Steinberg representation for finite groups VS for $p$-adic groups and parahorics

Question

The term "Steinberg representation" comes out in both the context of finite groups of Lie type (ie. connected reductive groups over a finite field) and of $p$-adic groups (ie. connected reductive groups over a non-archimedean local field of characteristic $0$).

For finite groups of Lie type, I see that the Steinberg representation can be defined in terms of the duality functor applied to the trivial representation, as is explained in Digne and Michel's book "Representations of finite groups of Lie type", Chapter 9 (and 8 for the duality).

For a quasi-split $p$-adic group $G$ over some finite extension $F/\mathbb Q_p$, once a rational Borel subgroup $B$ is fixed, I see that the Steinberg representation may be defined as the quotient $$\mathrm{Ind}_B^G\,\mathbf 1 \big/ \sum_{B\subsetneq P} \mathrm{Ind}_P^G \, \mathbf 1$$ where $P$ ranges over all parabolic subgroups of $G$ containing $B$ properly, including the case $P=G$. Here, $\mathbf 1$ denotes the trivial representation of the corresponding group and $\mathrm{Ind_P^G}$ denotes the (unnormalized) parabolic induction. Note also that in my notations, I identify the various algebraic groups with their groups of $F$-rational points. This definition is stated in Herzig's notes available here.

I see the similarity between both constructions, as the duality functor in the case of finite groups of Lie type also makes use of all standard parabolic subgroups and inductions in its definition. Yet, I can not quite spell out how both notions interact together in the following situation.

Let's consider our $p$-adic group $G$ from before. It gives rise to various finite groups of Lie type when considering the parahoric subgroups of $G$. Indeed, if $\mathcal P$ is a parahoric and $\mathcal P^+$ is its pro-$p$-radical, then the quotient $G_{\mathcal P} := \mathcal P/\mathcal P^+$ is a finite group of Lie type over the residue field $k$ of $F$. This is called the maximal reductive quotient of $\mathcal P$. Different conjugacy classes of parahoric subgroups $\mathcal P$ may give rise to different finite groups of Lie type $G_{\mathcal P}$.
Then, what happens if I start from the Steinberg representation $\mathrm{St}_{G_{\mathcal P}}$ of some maximal reductive quotient $G_{\mathcal P}$, inflate it to $\mathcal P$ and then apply smooth induction (with compact support) to it ? That is, does the induced representation $$\mathrm{c-Ind}_{\mathcal P}^G \, \mathrm{St}_{G_{\mathcal P}}$$ relate with the Steinberg representation of $G$ in any way ?
Further, how do these induced representations compare when choosing a different parahoric $\mathcal Q \not = \mathcal P$ ?

Any reference or any insight would be greatly appreciated !

Answer 1

The link between the Steinberg representation of $G$, denoted ${\rm St}_G$, and the ${\rm St}_{G_P}$, for the various parahoric $P$ is the following.

We first have a local point of view. For any parahoric subgroup $P$, the fixed vector space ${\rm St}_G^{P^+}$ is isomorphic to ${\rm St}_{G_P}$ as a $P/P^+$-module. So by Frobenius reciprocity we have a natural morphism from ${\rm c-Ind}_{G_P}^G \, {\rm St}_{G_P}$ towards ${\rm St}_G$.

Globally things go as follows. We may form a homological $G$-equivariant coefficient system ${\mathcal C}$ on the building $X$ of $G$, in the sense of Schneider-Stuhler IHES. If $\sigma$ is a simplex corresponding to a parahoric subgroup $P$, the stalk of $\mathcal C$ over $\sigma$ is ${\rm St}_{G_P}$. Then ${\rm St}_G$ is isomorphic to the homology $H_0 (X,{\mathcal C})$ of $X$ with coefficient in $\mathcal C$.

Please tell me if you need some more detail.