\(\As\) and \(\AXs\): three examples

[Sak, §1.7] Definition

\[\As:=\Hom(\LT,\C^\times),\qquad\chi\longmapsto\big(\lambda\mapsto\chi(\lambda(\vpi))\big).\]

\(\As\) is a complex algebraic torus of dimension \(\operatorname{rk}T\) (the maximal torus of the dual group \(\Gv\)), with coordinate ring \(\C[\As]=\C[\LT]\), the Laurent polynomials \(\sum c_\lambda x^\lambda\). The Weyl group \(W\) acts on \(\LT\), hence on \(\As\).

(\(\GL_n\))

\(\LT=\Z^n\), and a point of \(\As\) is \(x=(x_1,\dots,x_n)\in(\C^\times)^n\) with \(x_i=\chi_i(\vpi)\) for \(\chi=\chi_1\otimes\dots\otimes\chi_n\); \(x^\lambda=x_1^{\lambda_1}\cdots x_n^{\lambda_n}\). \(W=S_n\) permutes coordinates, and \(\C[\As]^W=\C[e_1,\dots,e_n,e_n^{-1}]\) for the elementary symmetric functions \(e_k\).

[Sak, §2.1, p. 8] Definition

\[\AXs:=\Hom(\LX,\C^\times),\qquad\AXs\longrightarrow\As,\quad y\longmapsto y\circ(\LT\to\LX).\]

So \(y\in\AXs\) is an unramified character of \(\AX(F)\), and its image is the character \(t\mapsto y(\bar t)\) of \(T(F)\), \(\bar t\) the image of \(t\) in \(\AX(F)\). Sakellaridis writes \(\AXs\) also for the image.

(when \(\AXs\subseteq\As\)) Proved here

The map \(\AXs\to\As\) is injective if and only if \(\LT\to\LX\) is surjective. Under (A5), Sakellaridis states \(\AXs\subseteq\As\) [Sak, §8, p. 63].

Proof

\(T\twoheadrightarrow\AX\) is onto, so the cokernel \(C\) of \(\LT\to\LX\) is finite. \(\Hom(-,\C^\times)\) is exact because \(\C^\times\) is divisible, so \(\ker(\AXs\to\As)=\Hom(C,\C^\times)\), which is \(0\) iff \(C=0\).

(\(G=\GL_2\), \(X=\Gm\), \(g\cdot x=\det(g)x\))

1. \(\As=(\C^\times)^2\ni(x_1,x_2)\).
2. \(\AX=\Gm\) via \(\det\), and \(\LT=\Z^2\to\LX=\Z\) is \((\lambda_1,\lambda_2)\mapsto\lambda_1+\lambda_2\) (torus page); onto.
3. \(\AXs=\C^\times\ni y\), and \(y\mapsto(y,y)\): the character \(\diag(t_1,t_2)\mapsto y^{\val(t_1t_2)}\) is \(\chi\otimes\chi\) with \(\chi(\vpi)=y\). \[\AXs=\{(y,y)\}\subseteq\As,\qquad\dXh\AXs=\{(q^{-1/2}y,\,q^{1/2}y)\}\quad\text{(computed on the }\delta(X)\text{ page)}.\]
4. Check: these are the Satake parameters of the characters \(\chi\circ\det\) of \(\GL_2(F)\), the representations with an \(\SL_2(F)\)-invariant functional (why δ(X)).

(\(G=\GL_2\), \(X=\GL_2/T\), pairs of lines)

1. \(\As=(\C^\times)^2\ni(x_1,x_2)\), the same as in the first example.
2. \(\AX=T/Z\) via \(t_1/t_2\), and \(\LT\to\LX\) is \((\lambda_1,\lambda_2)\mapsto\lambda_1-\lambda_2\); onto.
3. \(\AXs\ni y\mapsto(y,y^{-1})\): the character \(\diag(t_1,t_2)\mapsto y^{\val(t_1/t_2)}\). \[\AXs=\dXh\AXs=\{(y,y^{-1})\}\subseteq\As,\qquad\WX=\{1,w\},\ w(y)=y^{-1}.\]
4. Check: by the Theorem \(\CK\simeq\C[y^{\pm1}]^{y\leftrightarrow y^{-1}}=\C[y+y^{-1}]\); the operator \(T_1\) is the adjacency operator of the tree and acts as \(q^{1/2}(y+y^{-1})\) (worked example on the Theorem page).

(\(G=\GL_2\times\GL_2\), \(X=\GL_2\))

1. \(\As=(\C^\times)^2\times(\C^\times)^2\ni(x_1,x_2;\,x_1',x_2')\), \(W=S_2\times S_2\).
2. \(\AX=(T\times T)/\Delta T\simeq T\), and \(\LT\times\LT\to\LX=\Z^2\) is \((\lambda,\lambda')\mapsto\lambda-\lambda'\); onto.
3. \(\AXs=(\C^\times)^2\ni(y_1,y_2)\mapsto(y_1,y_2;\,y_1^{-1},y_2^{-1})\); \(\dX=1\) because \(\LevX=T\times T\) is a torus.
4. \(\WX=S_2\), permuting \((y_1,y_2)\) and \((y_1^{-1},y_2^{-1})\) together. So \[\C[\dXh\AXs]^{\WX}=\C[y_1^{\pm1},y_2^{\pm1}]^{S_2}\simeq\mathcal H_{\GL_2(\cO)},\qquad\Res(f\otimes g)(y)=f(y)\,g(y^{-1}).\]
5. Check: \(\CK=C_c(K_2\backslash\GL_2(F)/K_2)\) is the Hecke algebra of \(\GL_2\), and Satake identifies it with \(\C[y^{\pm1}]^{S_2}\); the right action of \(g\) is through \(g^\vee(h)=g(h^{-1})\), whose Satake transform is \(y\mapsto\Sat g(y^{-1})\).