The little Weyl group \(\WX\)
The cone \(\Vc\) of the torus page is a fundamental domain for a reflection group \(\WX\). The Theorem takes \(\WX\)-invariants, and whether \(\WX\)-orbits stay apart inside \(\As/W\) decides whether the basic function generates.
DefinitionWhich reflections of \(\LX\) does the geometry of \(X\) remember?
[SV, §2.1, p. 20; Sak, §2.2] Definition
The little Weyl group \(\WX\) is the finite group generated by reflections of \(\LX\otimes\Q\) that has the cone \(\Vc\) as a fundamental domain. Knop's description [Sak, §2.2, p. 9]: \(W\) acts on the set of \(B\)-orbits of maximal rank, transitively, and the stabilizer of the open orbit \(\Xo\) is
\[W(X)=\WX\ltimes W_{\PX},\]which acts on \(\Xc(X)\) through its quotient \(\WX\).
Two extreme cases. If \(\Vc\) is all of \(\LX\otimes\Q\), then \(\WX=1\); this happens exactly for \(X\) (\(\GL_2/\SL_2\), \(\Aff^n\smallsetminus\{0\}\)). If \(X\) is symmetric, \(\WX\) is the restricted Weyl group \(N_G(S)/Z_G(S)\) of a maximal \(\theta\)-split torus \(S\).
Spherical rootsWhat are the walls of \(\Vc\)?
[SV, §2.1–2.2; Sak, §6.1] Definition
The spherical roots \(\Sigma_X\subseteq\Xc(X)\) are the primitive elements on the extremal rays of the cone \(\{\chi:\ \langle v,\chi\rangle\le0\ \forall v\in\Vc\}\); so
\[\Vc=\{v\in\LX\otimes\Q:\ \langle v,\gamma\rangle\le0\ \ \forall\gamma\in\Sigma_X\}.\]The normalized spherical root is the positive multiple of \(\gamma\) that is a root of \(G\) or a sum of two orthogonal roots; the normalized spherical roots \(\Delta_X\) are the simple roots of a root system with Weyl group \(\WX\) [SV, Prop. 2.2.1].
Each spherical root has a type. Over \(\bar F\), for a simple root \(\alpha\) and a \(B\)-orbit \(Y\) of maximal rank, \(YP_\alpha/R(P_\alpha)\) is a homogeneous \(\PGL_2\)-variety of one of four types [Sak, §2.2, p. 9]: \(G\) (\(\PGL_2\backslash\PGL_2\)), \(U\) (\(SU\backslash\PGL_2\)), \(T\) (\(T\backslash\PGL_2\)) or \(N\) (\(N(T)\backslash\PGL_2\)). Type \(N\) is excluded by the Theorem's hypothesis (A4) (Hypotheses).
ExamplesWhat does \(\WX\) do to \(\AXs\)?
| \(X\) | \(\WX\) | action |
|---|---|---|
| \(\GL_2/\SL_2\), \(\Aff^n\smallsetminus\{0\}\) | \(1\) | — |
| \(\GL_2/T\) | \(\Z/2\) | \(y\mapsto y^{-1}\); root \(\epsilon_1-\epsilon_2\), type \(T\) |
| group \(H\) under \(H\times H\) | \(W_H\) | diagonally on \((y;y^{-1})\) |
| \(\GL_{2n}/\Sp_{2n}\) | \(S_n\) | permutes the pairs; restricted roots \(A_{n-1}\) |
| \(\GL_{p+q'}/(\GL_p\times\GL_{q'})\) | \(W(C_p)\) | permutations and \(y_i\mapsto y_i^{-1}\) |
| \(E_6/F_4\) | \(S_3\) | restricted roots \(A_2\), each of multiplicity \(8\) |
\(\WX\) and the surjectivity of \(\Res\)Why invariants under \(\WX\) and not under all of \(W\)?
The Theorem compares \(\HK\simeq\C[\As]^W\) with \(\C[\dXh\AXs]^{\WX}\) through restriction \(\Res\) (the Theorem). Rings of invariants of finite groups are coordinate rings of quotients, so \(\Res\) is the pullback along
\[\iota:\ \dXh\AXs/\WX\longrightarrow\As/W .\](surjectivity is geometric) Proved here
\(\Res\) is surjective if and only if \(\iota\) is a closed immersion. In particular, if two points of \(\dXh\AXs\) that are not \(\WX\)-conjugate are \(W\)-conjugate, then \(\Res\) is not surjective.
Proof
A morphism of affine varieties is a closed immersion iff the ring map is surjective; \(\Res\) is that ring map. A closed immersion is injective on points; two \(\WX\)-inequivalent points with the same image in \(\As/W\) violate injectivity.
Sakellaridis's Example 8.0.1 is this situation: \(\WX\) equals the normalizer of \(\AXs\) in \(W\), yet “\(\WX\)-nonconjugate points of \(\AXs\) … can be conjugate under \(W\)” [Sak, p. 64]. For \(E_6/F_4\), two such points are \(y_1\) and \(\sigma y_1\) (computing r). Injectivity on points is necessary, not sufficient: a closed immersion must also be injective on tangent vectors.