup of $G_2(F)$ stabilizing the line $Fx$. Consider $$ B=\{ (ax,0,0)|a\in F \text { and } a\neq 0\}. $$ Let $Q_1$ be the maximal parabolic of $GL_3(F)$ stabilizing $B$. Then $P_1 \times Q_1$ acts transitively on $B$ and $$ BB= (G_2(F)\times GL_3(F)) \times_{(P_1\times Q_1)} B. $$ Let $y$ and $z$ be two traceless and linearly independent Octonions such that $y^2=z^2=yz=0$. Let $P_2$ be the maximal parabolic subgroup of $G_2(F)$ stabilizing the space $Fy+Fz$. Consider $$ A=\{ (ay+bz,cy+dz,0)|a,b,c,d\in F {\text { and }} ad-bc\neq 0\}. $$ Let $Q_2$ be the maximal parabolic subgroup of $GL_3(F)$ stabilizing $A$. Then $P_2 \times Q_2$ acts transitively on $A$ and $$ AA = (G_2(F)\times GL_3(F)) \times_{(P_2 \times Q_2)} A. $$ The proposition is proved. \enddemo We can now summarize the structure of $V_U$ as a $G_2(F) \times GL_3(F)$-module. \proclaim{Theorem 5.3} $V_U$ has a filtration with three subquotients: $$ \ca(AA),\,\, \ca(BB),\, \text { and } V_N. $$ Moreover \roster \item $$ \ca(AA) = \ind_{P_2\times Q_2}^{G_2\times GL_3}(\ca(A))\otimes |\det|^4 $$ \item $$ \ca(BB)=\ind_{P_1\times Q_1}^{G_2\times GL_3}(\ca(B))\otimes |\det|^4 $$ \item $$ V_N \cong V_M \otimes |\det |^2 + 1\otimes |\det|^4 $$ \endroster as $G_2(F) \times GL_3(F)$-modules. Here $\det$ denotes the usual determinant of $3\times 3$ matrices. \endproclaim We are now ready to state and prove the main theorem. \proclaim{Theorem 5.4} Let $\Phi : G_2(\Bbb C) \to Spin_7(\Bbb C)$ be the standard inclusion of the L-groups of $G_2(F)$ and $PGSp_6(F)$; $G_2(\Bbb C)$ fixes a non-zero vector in the $8$-dimensional spin representation of $Spin_7(\Bbb C)$. Let $Y'$ be a tempered spherical representation of $PGSp_6(F)$. Then $\Theta(Y')$ is not empty only if the Satake parameter of $Y'$ is $\Phi(s)$ for some $s$, a Satake parameter of a tempered spherical representation $\tilde Y$ of $G_2(F)$. In that case $$ \Theta (Y') = \{\tilde Y\}. $$ \endproclaim \demo{Proof} Let $X'$ be a spherical tempered representation of $PGSp_6(F)$. Recall that every tempered spherical representation of $PGSp_6(F)$ is fully induced (see \cite{T1}). To simplify notation we work with $Sp_6(F)$. The restriction of $X'$ to $Sp_6(F)$ can be written as $\Ind _{Q}^{Sp_6}X\otimes |\det|^2$ where $X$ is a tempered spherical representation of $GL_3(F)$ (note that $\rho_U=|\det|^2$). Let $Y'$ be a spherical tempered representation of $PGSp_6(F)$ whose Satake parameter is $\Phi(s)$. This means that the restriction of $Y'$ to $Sp_6(F)$ can be written as $\Ind_{Q}^{Sp_6}(Y\otimes |\det|^2)$ where $Y$ is a tempered spherical representation of $PGL_3(F)$. Also $\tilde Y$, the representation of $G_2(F)$ whose parameter is $s$, is the lift of $Y$ from $PGL_3(F)$. By Theorem 5.3, $V_M\otimes |\det|^2$ is a quotient of $V_U$ ($M$ is a group of type $E_6$). It follows from the Frobenius reciprocity that $\tilde Y\in \Theta(Y')$. The rest of the theorem follows form the knowledge of $V_U$. Indeed, let $X'$ be a tempered spherical representation of $PGSp_6(F)$. Again, it is a fully induced representation so its restriction to $Sp_6(F)$ can be written as $\Ind _{Q}^{Sp_6}X\otimes |\det|^2$ where $X$ is a tempered spherical representation of $GL_3(F)$. If $\tilde Y\otimes X'$ is a quotient of $V$ for some irreducible representation $\tilde Y$ of $G_2(F)$, then $\tilde Y \otimes (X\otimes |\det|^2)$ is a quotient of $V_U$, i.e. it is a quotient of one of the three pieces in Theorem 5.3. For example, if it is a quotient of $\ca(AA)$ then $$ \tilde Y\otimes (X\otimes |\det|^2) = \Ind_{P_2}^{G_2}(\tilde X)\otimes(X\otimes |\det|^2) $$ where the parameter of $\tilde X$ is $(|\cdot|^{3/2}\chi^{-1}, |\cdot|^{3/2}\mu^{-1})$ and the parameter of $X$ is $(\chi,\mu, \chi\mu)$ for some unitary characters $\chi$ and $\mu$. Since $\Ind X \otimes |\det|^2 = \Ind Y\otimes |\det|^2$, where $Y$ is a representation of $PGL_3(F)$ with parameter $(\chi,\mu,\chi^{-1}\mu^{-1})$, we are back in the situation as in the beginning of the proof. The theorem is proved. \enddemo With only minor modifications of the preceeding proof one can easily show the following proposition. \proclaim{Proposition 5.5} Let $Y$ be a spherical representation of $PGSp_6(F)$ which appears as a quotient of $V$, the minimal representation of $E_7$. Then the Satake parameter of $Y$ is $\Phi(s)$ for some $s\in G_2(\Bbb C)$. \endproclaim \head 6. Some facts about Jordan algebras \endhead In this section we shall describe structure of exceptional groups using simple Jordan algebras of rank $3$. Let $D$ be a non-degenerate subalgebra of Octonions. Let $J_D \subseteq J$ be the corresponding Jordan algebra of rank $3$. The Dickson form on $J_D$ is given by $$6(x,y,z)= 2tr(xyz) - tr(x)tr(yz)-tr(y)tr(zx)-tr(z)tr(xy)-tr(x)tr(y)tr(z). $$ Define $$ \frak l_D = \{ x\in gl(J_D) \,|\, (xv,v,v)=0 \text { for all } v\in J_D\}. $$ Now assume that $D=F$, $F+F$, $M_2(F)$ ($2\times 2$-matrices) or $\Bbb O$. Let $G_D$ be a simply connected group of type $F_4$ or $E_n$, $n=6,7$ or $8$ respectively. Let $\frak g_D$ be its Lie algebra. Let $\Delta$ be the Dynkin diagram of $\frak g_D$. We shall identify it with a set of simple roots. Let $\tilde \alpha$ be the highest positive root. Let $\alpha$ be the unique simple root not perpendicular to $\tilde \alpha$. Let $\frak m_D\subset \frak g_D$ be the simple algebra whose Dynkin diagram is $\Delta \setminus \{\alpha\}$. Extend $\Delta$ by adding $-\tilde \alpha$. Let $\beta$ be the unique simple root not perpendicular to $\alpha$. Remove the vertex corresponding to the simple root $\beta$. The extended diagram breaks into several pieces, one of which is an $A_2$ diagram corresponding to $\{\alpha, -\tilde \alpha\}$. Then $\frak l_D\subset \frak g_D$ corresponds to the rest of the diagram. Consider the adjoint action of ${sl}(3) +\frak l_D$ on $\frak g_D$. One has a decomposition $$ \frak g_D=sl(3)+\frak l_D + V \otimes J_D + V^* \otimes J^*_D $$ where $V$ is the standard $3$-dimensional representation of $sl(3)$ (see \cite{HP}). Let $M_D$ and $L_D$ be the corresponding simply connected groups and let $H_D$ be a subgroup of $L_D$ centralizing the identity of $J_D$. The possible choices are given by the Freudenthal's magic square: $$ \matrix G_D & M_D & L_D & H_D & J_D\\ F_4 & Sp_6 & SL_3 & SO_3 & S^2 V \\ E_6 & SL_6 & SL_3 \times SL_3 & SL_3 & V\otimes V^* \\ E_7 & E_6 & SL_6 & Sp_6 & \wedge^2 F^6 \\ E_8 & E_7 & E_6 & F_4 & F^{27} \endmatrix $$ In the above table we have identified $J_D$ with a "usual" representation of $L_D$ whenever possible. Since the centralizer of $H_D$ in $J_D$ is one-dimensional, it follows that the centralizer of $H_D$ in $\frak g_D$ is (see \cite{FH}) $$ \frak g_2 = sl(3) + V + V^*. $$ Assume that $D\neq F$. We shall need few facts about the action of $H_D$ on $J_D^0$, the set of traceless elements in $J_D$. Recall that a point $x \in J^0_D$ is singular if $x^2=0$. The group $H_D$ acts transitively on the set of singular vectors in $J_D^0$. Denote by $Q_1$ a parabolic subgroup of $H_D$ stabilizing a singular line in $J_D^0$. Let $Fy +Fz \subset J^0_D$ be a singular space, i.e. each point is singular. This is equivalent to $$ y^2=z^2=yz=0. $$ We need to know $H_D$-orbits of singular two-dimensional subspaces of $J_D^0$. We have two different cases. If $\dim D=2$ then $J_D$ is isomorphic to the algebra of all $3\times 3$-matrices with coefficients in $F$ and $H_D=SL_3$ acts by conjugation. The restriction of the Dickson form to $J_D$ is given by $$ (x,x,x)=\det(x) $$ where $\det$ is the determinant of $3\times 3$ matrices. In this case, a singular point in $J_D^0$ is a nilpotent rank-one matrix. There are two $SL_3$-orbits of singular two-dimensional spaces. Let $Fy+Fz$ be a singular space. Then either the images or the kernels of $y$ and $z$ coincide. Denote by $Q^+$ or $Q^-$ the stabilizer of the singular space depending on the orbit. It is a maximal parabolic subgroup of $SL_3$. In other cases, a stabilizer of a singular two-dimensional space is a parabolic subgroup only if the space is "amber" (\cite{A}). \proclaim{Definition 6.1} Let $S\subset J_D^0$ be a singular two-dimensional space. We say that $S$ is amber if $S\subset xJ_D$ for every $x\in S$. \endproclaim \proclaim{Proposition 6.2} If $\dim D \geq 4$ then $H_D$ acts transitively on the space of amber two-dimensional subspaces of $J_D^0$. \endproclaim \demo{Proof} If $\dim D=8$ this is a result of Aschbacher, 9.3-5 \cite{A1}. Let $Fy +Fz=S$ be an amber space. Since $H_D$ acts transitively on the set of singular vectors, we can assume that $z=x$, where $x$ is a fixed singular vector in $J^0_D$. Let $Q_1$ be a parabolic subgroup of $H_D$ stabilizing the line $Fx$. Let $L_1$ be a Levi factor of $Q_1$. Let $I_x \subset J_D$ be an $L_1$-invariant space such that $$ xJ_D \cap J_D^0 = Fx + I_x . $$ To finish the proof, we have to show that $L_1$ acts transitively on the set of singular points in $I_x$. If $\dim D=4$ then $L_1= GL_2 \times SL_2$ and $I_x$ can be identified with the space of $2\times 2$-matrices. If $\dim D=8$ then $[L_1,L_1]=Spin_7$ and $I_x$ is the corresponding $8$-dimensional spin-representation. In both cases $L_1$ has two non-trivial orbits in $I_x$, the smaller consisting of singular points. The proposition is proved. \enddemo Let $Q_2$ be the stabilizer of an amber plane. We finish this section with a description of $Q_2$ in both cases. Identify the amber space with $F^2$. Let $L_2$ be a Levi factor of $Q_2$. If $\dim D=4$ then $Q_2$ is contained in a Siegel parabolic subgroup of $Sp(6)$ and $L_2=GL_2 \times GL_1$. The action of $L_2$ on the amber space is the standard action of $GL_2\times GL_1$ on the space of $2\times 1$ matrices. If $\dim D=8$. Then $L_2$ is a quotient of $GL_3\times GL_2$ obtained by identifying centers of both factors (consider $GL_3\times GL_2$ acting on $3\times 2$ matrices, for example). Let $(l_3 , l_2) \in GL_3\times GL_2$. The action of $L_2$ on the amber space is given by $$ \det(l_3)\det(l_2) \l_2 x\,\, x\in F^2. $$ In particular, the action of $L_2$ on the amber space induces a non-split exact sequence $$ 1\to SL_3 \to L_2 \to GL_2 \to 1. $$ We have $[L_2,L_2]=SL_3 \times SL_2$ where $SL_3$ is spanned by long roots and $SL_2$ by short roots. \head 7. Jacquet functor for $G_2$ - Heisenberg parabolic \endhead We continue to use the notation and definitions introduced in the previous section. Let $$ x= \pmatrix 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \endpmatrix \in sl(3)\subset \frak g_D. $$ Let $\frak g(k) = \{ y\in \frak g_D | [x,y]=ky\}$. Then $\frak g(k) \neq 0$ for $k=-2,-1,0,1,2$. Let $\frak p = \frak m + \frak n$ be the maximal parabolic subalgebra with $\frak m=\frak g(0)$ and $\frak n = \frak g(1) + \frak g(2)$. Note that $[\frak m, \frak m] = \frak m_D$ and $\frak n$ is a Heisenberg Lie algebra with center $\frak g(2)$. Let $P=M N$ be the corresponding maximal parabolic subgroup. Let $Z$ be the center of $N$ and let $N_D$ be the quotient of $N$ by $Z$. Then $N_D$ is an irreducible $M_D$-module. Since $N_D\cong \frak g(1)$, the restriction to $L_D$ is isomorphic to $$ F + J_D + J_D^* + F. $$ The centralizer of $H_D$ in $G_D$ is $G_2(F)$, and $$ G_2(F) \cap P = P_2 = GL_2(F) U_2 $$ is the Heisenberg parabolic of $G_2(F)$. Then $Z \subset U_2$ and let $U$ be the quotient of $U_2$ by $Z$. Let $\Omega$ be the set of singular vectors in $N_D$. Let $\overline{N}$ denote the opposite nilpotent subgroup. Let $\overline{N}_D$, $\overline{Z}$... be the corresponding objects attached to $\overline{N}$. Note that the Killing form on $\frak g_D$, the Lie algebra of $G_D$, defines a non-degenerate pairing $<\cdot ,\cdot>$ between $N_D$ and $\overline{N}_D$. \proclaim{Theorem 7.1} Assume that the residual characteristic of $F$ is odd. Let $V$ be the minimal representation of $G_D$. Let $Z$ be the center of $N$ as above. Then $V_Z$ has a $P$-invariant filtration $$ 0 \to {\Cal C}_{c}^{\infty}(\Oz) \to V_Z \to V_N \to 0 $$ where $\Cal C_{c}^{\infty}(\Oz)$ denotes the space of locally constant, compactly supported functions on $\Oz$. \roster \item Let $f\in \Cal C_{c}^{\infty}(\Oz)$. Then $$\ \pi(n)f(x) = \psi()f(x), \qquad n\in N $$ and $$ \pi(m)f(x) = |\det(m)|^{\frac{s}{d}}f(m^{-1}x), \qquad m\in M. $$ \item $$ V_N \cong V_M \otimes |\det|^{\frac{t}{d}} + |\det|^{\frac{s}{d}} $$ where $V_M$ is the minimal representation of $M$ (center acting trivially). \endroster In the above formulas $\det$ is the determinant of the representation of $M$ on $N_D$, $d$ is the dimension of $N_D$ and the values of $s$ and $t$ are given in the following table. $$ \matrix G & s & t \\ E_6 & 4 & 3 \\ E_7 & 6 & 4 \\ E_8 & 10 & 6 \endmatrix $$ \endproclaim \demo{Proof} Part (2) is Prop. 4.1 \cite{S}. Let $W$ be the kernel of the projection of $V_Z$ onto $V_{N}$. Let $x\in \overline N_D$ and let $$ \psi_x(y)=\psi() $$ be a character of $N$. The theorem of Moeglin and Waldspurger \cite{MW} implies that $\dim W_{N,\psi_x}=0$ or $1$ and it is one if and only if $x\in \Oz$. Let $x\in \Oz$. Let $M_x$ be the stabilizer of $x$ in $M$ and $\chi$ the character of $M_x$ describing the action of $M_x$ on $W_{N,\psi_x}$. By Frobenius reciprocity there exists a non-trivial $P$-homomorphism $$ T : W \to \Ind_{M_x N}^{P}(\chi \otimes \psi_x). $$ Let $\Cal C^{\infty}(\Oz)$ denote the space of locally constant functions on $\Oz$. Note that we have an inclusion $$ \Ind_{M_x N}^{P}(\chi \otimes \psi_x) \subseteq \Cal C^{\infty}(\Oz). $$ Let $w\in W$ and $f=T(w)$. We need to show that $f$ is a compactly supported function on $\Oz$. Let $N_{D,k}$, $k \in \Bbb Z$ be a family of lattices of $N_D$ such that $\cup_k N_{D,k}= N_D$ and $\cap_k N_{D,k}=0$. Let $\overline N_{D,k}$ be their dual lattices in $\overline N_D$. Since $W$ is a smooth module, there exists a small integer $k$ depending on $w$ such that $\pi(n)w=w$ for all $n\in N_{D,k}$. This implies that $f$ is supported inside $\overline N_{D,k}$. Since $W_{N}=0$ there exists an integer $l$ depending on $w$ such that $$ \int_{N_{D,l}}\pi(n)w dn = 0 $$ (see 2.33 \cite{BZ}). This implies that $f$ is supported outside $\overline N_{D,k}$. Since $\Oz$ is locally closed and the boundary is $\{ 0\}$, it follows that $f\in \Cal C_{c}^{\infty}(\Oz)$. Let $\ind$ denote smooth induction with compact support. By the Bernstein-Zelevinsky analogue of Mackey Theory (see \cite{BZ}, pages 46-47) $$ \ind_{M_x N}^{P}(\chi \otimes \psi_x)= \Cal C_{c}^{\infty}(\Oz) $$ is irreducible $P$-module. Hence $T(W)=\Cal C_{c}^{\infty}(\Oz)$. Let $W'$ be the kernel of $T$. Since $$ \dim W_{N, \psi_x}=\dim \Cal C_{c}^{\infty}(\Oz)_{N, \psi_x} $$ for any $x\in \overline N_D$, it follows that $W'_{N, \psi_x}=0$ for any $x\in \overline N_D$ (2.35 \cite{BZ}). Therefore $W'=0$ by 5.14 \cite{BZ}. Note that the inclusion $M_x \to M$ induces an isomorphism $M_x/[M_x ,M_x] \cong M/[M,M]$ This can be easily checked by choosing $x$ to be in $\frak g_{D,\alpha} \subset \frak g(1)$, $\alpha$ is defined in the previous section. Hence $\chi$ is a character of $M$ and to finish the proof we have to show that $$ \chi(m) =|\det(m)|^{\frac{s}{d}} \qquad m\in M. $$ Let $GL_2(F)$ be the Levi factor of $P_2$ defined above. The inclusion of $GL_2(F)$ into $M$ induces in isomorphism $GL_2(F)/SL_2(F) \cong M/[M,M]$. This can be seen by looking at the action of $GL_2(F)$ and $M$ on $Z$. Therefore it suffices to find the restriction of $\chi$ to $GL_2(F)$. In Section 8 we shall use the information on correspondences obtained in previous sections to find the character. \enddemo Let $NN$ be the orthogonal complement of $\overline{U}$ in $N_D$. Let $J_D^0$ be the set of traceless elements of $J_D$. Then $NN$ as a $GL_2(F)\times H_D$-module is isomorphic to $$ F^2 \otimes J_D^0. $$ Write $$ \Omega\Omega = \Omega \cap NN. $$ \proclaim{Proposition 7.2} \roster \item $\Om = \{(y,z)\neq (0,0)|~ y,z \in J_D^0 \text { and the space } Fy+Fz \text { is amber } \}$. \item Let $AA$ and $BB$ be the subsets of $\Om$ consisting of all pairs $(y,z)$ such that the space $Fy+Fz$ has dimension $2$ and $1$ respectively. Then $BB$ is a $GL_2(F)\times H_D$-orbit. If $\dim D\geq 4$ then $AA$ is a $GL_2(F)\times H_D$-orbit. If $\dim D=2$ then $AA$ is a union of two orbits. \endroster \endproclaim \demo{Proof} Recall that a two-dimensional space $S \subset J_D^0$ is amber if every element is singular and $$ S\subset xJ_D $$ for all $x\in S$. As before, write $$ N_D = F + J_D +J_D^* +F. $$ Let $Q_D=LU_D \subseteq M_D$ be the stabilizer of the last summand in the above decomposition. Then $[L,L]=L_D$, $U_D\cong J_D$ as an $L_D$-module and $Q_D$ stabilizes the partial flag $$ N_D \supset J_D +J_D^* + F \supset J_D^* +F \supset F. $$ More precisely, let $u\in U_D$, $(a,y,z,b)\in N_D$ and $(a',y',z',b')=u(a,y,z,b)$. Then $$ a'=a,\, y'= y + au,\, z'=z+2u\times y + au\times u, $$ $$ b'=b+tr(uy)+3(u,u,z)+a\det u, $$ where $\det u=(u,u,u)$ and $u\times y$, the cross product, is an element of $J_D$ such that $$ tr((u\times y)x) = 3(x,y,u) $$ for all $x\in J_D$ (see \cite{Ki} page 143). \proclaim{Lemma 7.3} The group $Q_D$ has $4$ orbits on $\Omega$. Their representatives are $$ v_1=(1,0,0,0),\, v_2=(0,x,0,0) $$ $$ v_3=(0,0,x,0),\, v_4=(0,0,0,1) $$ where $x$ is any singular element in $J_D$. \endproclaim \demo{Proof} Note that $\Bbb P(\Omega)=M_D/Q_D$. We have to compute $Q_D\backslash M_D/Q_D$ which is the same as $W_L\backslash W_M/W_L$, here $W_M$ and $W_L$ denote the Weyl groups of $M_D$ and $L_D$. Since $N_D$ is a miniscule representation i.e. weight vectors are all contained in one $W_M$-orbit, it follows that they are parametrized by $W_M/W_L$. On the other hand $J_D$ is a miniscule representation of $L_D$ so $W_L\backslash W_M/W_L$ has four orbits and it follows easily that $v_i$, $(1\leq i \leq 4)$ form a complete set of representatives of $Q_D$-orbits. The lemma is proved. \enddemo Let $(0,y,z,0) \in \Om$. If $y\neq 0$ then Lemma 7.3 implies that it is $Q_D$ conjugated to $v_2$. Therefore $y$ is in the $L$-orbit of $x$, hence $y$ is singular. Since the action of $GL_2(F)$ is $$ (y,z) \pmatrix a & b \\ c & d \endpmatrix =(ay+cz,by+dz) $$ the same argument implies that any element of $Fy+Fz$ is singular. Note that $(0,y,z,0)$ is $Q_D$, hence $U_D$-conjugated to $(0,y,0,0)$. But this means that $z=2 u \times y$ for some $u\in U_D\cong J_D$ such that $tr(uy)=0$. Since $ tr((u\times y)v)=3(u,y,v) $ for all elements $v\in J_D$ and $$ 6(u,y,v)=2tr(uyv)-tr(u)tr(yv) $$ (use $tr(y)=tr(uy)=0$), it follows that $$ tr((2u\times y)v)=tr(u^*yv) $$ for all $v\in J_D$, here $u^*=2u-tr(u)$. Therefore $z=u^*y\in yJ_D$. Since the same argument can be repeated for any linear combination of $y$ and $z$, the first part of the proposition follows. Let $x \in J_D^0$ such that $x^2=0$. Let $Q_1$ be the parabolic subgroup of $H_D$ stabilizing the line $Fx$. Consider $$ B=\{ (ax,0)|a\in F \text { and } a\neq 0\}. $$ Let $QQ\subset GL_2(F)$ be the Borel subgroup stabilizing the line $B$. Then $QQ\times Q_1$ acts transitively on $B$ and $$ BB= (GL_2(F)\times H_D) \times_{(QQ \times Q_1)} B. $$ Assume now that $\dim D\geq 4$. Let $y,z\in J_D^0$ such that the space $Fy+Fz$ is amber. Let $Q_2$ be the parabolic subgroup of $H_D$ stabilizing the space $Fy+Fz$. Consider $$ A=\{ (ay+bz,cy+zd)|a,b,c,d\in F \text { and } ad-bc\neq 0\}. $$ Then $GL_2(F)\times Q_2$ acts transitively on $A$ and $$ AA = H_D \times_{Q_2} A. $$ If $\dim D=2$ then we have two orbits of singular two-dimensional spaces. Let $Fy^++Fz^+$ and $Fy^-+Fz^-$ be their representatives and $Q^+$ and $Q^-$ their stabilizers in $SL_3(F)$. One can define $A^+$ and $A^-$ as above, hence $$ AA = SL_3(F) \times_{Q^+} A^+ \cup SL_3(F) \times_{Q^-} A^-. $$ The proposition is proved. \enddemo Since $\ca(\Oz)_{U_2}=\ca(\Om)$, the structure of $V_{U_2}$ as $GL_2(F) \times H_D$ is given as follows. \proclaim{Theorem 7.4} Assume that the residual characteristic of $F$ is odd. $V_{U_2}$ has a filtration with three quotients: $$ \ca(AA),\,\, \ca(BB),\, \text { and } V_{N}. $$ Moreover, as $GL_2(F) \times H_D$-modules, \roster \item If $\dim D\geq 4$ then: $$ \ca(AA) = |\det|^s\otimes \ind_{Q_2}^{H_D}(\ca(A)) $$ \item If $\dim D =2$ then: $$ \ca(AA) = |\det|^s\otimes \ind_{Q^+}^{SL_3}(\ca(A^+))+ |\det|^s\otimes\ind_{Q^-}^{SL_3}(\ca(A^-)) $$ \item $$ \ca(BB)= |\det|^s\otimes \ind_{QQ\times Q_1}^{GL_2(F) \times H_D}(\ca(B)) $$ \item $$ V_{N_D}\cong |\det|^t \otimes V_M + |\det|^s \otimes 1 $$ where $V_M$ is the minimal representation of $M_D$ (center acting trivially). \endroster In the above formulas $\det $ is the usual determinant of $2\times 2$ matrices, and $s$ and $t$ are given by the following table: $$ \matrix G & s & t \\ E_6 & 2 & 3/2 \\ E_7 & 3 & 2 \\ E_8 & 5 & 3 \endmatrix $$ \endproclaim {\it Remark:} In this section we have assumed that $G_D$, and hence $H_D$ are simply connected. In fact, the isogeny class of $H_D$ determines the isogeny class of $G_D$ and vice-versa. Note that the action of $H_D$ on $\ca(\Om)$ is geometric. It comes from the (algebraic) action of $H_D$ on $J_D^0$ which extends uniquely to the group of adjoint type. Therefore we have a canonical way of extending the representation $V$ to groups of adjoint type. \head 8. $\Theta$-lifts from $G_2(F)$ \endhead In this section we compute $\Theta$-lifts of spherical tempered representations of $G_2(F)$ in all three cases. In the process we also compute the normalizing factors (i.e. coefficients $s$) in Theorems 7.1 and 7.4. We assume that the residual characteristic of $F$ is odd. We study the dual pair $G_2(F)\times F_4(F)$ in a simple group $G$ of type $E_8$ first. Let $\tilde Y$ be a spherical tempered representation of $G_2(F)$. Then $\tilde Y=\Ind_{P_2}^{G_2}(\tilde X)$ where $\tilde X$ is a spherical representation of $GL_2(F)$ with the parameter $(\chi^{-1} |\cdot |^{3/2}, \mu^{-1} |\cdot |^{3/2})$. As before, $\chi$ and $\mu$ must be unitary characters. Let $Q_2=L_2V_2$ be the maximal parabolic subgroup of $F_4(F)$ stabilizing an amber projective line. In Section 6 we have shown that the action of $L_2$ on the corresponding amber line gives an exact sequence $$ 1\to SL_3 \to L_2 \to GL_2 \to 1. $$ One checks that $\rho_{V_2}=|\det|^7$, here $\det$ is the usual determinant on $GL_2$. Let $X$ be a spherical representation of $GL_2$ with the parameter $(\chi |\cdot |^{7/2}, \mu |\cdot |^{7/2}).$ Pull $X$ back to $L_2$. Let $$ W=\Ind_{Q_2}^{F_4}X. $$ Note that $W$ is a unitarizable representation of $F_4(F)$. It is quite possible that $W$ is always irreducible but we do not know. \proclaim{Theorem 8.1} Let $\tilde Y$ be a spherical tempered representation of $G_2(F)$ and $W$ the representation of $F_4(F)$ defined above. Then $\Theta(\tilde Y)\neq \{\}$ and if $W'\in \Theta(\tilde Y)$ then $W'$ is a summand of $W$. Let $s\in G_2(\Bbb C)$ be the Satake parameter of $\tilde Y$. Then the Satake parameter of $W$ is $\Psi(s\times \rho)$ where $\Psi : G_2(\Bbb C)\times SO_3(\Bbb C)\to F_4(\Bbb C)$ and $\rho \in SO_3(\Bbb C)$ is the Satake parameter of the trivial representation of $SL_2(F)$. \endproclaim \demo{Proof} Let $P=MN$ be the maximal parabolic subalgebra of $G$ studied in the last two sections. We described an embedding of the dual pair $G_2(F)\times F_4(F)$ in $G$ such that $G_2(F)\cap P$ is the Heisenberg maximal parabolic subgroup $P_2$. Yet another embedding of the dual pair $G_2(F)\times F_4(F)$ is given by the inclusion of Jordan algebras $J_F \to J$ (use the description of the exceptional Lie algebras given in Section 6). In this case, $G_2(F)\subset M$ and $F_4(F)\cap P=Q_4=L_4V_4$, the Heisenberg maximal parabolic subgroup of $F_4(F)$. The Levi component $L_4$ is isomorphic to $GSp_6(F)$. Note that the inclusion $GSp_6(F)\to M$ induces an isomorphism $GSp_6(F)/Sp_6(F)\cong M/[M,M]$. This is easily seen by considering the action of $GSp_6(F)$ and $M$ on $Z$, the center of both, $V_4$ and $N$. Let $V$ be the minimal representation of $G$. As an $G_2(F) \times GSp_6(F)$-module $$ V_N \cong V_M \otimes |\det|^3 + 1\otimes |\det|^5 $$ where $\det$ denotes the usual determinant on $GSp_6(F)$ (Prop. 4.1 \cite {S}). Let $\tilde Y=\Ind_{P_2}^{G_2}(\tilde X)$ be a spherical tempered representation of $G_2(F)$ and let $Y'$ be the Langlands lift of $\tilde Y$ to $PGSp_6(F)$. By Theorem 5.4 $\tilde Y\otimes Y'$ is a quotient of $V_M$. It follows from the Frobenius reciprocity that $\tilde Y\otimes W'$ is a quotient of $V$ for some $W'$, a subquotient of $$ \Ind_{Q_4}^{F_4} (Y'\otimes |\det|^3). $$ On the other hand, $\tilde X\otimes W'$ must be a quotient of $V_{U_2}$. By Theorem 7.4 $$ \ca(AA) = \delta(\det)\otimes \ind_{Q_2}^{H_D}(\ca(A)) $$ for a certain character $\delta$ which we shall now determine. Although we do not know $\delta$, for a generic choice of $\chi$ and $\mu$, $\tilde X\otimes W'$ will be a quotient of $\ca(AA)$. Since $\ca(A)$ is a regular representation of $GL_2(F)$ twisted by $\delta$, $\tilde X\otimes W'$ must be a quotient of $$ \tilde X\otimes \Ind_{Q_2}^{F_4}X $$ where $X$ is a representation of $L_2$ pulled back from a representation of $GL_2(F)$ with a parameter $(\chi\delta |\cdot |^{-3/2}, \mu\delta |\cdot |^{-3/2}).$ We get that $W'$ is a subquotient of both, $\Ind_{Q_4}^{F_4}(Y'\otimes |\det|^3)$ and $\Ind_{Q_2}^{F_4}X$. This immediately implies that $$ \delta(\det)=|\det |^5 . $$ The knowledge of $\delta$ implies that any $\tilde X$ is a quotient of $\ca(AA)$ only and, therefore $\Theta$-lifts of $\tilde Y$ are the summands of $W$. It remains to check the statement about Satake parameters. The $L$-group of $F_4(F)$ is $F_4(\Bbb C)$. Let $Q_3=L_3V_3$ be the maximal parabolic subgroup of $F_4$ such that $[L_3,L_3] =SL_2 \times SL_3$ where $SL_2$ is spanned by long roots and $SL_3$ by short roots. Since $L_3(\Bbb C)$ is the $L$-group of $L_2$, it fits into exact sequence $$ 1\to GL_2 \to L_3 \to PGL_3 \to 1. $$ Let $s\in GL_2(\Bbb C)$ be the parameter $(\chi, \mu)$. Obviously, the Satake parameter of $W$ is $s \times \rho \in GL_2 \times SO(3) \subset L_3$. Let $G_2$ be the centralizer of $SO(3)$ in $F_4$. Then $L_3 \cap G_2 = GL_2$ and $s$ is precisely the Satake parameter of $\tilde Y$ as in Section 4. \enddemo Theorem 7.4 can be used, in a similar way, to prove converses of Theorems 4.4 and 5.4. We state results without giving details of proofs. \proclaim{Theorem 8.2} Let $\Phi : SL_3(\Bbb C) \to G_2(\Bbb C)$ be the inclusion of the L-groups of $PGL_3(F)$ and $G_2(F)$. Let $\tilde Y$ be a tempered spherical representation of $G_2(F)$. The Satake parameter of $\tilde Y$ is $\Phi(s)$ for some $s$, a Satake parameter of a tempered spherical representation $\tilde Y$ of $G_2(F)$. Note that $\Phi(s)=\Phi(s^*)$ where $s^*$ is a Satake parameter of $Y^*$, the dual of $Y$. Then $$ \Theta (\tilde Y) = \{Y, Y^*\}. $$ \endproclaim \proclaim{Theorem 8.3} Let $\Phi : G_2(\Bbb C)\to Spin_7(\Bbb C)$ be the inclusion of the L-groups of $G_2(F)$ and $PGSp_6(F)$. Let $\tilde Y$ be a tempered spherical representation of $G_2(F)$. Then $$ \Theta (\tilde Y) = \{Y' \} $$ where $Y'$ is the spherical tempered representation of $PGSp_6(F)$ whose Satake parameter is $\Phi(s)$. \endproclaim \head Acknowledgments \endhead It is a pleasure to thank B. 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