[Paper Review] Involutions on pro-$p$-Iwahori Hecke algebras

Noriyuki Abe|arXiv (Cornell University)|Apr 3, 2017

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper computes the action of two key algebraic involutions—$ζ$ (anti-involution) and $ι$ (involution) — on simple modules over pro-$p$-Iwahori Hecke algebras in positive characteristic. It determines the twisted module $π^{\iota}$ and the dual module $π^{*}$ for simple modules $π$, using a three-step classification: parabolic inductions, generalized Steinberg modules, and supersingular modules, with the hardest case resolved via detailed braid and quadratic relations. The key result is an explicit isomorphism classifying the dual and twisted modules in terms of dual parameters and opposite parabolic subgroups.

ABSTRACT

The pro-$p$-Iwahori Hecke algebra has an involution $ι$ defined in terms of Iwahori-Matsumoto basis. Then for a module $π$ of pro-$p$-Iwahori Hecke, $π^ι= π\circ ι$ is also a module. We calculate $π^ι$ for simple modules $π$. We also calculate the dual of $π$.

Motivation & Objective

• To compute the action of the involution $ι$ on simple modules of pro-$p$-Iwahori Hecke algebras over a field of characteristic $p$.
• To determine the dual module $π^{*}$ for simple modules $π$ in the same setting.
• To extend the representation theory of $G$ over fields of positive characteristic by analyzing module structures under $ι$ and $ζ$.
• To provide foundational data for computing extensions between simple modules, as part of a broader program in representation theory.

Proposed method

• Uses the Iwahori-Matsumoto basis $π = \{T_w\}_{w\in W(1)}$ to define the involution $ι$ via $T_w \mapsto (-1)^{\ell(w)}T_w^{*}$, where $T_w^{*}$ is a dual basis.
• Applies the anti-involution $ζ$ defined by $T_w \mapsto T_{w^{-1}}$ to define the dual module $\pi^{*} = \mathrm{Hom}_C(\pi, C)$ with action $(fX)(v) = f(v\zeta(X))$.
• Classifies simple modules in three steps: parabolic inductions, generalized Steinberg modules (most complex), and supersingular modules, using the structure of the pro-$p$ Weyl group $W(1)$ and reflection data.
• Employs braid relations and quadratic relations in the Hecke algebra to compute module twists, particularly for generalized Steinberg modules via detailed analysis of the negative algebra and extension functors.
• Uses the adjoint action of Weyl group elements (e.g., $n_{w_G w_P}$) to relate modules over opposite parabolic subgroups and establish isomorphisms between twisted and dual modules.
• Applies pull-back constructions and compatibility of extension functors $j_{Q}^{P-*}$ to relate $\pi^{*}$ to modules over opposite parabolic subgroups with dual parameters.

Experimental results

Research questions

• RQ1 How does the involution $\iota$ act on simple modules of the pro-$p$-Iwahori Hecke algebra when the base field has characteristic $p$?
• RQ2 What is the structure of the dual module $\pi^{*}$ for a simple module $\pi$ over the pro-$p$-Iwahori Hecke algebra?
• RQ3 How does the action of $\iota$ on generalized Steinberg modules compare to that on parabolic inductions and supersingular modules?
• RQ4 Can the dual of a simple module be expressed in terms of a module over the opposite parabolic subgroup with dual parameters?
• RQ5 What is the relationship between $I(P;\chi,J,V;Q)^{*}$ and $I(P';\chi',J',V';Q')$ under the involution and duality?

Key findings

• For a simple module $\pi = I(P;\chi,J,V;Q)$, the twisted module $\pi^{\iota}$ is isomorphic to $I(P;\chi^{-1},J,V^{*};Q)$, where $V^{*}$ is the dual representation and $\chi^{-1}$ is the inverse character.
• The dual module $\pi^{*}$ is isomorphic to $I(P';\chi',J',V';Q')$, where $P' = n_{w_G w_P} P^{\mathrm{op}} n_{w_G w_P}^{-1}$, $Q' = n_{w_G w_Q} Q^{\mathrm{op}} n_{w_G w_Q}^{-1}$, and $(\chi',J',V')$ is the pull-back of $(\chi^{-1},J,V^{*})$ via $n_{w_G w_P}$.
• The computation of $\pi^{\iota}$ for generalized Steinberg modules is the most technical part, relying on detailed analysis of the negative algebra and compatibility of extension functors.
• The duality result is established via the adjoint action of Weyl group elements, which induces an isomorphism between $\mathrm{St}_Q^{P}(\sigma)^{*}$ and $\mathrm{St}_{Q'}^{P'}(n_{w_G w_P} \sigma^{*})$.
• The isomorphism $I(P;\chi,J,V;Q)^{*} \simeq I(P';\chi',J',V';Q')$ holds due to compatibility of the negative algebra and extension functors under conjugation by $n_{w_G w_P}$.
• The results are consistent with the structure of the pro-$p$ Weyl group $W(1)$ and the action of $W$ on $Z_\kappa$, ensuring that the parameter transformations are well-defined and preserve the module structure.