Geometry · 6

Computing \(\dX\)

In the first example of the previous page, \(\AXs\) was the diagonal of \(\As\), yet the Theorem uses the shifted line \(\{(q^{-1/2}y,q^{1/2}y)\}\). This page says what the shift is (half the modular character of the Borel of \(\LevX\)), how to compute it in six steps, and why it must be there.

Sections · arrows: “is used by”

Modular characters as points of \(\As\)How can a character of a group be a point of a complex torus?

Definition

For a parabolic \(P=MU_P\) of \(G\), the is \(\delta_P(p)=|\det\mathrm{Ad}(p)|_{\mathrm{Lie}\,U_P}|\). For \(B\subseteq\GL_n\),

\[\delta_B(\diag(t_1,\dots,t_n))=\prod_{i\lt j}|t_i/t_j|,\qquad\delta_B\big(\lambda(\vpi)\big)=q^{-\langle2\rho,\lambda\rangle},\]

with \(\rho\) the half-sum of positive roots. \(\delta_B\) is trivial on \(T(\cO)\), so \(\delta_B^{1/2}\) is a point of \(\As\):

\[\delta_B^{1/2}=q^{-\rho}:\quad\lambda\mapsto q^{-\langle\rho,\lambda\rangle};\qquad\GL_n:\ \delta_B^{1/2}=\big(q^{-\frac{n-1}2},q^{-\frac{n-3}2},\dots,q^{\frac{n-1}2}\big).\]

The definitionWhich modular character, of which group?

[Sak, §1.7, p. 7] Definition

\(\dX\) is the absolute value of the modular character of \(\LevX\cap B\), a Borel subgroup of the Levi \(\LevX\) of the geometry page. As a point of \(\As\),

\[\dXh=q^{-\rho_L},\qquad\rho_L=\text{half the sum of the positive roots of }\LevX .\]

(the sign does not matter) Proved here

\[q^{\rho_L}\AXs=w_L\big(q^{-\rho_L}\AXs\big),\qquad w_L\ \text{the longest element of }W_{\LevX}.\]

So \(\dXh\AXs\) and \(\dX^{-1/2}\AXs\) have the same image in \(\As/W\).

Proof

A point of \(\AXs\) is a character of \(\LevX\) (through \(\LevX\to\AX\)), so it is fixed by \(W_{\LevX}\). And \(w_L\rho_L=-\rho_L\). Hence \(w_L(q^{-\rho_L}y)=q^{\rho_L}y\).

Not to be confused with \(\delta_{\PX}\), the modular character of \(\PX\) itself, which appears in Sakellaridis's normalization of eigenfunctions, [Sak, (1.1)], and not in the Theorem.

The recipe, step by stepGiven \(X\), what do I actually compute?

Find the open \(B\)-orbit \(\Xo\).
Compute \(\PX=\{g:\ g\Xo=\Xo\}\).
Take its Levi \(\LevX\supseteq T\) (for symmetric \(X\): \(\LevX=Z_G(S)\), \(S\) a maximal \(\theta\)-split torus).
Write down \(\rho_L\), half the sum of the positive roots of \(\LevX\).
\(\dXh=q^{-\rho_L}\in\As\).
Translate: \(\dXh\AXs=\{q^{-\rho_L}y:\ y\in\AXs\}\).

For \(\GL_n\) the steps 3–5 are mechanical: if \(\LevX=\GL_{n_1}\times\dots\times\GL_{n_k}\), then \(q^{-\rho_L}\) is a string of exponents \(-\tfrac{n_j-1}2,\dots,\tfrac{n_j-1}2\) on each block.

Table 1. The recipe on seven varieties.
\(X\)	\(\LevX\)	\(\dXh\AXs\), up to \(W\)
group \(\GL_2\) under \(\GL_2^2\)	\(T\times T\)	\((y_1,y_2;y_1^{-1},y_2^{-1})\), no shift
\(\GL_2/T\)	\(T\)	\((y,y^{-1})\), no shift
\(\GL_2/\SL_2\)	\(\GL_2\)	\((q^{1/2}y,q^{-1/2}y)\)
\(\GL_{2n}/\Sp_{2n}\)	\(\GL_2^n\)	\((q^{1/2}y_i,q^{-1/2}y_i)_{i\le n}\)
\(\GL_{p+q'}/(\GL_p\times\GL_{q'})\), \(p\le q'\)	\(\Gm^{2p}\times\GL_{q'-p}\)	\((y_1,\dots,y_p,\ q^{\frac{q'-p-1}2},\dots,q^{-\frac{q'-p-1}2},\ y_p^{-1},\dots,y_1^{-1})\)
\(E_6/F_4\)	\(\Spin_8\cdot S\)	\(q^{\rho_L}y\), \(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\)
\(\Aff^n\smallsetminus\{0\}\)	\(\GL_{n-1}\times\GL_1\)	\((q^{\frac{n-2}2},\dots,q^{-\frac{n-2}2},\ y)\)

\(q'\) is the size of the second block (\(q\) is taken by the residue field). The \(\GL\) rows are from [Sak, Appendix A, rows 1–2] and the spherical-generation workspace's root data; the \(E_6/F_4\) row is from the satake-two-levels Proposition VII page.

A dot plot: for each of six varieties, dots on a horizontal line at the exponents of delta(X)^{1/2}; zeros in blue, nonzero exponents in orange at plus or minus one half or one. — The exponents of \(\dXh\) are the \(\rho\)-strings of the blocks of \(\LevX\): a torus Levi gives no shift, each \(\GL_2\) block gives \(\pm\tfrac12\).

Why the shift is thereWhich unramified representations have \(H\)-invariant functionals?

[Sak, Thm. 3.2.2], from [Sak08] Proved (cited)

If an irreducible unramified \(\pi\) is \(X\)-distinguished (\(\Hom_G(\pi,C^\infty(X(F)))\ne0\)), then \(\pi\) is the unramified subquotient of the normalized induction \(I_{\PX}(\chi)\) for some \(\chi\in\AXs\).

(Satake parameter of \(I_{\PX}(\chi)\)) Proved here

For \(\chi\in\AXs\), viewed as a character of \(\LevX\), the unramified subquotient of \(I_{\PX}(\chi)\) has Satake parameter \(\chi\cdot\dXh\) in \(\As/W\).

Proof

The trivial representation of \(L=\LevX\) is the constants in \(C^\infty(B_L\backslash L)\), the unnormalized induction of \(1\), which is the normalized induction \(I_{B_L}(\delta_{B_L}^{-1/2})\). So its Satake parameter is \(\delta_{B_L}^{-1/2}=q^{\rho_L}\), which by the Lemma above is \(W\)-conjugate to \(q^{-\rho_L}=\dXh\).
Twisting by the character \(\chi\) of \(L\) multiplies the parameter by \(\chi\).
Induction in stages, \(I_{\PX}(\chi)\subseteq I_B(\chi\,\delta_{B_L}^{-1/2})\), and the unramified subquotient of a principal series \(I_B(\mu)\) has parameter \(\mu\).

Together: the Hecke eigenvalues that can occur on \(C^\infty(X(F))^K\) are the points of \(\dXh\AXs/\WX\) (mapped to \(\As/W\)).

(\(\GL_2/\SL_2\), by hand)

The \(\SL_2(F)\)-distinguished unramified representations of \(\GL_2(F)\) are the characters \(\chi\circ\det\). On \(\chi\circ\det\), \(T_{(1,0)}=\one_{K\diag(\vpi,1)K}\) acts by \(|K\diag(\vpi,1)K/K|\,\chi(\vpi)=(q+1)y\). The Satake transform of \(T_{(1,0)}\) is \(q^{1/2}(x_1+x_2)\) (Satake), and at the shifted point

\[q^{1/2}\big(q^{-1/2}y+q^{1/2}y\big)=(1+q)\,y .\]

At the unshifted point \((y,y)\) one would get \(2q^{1/2}y\), which is wrong.

The dual-group readingIs the shift an accident of normalization?

No. In Sakellaridis–Venkatesh, a spherical variety comes with a dual group \(\GX\) and a morphism \(\GX\times\SL_2\to\Gv\) [SV, Thm. 2.2.3, under their axioms (GN2)–(GN5)]. The Satake parameters of \(X\)-distinguished unramified representations are expected to be the images of \((\,y,\ \diag(q^{1/2},q^{-1/2}))\), \(y\) in a maximal torus of \(\GX\) [SV, §16.3]; the \(\SL_2\) factor is what produces the shift.

\(\GL_{2n}/\Sp_{2n}\): \(\GX=\GL_n\to\GL_{2n}\) by tensoring with the standard representation of \(\SL_2\) [SV, Ex. 1.3.2], giving exactly the pairs \((q^{1/2}y_i,q^{-1/2}y_i)\) of .
\(\Aff^n\): \(\GX=\Gm\), and the \(\SL_2\) is principal in the Levi \(\GL_{n-1}\) [BZSV, Ex. 4.3.12], giving the string \(q^{\frac{n-2}2},\dots,q^{-\frac{n-2}2}\) of the last row.

In these examples the \(\SL_2\) is principal in the Levi dual to \(\LevX\), which is the recipe above. Read more on the Sakellaridis–Venkatesh page.