[Paper Review] Parahoric induction and chamber homology for SL2

Tyrone Crisp|arXiv (Cornell University)|Jan 3, 2013

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper establishes a commutative diagram linking parahoric induction in chamber homology to parabolic induction in representation theory for $\mathrm{SL}_2$ over a $p$-adic field. It shows that the map induced by parahoric induction on chamber homology complexes commutes with parabolic induction, extending Dat's result to higher homology degrees and realizing the Jacquet restriction functor in this setting.

ABSTRACT

We consider the special linear group G=SL2 over a p-adic field, and its diagonal subgroup M=GL1. Parabolic induction of representations from M to G induces a map in equivariant homology, from the Bruhat-Tits building of M to that of G. We compute this map at the level of chain complexes, and show that it is given by parahoric induction (as defined by J.-F. Dat).

Motivation & Objective

• To extend Dat's parahoric induction construction from degree zero to higher homology degrees in the context of $\mathrm{SL}_2$.
• To establish a commutative diagram between chamber homology complexes of $G = \mathrm{SL}_2(F)$ and $M \cong \mathrm{GL}_1(F)$, linking parahoric and parabolic induction.
• To realize the Jacquet restriction functor $\mathrm{r}^G_M$ in terms of chamber homology maps.
• To provide a homological interpretation of the compatibility between compact induction and parahoric induction in the $\mathrm{SL}_2$ case.
• To lay groundwork for generalizations to $\mathrm{SL}_n$ by analyzing the structure of chamber homology and induced maps.

Proposed method

• Constructs the chamber homology complex $C_*^G(X_G)$ for $G = \mathrm{SL}_2(F)$ using simplicial chains on the Bruhat-Tits building $X_G$ with isotropy group coefficients in representation rings.
• Defines a similar complex $C_*^M(X_M)$ for the Levi subgroup $M \cong \mathrm{GL}_1(F)$, with isotropy groups isomorphic to $L = \mathrm{GL}_1(\mathcal{O})$.
• Introduces a map $C_*^M(X_M) \to C_*^G(X_G)$ combining the inclusion $X_M \hookrightarrow X_G$ with parahoric induction from $L$ to isotropy subgroups of $G$.
• Uses Hochschild complexes $C_*(\mathrm{Mod}_f(G))$ and $C_*(\mathrm{Mod}_f(M))$ to model the homology of finitely generated smooth representations.
• Proves commutativity of the diagram (\ast) by showing that the induced map on homology respects parabolic induction in degree zero and one.
• Applies the injectivity of the restriction map $\mathrm{r}_c$ in degree one to identify $\mathrm{I} = \mathrm{i}_c$ on $\mathrm{H}_1^G(X)$, extending Dat's result to higher homology.

Experimental results

Research questions

• RQ1 Does parahoric induction extend to a compatible map between chamber homology complexes beyond degree zero?
• RQ2 Can the Jacquet restriction functor be realized as a map between chamber homology complexes?
• RQ3 How does the compatibility between parahoric and parabolic induction manifest in higher homology degrees for $\mathrm{SL}_2$?
• RQ4 What is the structure of $\mathrm{H}_1^G(X)$ in terms of induced representations and Weyl group actions?
• RQ5 To what extent can the $\mathrm{SL}_2$ results be generalized to $\mathrm{SL}_n$?

Key findings

• The map $C_*^M(X_M) \to C_*^G(X_G)$ induced by parahoric induction commutes with parabolic induction in homology, extending Dat's degree-zero result to higher degrees.
• The Jacquet restriction functor $\mathrm{r}^G_M$ is realized as the map $\mathrm{R} = \mathrm{r}_c$ on homology, with $\mathrm{r}_c$ injective in degree one.
• The first homology group $\mathrm{H}_1^G(X)$ admits a basis of cycles of the form $\mathrm{i}_{U,\overline{U}}(\rho) - \mathrm{i}_{U,\overline{U}}(\rho^w)$, indexed by $W$-orbits of irreducible $L$-representations.
• The induced map $\mathrm{I}: \mathrm{H}_0^M(Y) \to \mathrm{H}_0^G(X)$ agrees with parabolic induction $\mathrm{i}_c$ in degree zero, and $\mathrm{I}: \mathrm{H}_{n-1}^M(Y) \to \mathrm{H}_{n-1}^G(X)$ agrees in degree $n-1$ for $\mathrm{SL}_n$.
• For $\mathrm{SL}_n$, the maps $\mathrm{R}$ and $\mathrm{I}$ defined via parahoric induction satisfy $\mathrm{R} = \mathrm{r}_c$ and $\mathrm{I} = \mathrm{i}_c$ in degrees zero and $n-1$, respectively.
• The construction suggests that new tools are needed for larger Levi subgroups, as the compatibility with boundary maps fails in general.