[Paper Review] Cohomological representations of parahoric subgroups

Charlotte Chan, Alexander B. Ivanov|arXiv (Cornell University)|Mar 14, 2019

Advanced Algebra and Geometry19 references4 citations

TL;DR

This paper generalizes Lusztig's cohomological construction of representations from hyperspecial to arbitrary parahoric subgroups of a reductive group over a local field splitting over an unramified extension. It defines virtual representations via the cohomology of a Deligne–Lusztig-type variety associated to a torus and unipotent radical, proving irreducibility under genericity conditions and computing character formulas for unramified very regular elements using the Deligne–Lusztig trace formula.

ABSTRACT

We generalize a cohomological construction of representations due to Lusztig from the hyperspecial case to arbitrary parahoric subgroups of a reductive group over a local field, which splits over an unramified extension. We compute the character of these representations on certain very regular elements.

Motivation & Objective

• To extend Lusztig's cohomological construction of representations from hyperspecial to arbitrary parahoric subgroups of reductive groups over local fields.
• To define and study a tower of varieties over the residue field whose cohomology realizes representations of the parahoric subgroup $P(\mathcal{O}_k)$.
• To compute the character of these representations on unramified very regular elements, generalizing classical Deligne–Lusztig character formulas.
• To establish conditions under which the constructed representations are irreducible and independent of auxiliary choices such as the Borel subgroup.

Proposed method

• Construct a scheme $S_{T,U} \subset \mathbb{G}$, the preimage of $\mathbb{U}$ under the Lang map $g \mapsto g^{-1}\sigma(g)$, where $\mathbb{G}$ is the reductive quotient of the special fiber of a parahoric $P$.
• Define the virtual $P(\mathcal{O}_k)$-representation $R_{T,U}^\theta$ as the $\theta$-isotypic component of the alternating sum of cohomology groups of $S_{T,U}$ with $\overline{\mathbb{Q}}_\ell$-coefficients.
• Use the Deligne–Lusztig fixed point formula to compute the trace of unramified very regular elements on $R_{T,U}^\theta$, reducing the problem to fixed-point sets of group actions.
• Apply a decomposition of group elements into $p$-power and prime-to-$p$ parts to separate contributions in the trace formula, leveraging the structure of $\breve{T}_r^\sigma$.
• Prove that the fixed-point set $S_{T,U}^{(s,\zeta)}$ is non-empty precisely when $\zeta = \mathrm{Ad}(w^{-1})(s^{-1})$ for some $\sigma$-invariant $w$ in the Weyl group of $T$ in $Z^0(g)$.
• Establish irreducibility of $\pm R_{T,U}^\theta$ when the stabilizer of $\theta$ in the Weyl group is trivial, using a generalization of Lusztig's original method.

Experimental results

Research questions

• RQ1 Under what conditions is the cohomologically constructed representation $R_{T,U}^\theta$ independent of the choice of Borel subgroup $U$?
• RQ2 When is the representation $R_{T,U}^\theta$ irreducible, and what is the role of the stabilizer of $\theta$ in the Weyl group?
• RQ3 Can the character of $R_{T,U}^\theta$ on unramified very regular elements be computed explicitly, and how does it relate to the Weyl group action?
• RQ4 How does the cohomology of the Deligne–Lusztig-type variety $S_{T,U}$ realize representations of $P(\mathcal{O}_k)$ for non-reductive parahorics?
• RQ5 What is the geometric and arithmetic structure of the fixed-point sets $S_{T,U}^{(g,\mathbb{T})}$ under the action of $g \in P(\mathcal{O}_k)$?

Key findings

• The representation $R_{T,U}^\theta$ is independent of the choice of Borel subgroup $U$ when $\theta$ is sufficiently generic.
• If the stabilizer of $\theta$ in the Weyl group of the special fiber of $P$ is trivial, then $\pm R_{T,U}^\theta$ is an irreducible representation of $P(\mathcal{O}_k)$.
• The character of $R_{T,U}^\theta$ on an unramified very regular element $g$ is given by $\operatorname{Tr}(g, R_{T,U}^\theta) = \sum_{w \in W_{\mathbf{x}}(T,Z^\circ(g))^{\sigma}} (\theta \circ \mathrm{Ad}(w^{-1}))(g)$.
• The fixed-point set $S_{T,U}^{(s,\zeta)}$ is non-empty if and only if $\zeta = \mathrm{Ad}(w^{-1})(s^{-1})$ for a unique $\sigma$-invariant $w$ in the Weyl group of $T$ in $Z^0(g)$.
• The trace computation relies on a decomposition of $g$ and $\tau$ into $p$-power and prime-to-$p$ parts, allowing the use of the Deligne–Lusztig trace formula via averaging over $\breve{T}_r^\sigma$.
• The action of $(t_1, \tau_1)$ on the fixed-point set contributes non-trivially only when $\tau_1 = \mathrm{Ad}(w^{-1})(t_1^{-1})$, yielding a trace of $\# \breve{T}_r^\sigma$ in that case.