A Satake isomorphism in characteristic p

Abstract

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic Langlands correspondences for GL_2(Q_p), in generalisations of Serre's conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

24 pages, revised

References in corpus (2)

• The classification of irreducible admissible mod p representations of a p-adic GL_n
• R'esolutions flasques des groupes lin'eaires connexes

Cited by in corpus (31)

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• Eigenvarieties and invariant norms: Towards p-adic Langlands for U(n)
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• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• The geometric Satake equivalence for integral motives
• Freeness of spherical Hecke modules of unramified $U(2,1)$ in characteristic $p$
• A classification of the irreducible admissible genuine mod p representations of the metaplectic cover of p-adic SL(2)
• Arithmetic Satake compactifications and algebraic Drinfeld modular forms
• Generic and Mod p Kazhdan-Lusztig Theory for GL_2
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• A Satake homomorphism for the $\mathrm{mod}p$ derived Hecke algebra
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• Geometrization of the Satake transform for mod $p$ Hecke algebras
• The Left Adjoint of Derived Parabolic Induction
• Ramified Satake Isomorphisms for strongly parabolic characters
• Inverse Satake isomorphism and change of weight
• Serre's Modularity Conjecture
• Mod $p$ Bernstein centres of $p$-adic groups
• Irreducible $p$-modular representations of unramified $U(2,1)$
• On the pro-$p$-Iwahori invariants of supersingular representations of unramified $U(2,1)$
• Restriction of $p$-modular representations of $U(2,1)$ to a Borel subgroup
• Mod-$p$ maximal compact inductions do not have irreducible admissible subrepresentations
• On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras
• Irreducible admissible mod-p representations of metaplectic groups