Symmetric varieties

(\(X=\GL_2/T\), \(T\) the diagonal torus; workspace “Proposition F”) Proved here

(SG) holds, and \(\HI\ast\phiz\subsetneq\CI\): the functional

\[\lambda:\St\to\C,\qquad\lambda([f]):=f([1:0])-f([0:1]),\]

is nonzero and \(T(F)\)-invariant, so \(\St\) is \(T\)-distinguished.

Proof

1. The model. \(X(F)=\GL_2(F)/T(F)\) (Hilbert 90) and \(X(\cO)=K/T(\cO)\) (\(\operatorname{Pic}\cO=0\)). (SG): computing r, row \(\GL_2/T\) (adjacency on the tree).
2. \(\lambda\). It is well defined on \(\St=C^\infty(\mathbb P^1)/\C\one\), because it vanishes on constants; nonzero; and \(T\)-invariant, since \(T\) fixes the two points \([1:0]\) and \([0:1]\).
3. The witness. \(\St\) is self-contragredient: it is the unique irreducible quotient of \(C^\infty(\mathbb P^1)\) and the unique irreducible subrepresentation of its dual. The witness Proposition with \(H=T\): \(\Hom_G(\Sw(X),\St)\simeq\Hom_T(\St,\C)\ni\lambda\ne0\), and \(\St^I\ne0=\St^K\) (the Steinberg of \(\GL_2\)).

(Steinberg sweep) [Shtotland 2025, Thm. 1.5, Rem. 1.6, Prop. 2.7] Proved (cited)

Let \(G\) be split and \(X=G/G^\sigma\) split symmetric, and let \(\chi_0\) be the unramified quadratic character trivial on parahorics. \(\St\otimes\chi_0\) is \(G^\sigma\)-distinguished iff \(X\) is quasi-split and has no factor \(\PGL_{2n+1}/P(\GL_n\times\GL_{n+1})\). Hence, by the witness Proposition, \(\QX\ne0\) for every such \(X\).

(principal series, Speh representations, symplectic distinction) Definition

1. \(\chi_1\times\dots\times\chi_m\): normalized parabolic induction of characters to \(\GL_m(F)\); \(\nu:=|\det|\). An irreducible \(\pi\) has \(\pi^I\ne0\) iff it is a subquotient of an unramified principal series [Casselman 1980, Prop. 2.6–2.7; Borel 1976, 4.9 (∗)]. \(\dim(\chi_1\times\dots\times\chi_m)^K=1\) by \(G=PK\), so exactly one subquotient has a \(K\)-fixed vector. \(\St_2\subseteq\nu^{1/2}\times\nu^{-1/2}\).
2. Tadić's \(U(\delta,m)\), for \(\delta\) square-integrable on \(\GL_a\): the unique irreducible quotient of \(\nu^{(m-1)/2}\delta\times\dots\times\nu^{-(m-1)/2}\delta\), irreducible and unitary [Tadić 1986, Thm. D, Thm. A.8; \(\operatorname{char}F=0\)]. \(U(\one_{\GL_1},m)=\one_{\GL_m}\), and \(U(\St_2,2)\) is a subquotient of \(\nu^1\times\nu^0\times\nu^0\times\nu^{-1}\).
3. \(\pi\) is \(\Sp_{2n}\)-distinguished if \(\Hom_{\Sp_{2n}(F)}(\pi,\C)\ne0\); this space is at most one-dimensional [Heumos–Rallis, Thm. 2.4.2]. Every \(\sigma_1\times\dots\times\sigma_t\times\tau_{t+1}\times\dots\times\tau_s\) with \(\sigma_i=U(\delta_i,2m_i)\) and \(\tau_i=\nu^{\alpha_i}U(\delta_i,2m_i)\times\nu^{-\alpha_i}U(\delta_i,2m_i)\) is distinguished [Offen–Sayag, JNT 2007, Thm. 1], and these are all the distinguished irreducible unitary representations [Offen–Sayag, IMRN 2008, Thm. 8, Cor. 2; JFA 2008, Cor. 1].

(\(X_n=\GL_{2n}/\Sp_{2n}\); workspace “Proposition G”) Proved here, from cited results

\(n=1\): \(\HI\ast\phiz=\CI\). \(n\ge2\), \(\operatorname{char}F=0\): \(\HI\ast\phiz\subsetneq\CI\), witnessed by the Speh-type representation

\[\pi_n:=U(\St_2,2)\times\underbrace{\one_{\GL_2}\times\dots\times\one_{\GL_2}}_{n-2},\qquad\pi_n^I\ne0,\qquad\pi_n^K=0,\qquad\pi_n\ \text{is}\ \Sp_{2n}\text{-distinguished}.\]

Proof

1. \(n=1\). \(\GL_2\) acts on \(X_1=\Gm\) through \(\det\), and \(\det I=\det K=\cO^\times\). So \(I\backslash X_1(F)=K\backslash X_1(F)=F^\times/\cO^\times\simeq\Z\), \(\CI=\CK=C_c(\Z)\), and the \(T_{t^\lambda}\ast\one_{\cO^\times}=[It^\lambda I:I]\,\one_{\vpi^{\langle\det,\lambda\rangle}\cO^\times}\) span it.
2. Iwahori-spherical. Induction in stages makes \(\pi_n\) a subquotient of an unramified principal series of \(\GL_{2n}\), so \(\pi_n^I\ne0\) (Definition, item 1). It is irreducible and unitary, since products of elements of Tadić's set \(B\) are [Tadić 1986, Thm. D(i)].
3. No spherical vector. \(\pi_n\) is a subquotient of \(J_n:=\nu^{1/2}\St_2\times\nu^{-1/2}\St_2\times\one_{\GL_2}^{\times(n-2)}=\Ind_P^G\sigma\). By \(G=PK\), \((\Ind_P^G\sigma)^K\simeq\sigma^{M(\cO)}=0\), because \(\St_2^{\GL_2(\cO)}=0\) (the Steinberg of \(\GL_2\)). \(K\)-invariants are exact, so \(\pi_n^K=0\).
4. Distinguished. [Offen–Sayag, JNT 2007, Thm. 1] with \(\sigma_1=U(\St_2,2)\) and \(\sigma_2=\dots=\sigma_{n-1}=U(\one_{\GL_1},2)=\one_{\GL_2}\); for \(n=2\) also [Heumos–Rallis, Thm. 11.1]. \(\pi_n\) is unitary, so \(\tilde\pi_n\simeq\bar\pi_n\), and complex conjugation carries an invariant functional on \(\pi_n\) to one on \(\bar\pi_n\): \(\tilde\pi_n\) is distinguished. The witness Proposition.