[Paper Review] Homological properties of representations of p-adic groups related to geometry of the group at infinity

Roman Bezrukavnikov|ArXiv.org|Jun 10, 2004

Advanced Algebra and Geometry11 references16 citations

TL;DR

This paper establishes a canonical isomorphism between homological duality on the derived category of smooth representations of a p-adic group and the composition of the Deligne-Lusztig functor with Grothendieck-Serre duality, using a novel functor linking equivariant sheaves on the Bruhat-Tits building to the Jacquet functor via the Borel-Serre compactification. It further proves Kazhdan's conjecture on the Euler characteristic of Yoneda Ext groups as an integral of character products over elliptic conjugacy classes.

ABSTRACT

Geometry of buildings is used to prove some homological properties of the category of smooth representations of a reductive p-adic group (Kazhdan's "pairing conjecture", Bernstein's description of homological duality in terms of Deligne-Lusztig duality). A different proof had been obtained a little earlier by Schneider and Stuhler.

Motivation & Objective

• To establish a canonical isomorphism between homological duality on smooth G-modules and the composition of the Deligne-Lusztig functor with Grothendieck-Serre duality.
• To prove Kazhdan's conjecture expressing the Euler characteristic of Yoneda Ext groups between admissible representations as an integral of character products over elliptic conjugacy classes.
• To construct a functor L on the opposite category of smooth representations that realizes the boundary restriction of the direct image of equivariant sheaves on the Bruhat-Tits building under the Borel-Serre compactification.
• To clarify the geometric and homological structure of representations via sheaf-theoretic localization and Verdier duality on the building.

Proposed method

• Constructs an exact functor L: M^opp → Sh_G(∂X) such that i*∘j_*^G ≅ L∘Γ, where j: X → X̄ is the Borel-Serre compactification and i: ∂X → X̄ is the boundary inclusion.
• Uses the derived functor of compactly supported sections RΓ_c and Verdier duality RΓ to relate sheaf cohomology on the building X to derived categories of smooth G-modules.
• Establishes that the stalk of L(M) at a boundary point y ∈ ∂X is isomorphic to the Jacquet functor r_{P_y}(M), where P_y is the stabilizer of y.
• Applies a 'general nonsense' localization argument to show that for any B ∈ D^b(M) and i ∈ ℤ, there exists A ∈ D^b(S𝔥) and a morphism RΓ_c(A) → B inducing isomorphism on cohomology H^j for j > i.
• Uses the Bernstein-Bezrukavnikov duality isomorphism D_h ≅ DL∘D_Gr, where D_h is homological duality and D_Gr is Grothendieck-Serre duality.
• Employs Harish-Chandra's integrability of characters and invariant generalized functions to reduce the Euler characteristic to integrals over regular elliptic and non-elliptic sets, showing vanishing on non-elliptic parts via a homological argument involving non-compact centers.

Experimental results

Research questions

• RQ1 Is there a canonical isomorphism between homological duality on smooth G-modules and the composition of the Deligne-Lusztig functor with Grothendieck-Serre duality?
• RQ2 Can the Euler characteristic of Yoneda Ext groups between admissible representations be expressed as an integral of character products over elliptic conjugacy classes?
• RQ3 How does the boundary of the Borel-Serre compactification of the Bruhat-Tits building relate to the Jacquet functor on representations?
• RQ4 What is the precise functorial relationship between equivariant sheaves on the building and the derived category of smooth representations?
• RQ5 Does the vanishing of character integrals over non-elliptic sets imply the vanishing of Ext groups in the non-compact center case?

Key findings

• The paper establishes a canonical isomorphism D_h ≅ DL∘D_Gr between homological duality and the composition of Deligne-Lusztig and Grothendieck-Serre duality functors on the derived category of smooth G-modules.
• It proves that the restriction of the direct image of a G-equivariant simplicial sheaf on the Bruhat-Tits building to the Borel-Serre boundary is isomorphic to the functor L applied to the global sections of the sheaf.
• The functor L is shown to be exact and to act on admissible representations by taking the Jacquet functor at each boundary point, with the stabilizer parabolic P_y determining the Jacquet module r_{P_y}(M).
• The Euler characteristic of Ext groups between two admissible representations is given by ∫_Ell χ_ρ₁(g⁻¹)χ_ρ₂(g) dμ(g), proving Kazhdan’s conjecture.
• The integral of any invariant generalized function supported on regular non-elliptic elements against the character of an admissible representation vanishes, which is used to eliminate the non-elliptic contribution to the Euler characteristic.
• In the case where G has non-compact center, the class of any admissible representation in the Grothendieck group K^0(M) vanishes, implying that the alternating sum of Ext dimensions is zero for such representations.