Spherical roots, restricted roots, and the dual group of a symmetric variety

Question

Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a semisimple simply-connected group over $k$. Consider a symmetric variety of the form $X=G/H$, for $H=G^\theta$ the fixed point subgroup of an involution $\theta \in \mathrm{Aut}_2(G)$.

There are several root systems that can be attached to the symmetric variety $X$.

1. The restricted roots. These are the roots of the form $\tfrac{1}{2}(\alpha-\alpha^\theta)$, for $\alpha$ the roots of $G$. (Here $\alpha^\theta$ denotes the image of $\alpha$ under the natural action of $\theta$ on the roots. I will denote this root system by $\Phi_\theta$ (and the corresponding simple roots by $\Delta_\theta$). This root system is not reduced in general, but one can take the "shortest" elements to make it reduced.

2. The spherical root system. Following Sakellaridis-Venkatesh (Section 2.1), the spherical root system of $X$ is spanned by the generators of the intersections of the extremal rays of the dual cone to the valuation cone of $X$ with the weight lattice of $X$. Sakellaridis and Venkatesh denote this set of simple roots by $\Sigma_X$.

3. Sakellaridis and Venkatesh further construct from $\Sigma_X$ another system of simple roots, that they denote by $\Delta_X$, and write $\Phi_X$ for the corresponding root system. I think this is what is called the "minimal root system" in Timashev's book, and the one originally considered by Brion and later generalized by Knop.

Conjecturally (from Knop-Schalke), the Gaitsgory-Nadler dual group $G^\vee_X$ of the variety $X$ is a reductive group with root system $\Phi_X^\vee$, the dual of $\Phi_X$. According to Gaitsgory-Nadler, since $X$ is a symmetric variety, $G^\vee_X$ coincides with the dual group $\check{H}_\theta$ associated to the real form $G_\mathbb{R}$ of $G$ corresponding to $\theta$.

I am confused about these root systems, when they are equal and when they differ, and how do they differ. In particular, I am trying to understand some simple examples, where the computations should be quite explicit.

I would like to understand the example where $G=\mathrm{SL}_n$, for $n=2m$ and $\theta$ is the inner involution given by conjugation by the diagonal matrix with $m$ $1$'s and $m$ $-1$'s. This involution is quasi-split, so the Levi subgroup appearing in Nadler's paper should be the entire Langlands dual group $G^\vee=\mathrm{PSL}_n$. The restricted root system $\Phi_\theta$ associated to this $\theta$ is a reduced root system of type $C_m$, and thus its Langlands dual is $B_m$. However, in Nadler's table, the dual group of this $G/G^\theta$ is precisely of type $C_m$, so $\Phi_\theta$ cannot be isomorphic to the minimal root system $\Phi_X$ yielding the dual group (in fact it is its dual). How can I obtain the system $\Phi_X$ in this case?

Answer 1

You might already have found an answer, but hopefully this helps.

The way I think about this is that there are always (at least) three normalizations of the roots to a spherical variety $X$. Some quick notation: let $A\subset B$ be a maximal torus and Borel subgroup of $G$. Let $X=G/H$ be our homogeneous spherical variety, with the following two lattices: $\mathfrak{X}\subset X^\ast(A)$ is the lattice of characters arising from $B$-semi-invariant functions and $\Lambda_X$ is the ``root lattice'' of $X$. This is the sublattice of $\mathfrak{X}$ such that $\mathfrak{X}/\Lambda_X$ is the character group of the automorphism group $Aut^G(X)$.

1. The minimal roots $\Delta_{X}$, which is defined by taking ray generators of the dual cone in $\mathfrak{X}$ to the cone of invariant valuations.
2. The normalized roots $\Delta_X^n$: this set of roots is a set of additive generators of the root lattice $\Lambda_X$ and is crucial to the calculation of $Aut^G(X)$ due to Losev.
3. The Sakellaridis-Venkatesh renormalized roots $\Delta_X^{SV}$: I think this is what you refer to in (2) of your question, which isn't quite stated correctly. What they do is take $\Delta_X$ and renormalize through elements of the root lattice of the group $G$. Sometimes this requires scaling a spherical root by $1/2$ (such as $SL_2/N$) and sometime by $2$ (such as $Spin_6/Spin_5$). The point is now the spherical roots of $X$ lie in the root lattice of $G$. By Knop--Schalke, this is the root system that $\check{G}_X$ is dual to.

Now the point is that when $X=G/G^\theta$ is symmetric, the restricted roots $\Delta_\theta$ is exactly (up to the scale by $1/2$ which is not really necessary) the set of normalized roots $\Delta_X^n$. Namely if you instead set of elements $\alpha-\theta(\alpha)$ for $\alpha\in\Delta,$ then you recover the normalized roots as a subset of the character lattice of $A$. This is explained in Knop's ``Automorphisms, root systems, and compactifications of homogeneous varieties'' paper, and also in Timashev's book.

Calculationally, we can start by calculating $\Delta_X^n=\Delta_\theta$ as above (though you need to make sure that $A$ is in the correct relative position with $\theta$: it needs to be a maximally $\theta$-split maximal torus). Now we can recover $\Delta_X$ by locating which normalized roots are twice a ``distinguished root'' and rescaling it with $1/2$: this has to do with locating which roots in $\Delta_X^n$ which can be halved without violating Luna's axioms. For example, in the example you give with $G=SL_{2m}$ and $X=SL_{2m}/S(GL_m\times GL_m)$, if we use the standard numbering of the roots, then $$\Delta_X^n = \{\alpha_1+\alpha_{2m-1},\ldots, \alpha_{m-1}+\alpha_{m+1}, 2\alpha_m\}$$ is of Cartan type $C_m$ while $$\Delta_X = \{\alpha_1+\alpha_{2m-1},\ldots, \alpha_{m-1}+\alpha_{m+1}, \alpha_m\}$$ is type $B_m$. Here the root $\alpha_m$ is distinguished in the sense of Losev. In this case $\Delta_X=\Delta_X^{SV}$, so $\check{G}_X=Sp_{2m}$.

In general, computing $\Delta_X^{SV}$ from $\Delta_X$ requires further scaling if there are spherical roots of the form $2\alpha$ which are not distinguished (replacing $SL_{2m}/S(GL_m\times GL_m)$ with the (disconnected) normalizer of the Levi is an example of this) as well as sometimes scaling a sum $\frac{1}{2}(\alpha_1+\alpha_2)$ by $2$. This ensures the existence of a morphism $\check{G}_X\to \check{G}$, but means that this map will not be injective in general.

Answer 2

There is a small survey published in an Oberwolfach report by Bart Van Steirteghem comparing the different normalizations of spherical roots in more detail. The standard normalization is obtained by decreeing spherical roots to be primitive in the weight lattice of the spherical variety of $X$. On the other hand, Sakellaridis and Venkatesh want spherical roots to be primitive in the root lattice of $G$. As opposed to the standard normalization this leads to a dual group of $X$ which maps to the dual group of $G$. On the other hand, SV-spherical roots are in general not weights of $X$. A third difference is that the SV-root system is invariant under finite maps $X\to X'$ while the standard root system is not.