[Paper Review] A note on Integral Satake isomorphisms

Xinwen Zhu|arXiv (Cornell University)|May 26, 2020

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper formulates a canonical integral Satake isomorphism for the spherical Hecke algebra of an unramified p-adic group G, identifying it with a Z-algebra associated to an affine monoid derived from the Langlands dual group and the cocharacter ρad. The construction generalizes to Hecke algebras of arbitrary weights via Vinberg monoids, providing a uniform integral description that specializes to both the classical and mod p Satake isomorphisms, and is compatible with the geometric Satake equivalence over Fp.

ABSTRACT

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.

Motivation & Objective

• To formulate a canonical integral Satake isomorphism for the spherical Hecke algebra HG of an unramified p-adic group G over Z, avoiding dependence on q^{1/2}.
• To generalize the Satake isomorphism to Hecke algebras HG(V) for lattices V in irreducible algebraic representations of G, parameterized by highest weights λ.
• To establish compatibility with the geometric Satake equivalence, particularly in the mod p setting, by relating the Hecke algebra to representations of an affine monoid V_{Ĝ,ρad}.
• To unify the classical Satake isomorphism (over Z[q±1/2]) and the mod p Satake isomorphism via a single integral framework using the Vinberg monoid construction.

Proposed method

• Define the C-group CG = Ĝ ⋊ (Gm × Γ̃F/F) as a modification of the Langlands dual group, incorporating the cocharacter ρad and Galois action.
• Construct the affine monoid V_{Ĝ,ρad} as the pullback of the Vinberg monoid of Ĝ via the homomorphism dρad : V_{Ĝ,ρad} → A1, generalizing the dual group ĜT.
• Use the representation theory of V_{Ĝ,ρad} to describe the Grothendieck group of representations, identifying it with the integral Hecke algebra HG via trace maps.
• Establish a canonical isomorphism between the Grothendieck group of integral representations of CG and the integral Hecke algebra HG, using the trace of Frobenius and compatibility with geometric Satake.
• Prove that the ring of invariant functions on V_{Ĝ,ρad} specializes to all HG(V) for lattices V in irreducible representations of G.
• Verify compatibility with the geometric Satake equivalence by showing that the category of integral representations of CG maps fully faithfully to the category of perverse sheaves on the affine Grassmannian, with trace maps inducing the Satake isomorphism.

Experimental results

Research questions

• RQ1 Can the classical Satake isomorphism be formulated integrally over Z, independent of q^{1/2}?
• RQ2 How can the Satake isomorphism be generalized to Hecke algebras of arbitrary weights, not just the spherical case?
• RQ3 What is the role of the Vinberg monoid V_{Ĝ,ρad} in realizing the integral Satake isomorphism?
• RQ4 How does the integral Satake isomorphism relate to the geometric Satake equivalence over Fp, as studied by Cass?
• RQ5 Is there a canonical Z-algebra description of HG(V) that specializes to both the classical and mod p Satake isomorphisms?

Key findings

• The paper constructs a canonical integral Satake isomorphism HG ≃ Z[V_{Ĝ,ρad}|dρad=q]^{cσ(Ĝ)}, where V_{Ĝ,ρad} is an affine monoid associated to the Langlands dual group and the cocharacter ρad.
• The isomorphism is independent of q^{1/2} and specializes to the classical Satake isomorphism after inverting q^{1/2}, and to the mod p Satake isomorphism after reduction modulo p.
• The construction generalizes to Hecke algebras HG(V) for lattices V in irreducible representations of G, with the isomorphism taking the form HG(V) ≃ Z[V_{Ĝ,λad+ρad}|dρad=q]^{cσ(Ĝ)}.
• The ring of invariant functions on V_{Ĝ,ρad} surjects onto all HG(V), showing that this single monoid encodes the integral Hecke algebras for all weights.
• The geometric Satake equivalence induces an equivalence between the category of integral representations of CG and the category of perverse sheaves on the affine Grassmannian with integral structure, compatible with the trace map.
• The trace map from the Grothendieck group of integral representations of CG to HG is surjective, and the diagram involving geometric Satake and classical Satake commutes, confirming integrality and compatibility.