Localization of Singularities and Universal Geometric Rank Bounds in the Satake Correspondence
George Petroulakis
TL;DR
This work develops a Geometric Satake–inspired framework to localize and isolate singularities in the affine Grassmannian by factorizing the transition matrix C between the MV basis and the convolution basis as C = P · M · A · Q^{-1}. The key novelty is separating local IC contributions (M) from global combinatorics (A, Q, P) via Beilinson–Drinfeld fusion and nearby cycles, with M computed through Braden–MacPherson stalks on a moment graph. A universal geometric rank bound is proved: rank(C) is bounded by the dimension of local BMP stalks, captured by a geometric efficiency η(α,ν) = dim(M_α(ν) ⊗ C)/dim(V_λ ⊗ V_μ)_ν, which implies asymptotic sparsity η ≤ ℓ/(ℓ^2 + |Φ|) for simply-laced G. In the E6 case, the adjoint model demonstrates strong filtering (η = 1/18), and the bounds extend to E6–E8 with explicit constants, while Aℓ and Dℓ exhibit vanishing η as ℓ → ∞. Overall, the paper provides a modular, geometry-driven route to compute and bound transition matrices in the Satake context, revealing that local singularities govern global multiplicities and yield sparser structures in exceptional types.”
Abstract
This article introduces a framework for the localization and isolation of singularities in the affine Grassmannian. Our primary result is a structural factorization of the transition matrix $C$ between the Mirković--Vilonen (MV) basis and the convolution basis into $C = P \cdot M \cdot A \cdot Q^{-1}$, where the four factors represent: equivariant localization ($Q$), fusion via nearby cycles ($A$), local intersection cohomology stalks ($M$), and diagonal normalization ($P$). Utilizing this factorization, and by introducing the Geometric Efficiency metric ($η$) we establish a Universal Geometric Rank Bound, proving that the rank of the transition matrix $C$ is bounded by the dimension of the local Braden--MacPherson (BMP) stalks.