The objects of interest in this talk are the unitary representations of a locally compact group \( G \). These are homomorphisms \( \pi: G \to \mathrm U(\mathcal H) \) where \( \mathcal U(\mathcal H \) is the group of unitary operators on a Hilbert space \( \mathcal H \). It will be required that they be continuous in the following sense: for every \( v \in \mathcal H \) the map \( G \to \mathcal H, g \mapsto \pi(g)v \) is continuous. Basic examples are the following:

1. The trivial representation;
2. The left-regular representation \( \lambda: G \to \mathcal U(L^2(G, \mu_{Haar})) \) acting by \( \lambda(g)f(x) = f(g^{-1}x) \).

The coefficients of such a representation are the function on \( G \) of the form
\[
c_{v, w}(g) = \langle \pi(g)v, w \rangle_{\mathcal H}
\]
for \( v, w \in \mathcal H \).

The basic problem is to classify those (irreducible) representations for a given group \( G \). It is not solved in general, for example there is no complete classification of unitary representation of the group \( \mathrm{SL}_2(\mathbb Q_p) \).

Classes of groups for which this is of interest include the following:

1. Semisimple Lie groups such as \( \mathrm{Sl}_m({\mathbb R}) \);
2. Semisimple algebraic groups over non-archimedean local fields, such as \( \mathrm{SL}_m(\mathbb Q_p) \);
3. Closed subgroups of the group \( \mathrm{Aut}(T_d) \) of automorphisms of a d-regular tree \( T_d \).

Unitary representations can also be an interesting object of study for some non-locally compact groups such as groups of infinitely-supported permutations.

Before attempting a classification it is useful to look at the behaviour of the matrix coefficients of unitary representations. In particular, representations with the following property are of particular interest: say that a representation is \( C_0 \) if all its coefficients vanish at infinity, that is
\[
\forall v, w \in \mathcal H, \lim_{g\to+\infty} \langle \pi(g)v, w \rangle = 0
\]
where \( g \to +\infty \) means that \( g \) escapes every compact subset of \( G \).

Definition:

• Say that a group has the Howe–Moore property if every unitary representation which does not contain (nonzero) invariant vectors is \( C_0 \).
• Say that a group \( G \) is type I if for every unitary representation \( \pi \) of \( G \) the von Neumann algebra generated by \( \pi(G) \) is of type I (every factor is the algebra of bounded operators on some Hilbert space).

For example:

• Simple Lie groups and algebraic groups over local fields have the Howe–Moore property and are type I (note that in the general semisimple case the property must be defined with more care to take into account representations factoring through a factor);
• If \( H \le \mathrm{Aut}(T_d) \) is a closed, non-ccompact subgroups which acts 2-transitively on the boundary \( \partial T_d \) then it is HM and Type I.

Groups acting on regular tress

Recall that the topology on \( \mathrm{Aut}(T_d) \) is defined by taking pointwise stabilisers of finite subtrees as a basis of neighbourhoods for the identity. In this topology the stabiliser of a vertex is a maximal compact subgroup.

Geometric properties	Analytic properties
2-transitivity on \( \partial T_d \)	Howe–Moore property
2-transitivity and Tits independence	Type I
2-transitivity	\( (H, H_x) \) is a Gelfand pair for any vertex \( x \in T_d \).

here are some explanations:

• A subgroup \( H \subset \mathrm{Aut}(T_d) \) satisfies Tits’ independence condition if for any edge \( e \) separating \( T_d \) into two half-trees \( T^+, T^- \) the stabiliser \( H_e \) splits as the direct product \( H_{T^+} \times H_{T^-} \) (where \( H_{T^\pm} \) is the stabiliser of the half-tree).
• If \( G \) is a locally compact group and \( K \) a compact subgroup then \( (G, K) \) is called a Gelfand pair if for every irreducible unitary representation \( \pi \) of \( G \) the space \( \mathcal H_\pi^K \) of \( K \)-foxed vectors is of dimension at most 1. This is equivalent to the Hecke algebra of bi-\( K \)-invariant functions being commutative.

It is open whether the Howe–Moore property implies 2-transitivity on the boundary. On the other hand if \( H \) is Type I then it is 2-transitive and has Tits’ independence property (Houdayer–Raum), and if \( (H, H_x) \) is a Gelfand pair then H is 2-transitive (this is due to Caprace–Ciobotaru in the larger context of Euclidean buildings).

Burger–Mozes groups

Let \( F \subset \mathfrak S(d) \) be a non-2-transitive subgroup. Then the Burger–Mozes group \( U(F)^+ \) (finite-index subgroup in the subgroup of automorphisms acting locally by \( F \) is not 2-transitive on the boundary.

Question: Doe \( U(F)^+ \) have the Howe–Moore property?

This group is not type I since it has Tits’ independence property; also, since \( (U(F)^+, U(F)_x^+ \) is not a Gelfand pair there exists an irreducible representation of \( H \) where \( U(F)_x^+ \) has a 2-dimensional subspace of fixed vectors.

Problem: Is it possible to construct this explicitely this representation? Is it \( C_0 \)?

Theorem: If \( F \) is primitive then all its representations induced from « parabolic subgroups » (closed subgroups stabilising a vertex at infinity) are \( C_0 \).

If in addition \( F \) is not 2-transitive then teh Hecke algebra is infinitely generated and infinitely presented.