Locally profinite group

Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and $F^\backslash times$ are locally profinite. More generally, the matrix ring $\backslash operatorname\{M\}\_n(F)$ and the general linear group $\backslash operatorname\{GL\}\_n(F)$ 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).

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

Let $(\backslash rho,\; V)$ be a complex representation of G.We do not put a topology on V; so there is no topological condition on the representation. $\backslash rho$ is said to be smooth if V is a union of $V^K$ where K runs over all open compact subgroups K. $\backslash rho$ is said to be admissible if it is smooth and $V^K$ is finite-dimensional for any open compact subgroup K.

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

The dual space $V^*$ carries the action $\backslash rho^*$ of G given by $\backslash left\backslash langle\; \backslash rho^*(g)\; \backslash alpha,\; v\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash alpha,\; \backslash rho^*(g^\{-1\})\; v\; \backslash right\backslash rangle$. In general, $\backslash rho^*$ is not smooth. Thus, we set $\backslash widetilde\{V\}\; =\; \backslash bigcup\_K\; (V^*)^K$ where $K$ is acting through $\backslash rho^*$ and set $\backslash widetilde\{\backslash rho\}\; =\; \backslash rho^*$. The smooth representation $(\backslash widetilde\{\backslash rho\},\; \backslash widetilde\{V\})$ is then called the contragredient or smooth dual of $(\backslash rho,\; V)$.

The contravariant functor

:$(\backslash rho,\; V)\; \backslash mapsto\; (\backslash widetilde\{\backslash rho\},\; \backslash widetilde\{V\})$

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

• $\backslash rho$ is admissible.
• $\backslash widetilde\{\backslash rho\}$ is admissible.Blondel, Corollary 2.8.
• The canonical G-module map $\backslash rho\; \backslash to\; \backslash widetilde\{\backslash widetilde\{\backslash rho\}\}$ is an isomorphism.

When $\backslash rho$ is admissible, $\backslash rho$ is irreducible if and only if $\backslash widetilde\{\backslash rho\}$ is irreducible.

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

:{{see also|Hecke algebra of a locally compact group}}

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

:$(f\; *\; h)(x)\; =\; \backslash int\_G\; f(g)\; h(g^\{-1\}\; x)\; d\; \backslash mu(g)$

$C^\backslash infty\_c(G)$ becomes not necessarily unital associative $\backslash mathbb\{C\}$-algebra. It is called the Hecke algebra of G and is denoted by $\backslash mathfrak\{H\}(G)$. 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 $(\backslash rho,\; V)$ of G, we define a new action on V:

:$\backslash rho(f)\; =\; \backslash int\_G\; f(g)\; \backslash rho(g)\; d\backslash mu(g).$

Thus, we have the functor $\backslash rho\; \backslash mapsto\; \backslash rho$ from the category of smooth representations of $G$ to the category of non-degenerate $\backslash mathfrak\{H\}(G)$-modules. Here, "non-degenerate" means $\backslash rho(\backslash mathfrak\{H\}(G))V=V$. Then the fact is that the functor is an equivalence.Blondel, Proposition 2.16.

{{reflist}}

• Corinne Blondel, [https://webusers.imj-prg.fr/~corinne.blondel/Blondel_Beijin.pdf Basic representation theory of reductive p-adic groups]
• {{Citation | last1=Bushnell | first1=Colin J. |author1link = Colin J. Bushnell| last2=Henniart | first2=Guy |authorlink2=Guy Henniart | title=The local Langlands conjecture for GL(2) | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-31486-8 | doi=10.1007/3-540-31511-X | MR=2234120 | year=2006 | volume=335}}
• {{citation |last1=Milne |first1=James S. |authorlink1=James Milne (mathematician) |title=Canonical models of (mixed) Shimura varieties and automorphic vector bundles, |year=1988 |MR=1044823}}

Category:Topological groups