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Locally profinite group

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In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and the p-adic Lie groups. Non-examples are real Lie groups, which have the no small subgroup property.

In a locally profinite group, a closed subgroup is locally profinite, and every compact subgroup is contained in an open compact subgroup.

Examples

Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and F × {\displaystyle F^{\times }} {\displaystyle F^{\times }} are locally profinite. More generally, the matrix ring M n ⁡ ( F ) {\displaystyle \operatorname {M} _{n}(F)} {\displaystyle \operatorname {M} _{n}(F)} and the general linear group GL n ⁡ ( F ) {\displaystyle \operatorname {GL} _{n}(F)} {\displaystyle \operatorname {GL} _{n}(F)} are locally profinite. Another example of a locally profinite group is the absolute Weil group of a non-archimedean local field: this is in contrast to the fact that the absolute Galois group of such is profinite (in particular compact).

Representations of a locally profinite group

Let G be a locally profinite group. Then a group homomorphism ψ : G → C × {\displaystyle \psi :G\to \mathbb {C} ^{\times }} {\displaystyle \psi :G\to \mathbb {C} ^{\times }} is continuous if and only if it has open kernel.

Let ( ρ , V ) {\displaystyle (\rho ,V)} {\displaystyle (\rho ,V)} be a complex representation of G.[1] ρ {\displaystyle \rho } {\displaystyle \rho } is said to be smooth if V is a union of V K {\displaystyle V^{K}} {\displaystyle V^{K}} where K runs over all open compact subgroups K. ρ {\displaystyle \rho } {\displaystyle \rho } is said to be admissible if it is smooth and V K {\displaystyle V^{K}} {\displaystyle V^{K}} is finite-dimensional for any open compact subgroup K.

We now make a blanket assumption that G / K {\displaystyle G/K} {\displaystyle G/K} is at most countable for all open compact subgroups K.

The dual space V ∗ {\displaystyle V^{*}} {\displaystyle V^{*}} carries the action ρ ∗ {\displaystyle \rho ^{*}} {\displaystyle \rho ^{*}} of G given by ⟨ ρ ∗ ( g ) α , v ⟩ = ⟨ α , ρ ∗ ( g − 1 ) v ⟩ {\displaystyle \left\langle \rho ^{*}(g)\alpha ,v\right\rangle =\left\langle \alpha ,\rho ^{*}(g^{-1})v\right\rangle } {\displaystyle \left\langle \rho ^{*}(g)\alpha ,v\right\rangle =\left\langle \alpha ,\rho ^{*}(g^{-1})v\right\rangle }. In general, ρ ∗ {\displaystyle \rho ^{*}} {\displaystyle \rho ^{*}} is not smooth. Thus, we set V ~ = ⋃ K ( V ∗ ) K {\displaystyle {\widetilde {V}}=\bigcup _{K}(V^{*})^{K}} {\displaystyle {\widetilde {V}}=\bigcup _{K}(V^{*})^{K}} where K {\displaystyle K} {\displaystyle K} is acting through ρ ∗ {\displaystyle \rho ^{*}} {\displaystyle \rho ^{*}} and set ρ ~ = ρ ∗ {\displaystyle {\widetilde {\rho }}=\rho ^{*}} {\displaystyle {\widetilde {\rho }}=\rho ^{*}}. The smooth representation ( ρ ~ , V ~ ) {\displaystyle ({\widetilde {\rho }},{\widetilde {V}})} {\displaystyle ({\widetilde {\rho }},{\widetilde {V}})} is then called the contragredient or smooth dual of ( ρ , V ) {\displaystyle (\rho ,V)} {\displaystyle (\rho ,V)}.

The contravariant functor

( ρ , V ) ↦ ( ρ ~ , V ~ ) {\displaystyle (\rho ,V)\mapsto ({\widetilde {\rho }},{\widetilde {V}})} {\displaystyle (\rho ,V)\mapsto ({\widetilde {\rho }},{\widetilde {V}})}

from the category of smooth representations of G to itself is exact. Moreover, the following are equivalent.

  • ρ {\displaystyle \rho } {\displaystyle \rho } is admissible.
  • ρ ~ {\displaystyle {\widetilde {\rho }}} {\displaystyle {\widetilde {\rho }}} is admissible.[2]
  • The canonical G-module map ρ → ρ ~ ~ {\displaystyle \rho \to {\widetilde {\widetilde {\rho }}}} {\displaystyle \rho \to {\widetilde {\widetilde {\rho }}}} is an isomorphism.

When ρ {\displaystyle \rho } {\displaystyle \rho } is admissible, ρ {\displaystyle \rho } {\displaystyle \rho } is irreducible if and only if ρ ~ {\displaystyle {\widetilde {\rho }}} {\displaystyle {\widetilde {\rho }}} is irreducible.

The countability assumption at the beginning is really necessary, for there exists a locally profinite group that admits an irreducible smooth representation ρ {\displaystyle \rho } {\displaystyle \rho } such that ρ ~ {\displaystyle {\widetilde {\rho }}} {\displaystyle {\widetilde {\rho }}} is not irreducible.

Hecke algebra of a locally profinite group

Let G {\displaystyle G} {\displaystyle G} be a unimodular locally profinite group such that G / K {\displaystyle G/K} {\displaystyle G/K} is at most countable for all open compact subgroups K, and μ {\displaystyle \mu } {\displaystyle \mu } a left Haar measure on G {\displaystyle G} {\displaystyle G}. Let C c ∞ ( G ) {\displaystyle C_{c}^{\infty }(G)} {\displaystyle C_{c}^{\infty }(G)} denote the space of locally constant functions on G {\displaystyle G} {\displaystyle G} with compact support. With the multiplicative structure given by

( f ∗ h ) ( x ) = ∫ G f ( g ) h ( g − 1 x ) d μ ( g ) {\displaystyle (f*h)(x)=\int _{G}f(g)h(g^{-1}x)d\mu (g)} {\displaystyle (f*h)(x)=\int _{G}f(g)h(g^{-1}x)d\mu (g)}

C c ∞ ( G ) {\displaystyle C_{c}^{\infty }(G)} {\displaystyle C_{c}^{\infty }(G)} becomes not necessarily unital associative C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }-algebra. It is called the Hecke algebra of G and is denoted by H ( G ) {\displaystyle {\mathfrak {H}}(G)} {\displaystyle {\mathfrak {H}}(G)}. The algebra plays an important role in the study of smooth representations of locally profinite groups. Indeed, one has the following: given a smooth representation ( ρ , V ) {\displaystyle (\rho ,V)} {\displaystyle (\rho ,V)} of G, we define a new action on V:

ρ ( f ) = ∫ G f ( g ) ρ ( g ) d μ ( g ) . {\displaystyle \rho (f)=\int _{G}f(g)\rho (g)d\mu (g).} {\displaystyle \rho (f)=\int _{G}f(g)\rho (g)d\mu (g).}

Thus, we have the functor ρ ↦ ρ {\displaystyle \rho \mapsto \rho } {\displaystyle \rho \mapsto \rho } from the category of smooth representations of G {\displaystyle G} {\displaystyle G} to the category of non-degenerate H ( G ) {\displaystyle {\mathfrak {H}}(G)} {\displaystyle {\mathfrak {H}}(G)}-modules. Here, "non-degenerate" means ρ ( H ( G ) ) V = V {\displaystyle \rho ({\mathfrak {H}}(G))V=V} {\displaystyle \rho ({\mathfrak {H}}(G))V=V}. Then the fact is that the functor is an equivalence.[3]

Notes

  1. We do not put a topology on V; so there is no topological condition on the representation.
  2. Blondel, Corollary 2.8.
  3. Blondel, Proposition 2.16.

References