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Classify all representation which contains a Steinberg in its hyperspecial subgroup

Runze Wang

Abstract

This paper addresses Question 1 posed by Dipendra Prasad in his recent problem list: classify all irreducible smooth representations of an unramified reductive p-adic group such that the space of vectors fixed by the pro-unipotent radical of a hyperspecial maximal compact subgroup, viewed as a representation of the finite reductive group obtained as the quotient of that hyperspecial subgroup by its pro-unipotent radical, contains the Steinberg representation. We prove that any such representation must be Iwahori-spherical, hence a subquotient of some unramified principal series. By a detailed analysis of the action of the Iwahori--Hecke algebra on the Iwahori-fixed space, we show that in each principal series there exists exactly one irreducible subquotient containing the Steinberg representation, and we give an explicit construction of this subquotient. Hence this gives a bijection between Weyl group orbits of unramified characters and irreducible representations containing the Steinberg representation in their hyperspecial subgroup. This classification is richer than Prasad's expectation: besides twists of the Steinberg representation and generic unramified representations, every principal series contributes a unique subquotient with the required property. The proof relies on establishing an isomorphism between the finite Iwahori--Hecke algebra of the finite Lie group obtained as the quotient of the hyperspecial subgroup by its pro-unipotent radical and the finite Hecke algebra of the p-adic group consisting of compactly supported I-bi-invariant functions on the hyperspecial subgroup, and comparing their actions on the Iwahori-fixed vectors.

Classify all representation which contains a Steinberg in its hyperspecial subgroup

Abstract

This paper addresses Question 1 posed by Dipendra Prasad in his recent problem list: classify all irreducible smooth representations of an unramified reductive p-adic group such that the space of vectors fixed by the pro-unipotent radical of a hyperspecial maximal compact subgroup, viewed as a representation of the finite reductive group obtained as the quotient of that hyperspecial subgroup by its pro-unipotent radical, contains the Steinberg representation. We prove that any such representation must be Iwahori-spherical, hence a subquotient of some unramified principal series. By a detailed analysis of the action of the Iwahori--Hecke algebra on the Iwahori-fixed space, we show that in each principal series there exists exactly one irreducible subquotient containing the Steinberg representation, and we give an explicit construction of this subquotient. Hence this gives a bijection between Weyl group orbits of unramified characters and irreducible representations containing the Steinberg representation in their hyperspecial subgroup. This classification is richer than Prasad's expectation: besides twists of the Steinberg representation and generic unramified representations, every principal series contributes a unique subquotient with the required property. The proof relies on establishing an isomorphism between the finite Iwahori--Hecke algebra of the finite Lie group obtained as the quotient of the hyperspecial subgroup by its pro-unipotent radical and the finite Hecke algebra of the p-adic group consisting of compactly supported I-bi-invariant functions on the hyperspecial subgroup, and comparing their actions on the Iwahori-fixed vectors.
Paper Structure (14 sections, 17 theorems, 28 equations)

This paper contains 14 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Unrefined minimal $K$-types and the Moy–Prasad filtration
  3. Moy–Prasad filtrations
  4. Unrefined minimal $K$-types
  5. The main theorem on existence and depth
  6. The cuspidal support of the irreducible components of $K/K_{+}$
  7. Some facts about the unramified principal series
  8. Iwahori fixed subspace of an unramified principal series
  9. Effect of affine reflections on the Iwahori-fixed vectors
  10. Representation theory of the finite Lie group
  11. Relations between finite Lie group and $p$-adic group
  12. The classification of irreducible representations containing Steinberg
  13. Main Theorem
  14. Another Perspective: generic representation

Key Result

Theorem 2.4

Assume that $k$ is of characteristic zero. Let $(\pi,V_\pi)$ be an irreducible admissible complex representation of $G(k)$. Then there exists a non-negative rational number $\rho(\pi)$ with the following properties: Moreover, any two minimal $K$-types contained in $\pi$ are associates of each other.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2: MoyPrasad1994, 5.1
  • Definition 2.3: MoyPrasad1996
  • Theorem 2.4: MoyPrasad1994
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3: Borel–Casselman, Borel1976
  • Corollary 3.4
  • ...and 19 more