Spherical level · 2

The Satake isomorphism

In the lab, the image \(\Res(h)\) of a Hecke word was a polynomial. The Satake isomorphism turns Hecke operators into polynomials on \(\As\); it links the left side of the Theorem to the right.

Sections · arrows: “is used by”

The statementWhat is \(\HK\) as an algebra?

(Satake) [Cartier, §4; Satake 1963; Macdonald 1971] Proved (cited)

The Satake transform

\[\Sat f(\chi):=\int_{T(F)}\delta_B^{1/2}(t)\,\chi(t)\int_{N(F)}f(tn)\,dn\,dt,\qquad\chi\in\As,\]

is an isomorphism of algebras \(\Sat:\HK\xrightarrow{\sim}\C[\As]^W\), and

\[\Sat(T_\lambda)=q^{\langle\rho,\lambda\rangle}x^\lambda+\big(\text{monomials }x^\mu,\ \mu\lt\lambda\ \text{in the dominance order}\big).\]

Consequences: \(\HK\) is commutative; its characters are the points of \(\As/W\); and the leading-term formula makes the \(\Sat(T_\lambda)\) triangular in the monomial basis, which Sakellaridis's proof uses (the Theorem).

\(\GL_2\) by handWhere does the \(q^{1/2}\) come from?

(\(\GL_2\))

\[\Sat(T_{(1,0)})=q^{1/2}(x_1+x_2),\qquad\Sat(Z)=\Sat(T_{(1,1)})=x_1x_2 .\]
How this is computed
  1. With \(\vol N(\cO)=1\), \(\int_N\one_{KtK}(t^\mu n)\,dn\) is the volume of \(\{n:\ t^\mu n\in KtK\}\) for \(t=\diag(\vpi,1)\).
  2. \(\mu=(1,0)\): \(\begin{pmatrix}\vpi&\vpi x\\0&1\end{pmatrix}\in KtK\) iff \(\vpi x\in\cO\), a set of volume \(q\). Weight \(\delta_B^{1/2}(t^\mu)=|\vpi|^{1/2}=q^{-1/2}\). Contribution \(q\cdot q^{-1/2}x_1\).
  3. \(\mu=(0,1)\): \(\begin{pmatrix}1&x\\0&\vpi\end{pmatrix}\in KtK\) iff \(x\in\cO\), volume \(1\). Weight \(|\vpi|^{-1/2}=q^{1/2}\). Contribution \(q^{1/2}x_2\).
  4. Sum: \(q^{1/2}x_1+q^{1/2}x_2\). For \(Z=\one_{\vpi K}\) only \(\mu=(1,1)\) contributes, with weight \(1\).

For \(\GL_n\) and the minuscule \(\omega_k=(1^k,0^{n-k})\), the same count gives \(\Sat(T_{\omega_k})=q^{k(n-k)/2}e_k(x)\); these and \(Z^{-1}\) generate \(\HK\). The 𝔸ⁿ page uses these.

Why functions on \(\As/W\)How can a space of functions on \(X(F)\) become a space of functions on \(\As\)?

Because \(\HK\) is commutative, \(\CK\) is a module over the coordinate ring \(\C[\As]^W\) of \(\As/W\), and a module over a coordinate ring is a “sheaf” on that space. The Theorem says this sheaf is, under (A1)–(A5), the structure sheaf of the subvariety \(\dXh\AXs/\WX\), pushed to \(\As/W\). Sakellaridis's notation [Sak, §8, pp. 63–64]:

\[\HK\xrightarrow{\ \Sat\ }\C[\As]^W\xrightarrow{\ \Res\ }\C[\dXh\AXs]^{\WX},\qquad\mathcal H_X:=\im(\Res\circ\Sat),\]

and \(\HK\) acts on \(\CK\) through its quotient \(\mathcal H_X\). On the support: an unramified \(\pi\) with Satake parameter \(\chi\) has \(H\)-invariant functionals only if \(\chi\in\dXh\AXs\) up to \(W\) (δ(X)). Over \(\mathcal H_X\), \(\CK\) is torsion-free [Sak08, Thm. 6.2.1], quoted in [Sak, §8] and [Sak-IS, Thm. 3.2].

The other half of Sakellaridis's paperWhat are “spherical functions on spherical varieties”?

The title of [Sak] refers to the eigenfunctions of \(\HK\) on \(C^\infty(X(F))^K\) (not compactly supported), the analogues of Macdonald's spherical functions and of the Casselman–Shalika formula. For \(\chi\) in an open dense set of \(\dXh\AXs\), Theorem 1.2.1 of [Sak] gives a basis of each eigenspace by an explicit sum over \(\WX\), and Theorem 7.2.1 rewrites its value at \(x_0\) as a quotient of L-values. The module Theorem of these pages, [Sak, Thm. 8.0.2], is deduced from that formula in Part 3 of the paper.