Derived right adjoints of parabolic induction: an example

TL;DR

This paper computes the derived functors of the right adjoint of parabolic induction for specific representations of SL_2 over p-adic numbers, providing explicit calculations in the mod p setting.

Contribution

It provides explicit calculations of derived functors of the right adjoint of parabolic induction for SL_2(Q_p) in the mod p representation theory context.

Findings

Explicit formulas for R^n R_B^G(π) for G=SL_2(Q_p)

Calculations applicable to smooth, irreducible mod p representations

Advances understanding of the structure of derived functors in p-adic representation theory

Abstract

Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(\pi)$, where $B$ is a Borel subgroup of $G$, $\mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vign\'eras, and $\pi$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.