[Paper Review] Satellites of spherical subgroups

Victor V. Batyrev, Anne Moreau|arXiv (Cornell University)|Oct 24, 2016

Advanced Algebra and Geometry29 references3 citations

TL;DR

This paper introduces the concept of 'satellites'—spherical subgroups $ H_I \subset G $ of the same dimension as a given spherical subgroup $ H \subset G $, defined via subsets $ I $ of the spherical roots of $ G/H $. It establishes a deep connection between the Poincaré polynomials of $ G/H $ and $ G/H_I $, showing that their ratio is a polynomial in $ t^{-1} $ with integer coefficients, and provides explicit formulas for this ratio in various cases, including symmetric spaces and rank-one spherical varieties.

ABSTRACT

Let $G$ be a complex connected reductive algebraic group. Given a spherical subgroup $H \subset G$ and a subset $I$ of the set of spherical roots of $G/H$, we define, up to conjugation, a spherical subgroup $H_I \subset G$ of the same dimension of $H$, called a satellite. We investigate various interpretations of the satellites. We also show a close relation between the Poincaré polynomials of the two spherical homogeneous spaces $G/H$ and $G/H_I$.

Motivation & Objective

• To define and systematically study a new class of spherical subgroups, called satellites, associated with subsets of spherical roots of a given spherical homogeneous space $ G/H $.
• To establish a combinatorial and geometric framework for understanding how the spherical data of $ H $ and its satellite $ H_I $ relate under conjugation.
• To investigate the relationship between the Poincaré polynomials of $ G/H $ and $ G/H_I $, particularly their ratio, which is shown to be a polynomial in $ t^{-1} $ with integer coefficients.
• To provide explicit computations of the Poincaré polynomial ratio $ R_\varnothing(t^{-1}) $ for various spherical varieties, including symmetric spaces and rank-one cases.
• To unify and extend previous results on spherical varieties using the satellite construction, especially in the context of wonderful embeddings and parabolic stabilizers.

Proposed method

• Define the satellite $ H_I \subset G $ of a spherical subgroup $ H \subset G $ for any subset $ I $ of the spherical roots of $ G/H $, up to conjugation, preserving the dimension of $ H $.
• Use the Luna-Vust theory of colored fans and the valuation cone $ \mathcal{V} \subset N_{\mathbb{Q}} $ to describe the combinatorial structure of $ H_I $, leveraging the spherical roots $ s_1, \dots, s_k $ as generators of the dual cone.
• Relate the Poincaré polynomial $ \tilde{P}_{G/H}(t) $ of $ G/H $ to that of its satellite $ G/H_I $ via the formula $ \tilde{P}_X(t) = \tilde{P}_{G/H}(t) + \tilde{P}_{X'}(t) = \tilde{P}_{G/H}(t) + \frac{\tilde{P}_{G/H_\varnothing}(t)}{t-1} $, where $ X $ is a wonderful embedding with closed orbit $ X' \cong G/P $.
• Compute $ \tilde{P}_{G/H_\varnothing}(t) $ using the relation $ \tilde{P}_{G/H_\varnothing}(t) = \tilde{P}_{G/P}(t)(t-1) $, where $ P $ is a parabolic subgroup determined by dimension and Levi-type constraints.
• Apply known formulas for $ \tilde{P}_{G/B}(t) $ and $ \tilde{P}_{P/B}(t) $ in terms of exponents of the root system to derive $ \tilde{P}_{G/P}(t) $, and hence $ \tilde{P}_{G/H_\varnothing}(t) $, in specific cases.
• Derive the ratio $ R_\varnothing(t^{-1}) = \frac{\tilde{P}_{G/H_\varnothing}(t)}{\tilde{P}_{G/H}(t)} $ explicitly for 15 rank-one spherical varieties, showing it is always a polynomial in $ t^{-1} $ with integer coefficients.

Experimental results

Research questions

• RQ1 How can one construct, up to conjugation, a spherical subgroup $ H_I \subset G $ of the same dimension as a given spherical subgroup $ H \subset G $, indexed by a subset $ I $ of the spherical roots of $ G/H $?
• RQ2 What is the precise relationship between the Poincaré polynomials of $ G/H $ and its satellite $ G/H_I $, particularly in terms of their ratio?
• RQ3 In which cases is the ratio $ R_\varnothing(t^{-1}) $ of the Poincaré polynomials a polynomial in $ t^{-1} $ with integer coefficients?
• RQ4 Can the Poincaré polynomial of the satellite $ G/H_\varnothing $ be computed directly from the geometry of the wonderful embedding of $ G/H $, especially via the closed orbit $ X' \cong G/P $?
• RQ5 How do the combinatorial invariants—spherical roots, valuation cone, colored fan—of $ G/H $ and $ G/H_I $ relate under the satellite construction?

Key findings

• The satellite $ H_I $ of a spherical subgroup $ H \subset G $ is well-defined up to conjugation and has the same dimension as $ H $, with its spherical data determined by a subset $ I $ of the spherical roots of $ G/H $.
• The ratio $ R_\varnothing(t^{-1}) = \frac{\tilde{P}_{G/H_\varnothing}(t)}{\tilde{P}_{G/H}(t)} $ is a polynomial in $ t^{-1} $ with integer coefficients for all 15 rank-one spherical varieties listed in Table 1.
• For the case $ G = \mathbf{F}_4 $, $ H = \mathbf{B}_4 $, the ratio is $ R_\varnothing(t^{-1}) = 1 - t^{-8} $, computed via dimension constraints on the parabolic subgroup $ P $ with $ \dim G/P = 15 $.
• In the case $ G = \mathbf{G}_2 $, $ H = \mathrm{GL}_2 \ltimes (\mathbb{C} \oplus \mathbb{C}^2) \otimes \wedge^2 \mathbb{C}^2 $, the ratio is $ R_\varnothing(t^{-1}) = 1 - t^{-2} $, derived from the Poincaré polynomial of $ G/H_\varnothing $ and $ G/H $.
• For symmetric spaces such as $ G = \mathrm{SL}_n $, $ H = S(L_1 \times L_{n-1}) $, the ratio is $ R_\varnothing(t^{-1}) = 1 - t^{-(n-1)} $, showing a general pattern for this family.
• The construction of satellites and the resulting polynomial ratio $ R_\varnothing(t^{-1}) $ is consistent across cases with connected $ H $, and can be computed either via the wonderful embedding or directly from the parabolic stabilizer $ P $ of the closed orbit $ X' \cong G/P $.