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Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of groups but for the sake of convenience let us make the following definition:

Let $(V,\langle\ ,\ \rangle)$ be an Hilbert space over $\mathbf{C}$ of countable (finite or infinite) orthonormal Schauder basis. We let Let $\rho:G\rightarrow GL_{cont}(V)$ be a group homomorphism where $GL_{cont}(V)$ stands for the set of bounded (with respect to the operator norm) linear operators on $V$. We may view $GL_{cont}(V)$ as a topological group via the discrete topology. Now let $M$ be a $\mathbf{C}$ vector space with a linear $G$-action. We will say that $M$ is a unitary irreducible representation of $G$ if there exists an abstract isomorphism of $\mathbf{C}$ vector spaces $f:M\rightarrow V$ (where $V$ is chosen as above) such that the natural map $GL(M)\rightarrow GL(V)$

(1) factors through $GL_{cont}(V)$

(2) $V$ is irreducible as a $G$-module

(3) For all $g\in G$ and all $v,w\in V$ one has that $\langle\rho(g)v,\rho(g)w\rangle=\langle v,w\rangle$.

Now let us consider the space $L^2(G)$ of all functions $f:G\rightarrow\mathbf{C}$ where $f$ is measurable and square integrable with respect to the Haar measure. Note that this space has a natural structure of a $G$-module through left action.

Now in the special case where $G$ is a compact Lie group ($G$ is not necessarily connected so in particular this covers all finite groups) then all irreducible representation are unitary (the average trick) and finite dimensional (this I think is non-trivial and follows from Peter-Weyl, actually I never looked at the proof of this result). Moreover, if $\widehat{G}$ denote a complete set irreducible $\mathbf{C}$ representations of $G$ (up to isomorphisms as (unitary) $G$-modules) then one has that

$L^2(G)=\bigoplus_{\phi\in\widehat{G}}\oplus_{i,j}\sqrt{n_{\phi}}\phi_{ij}$ where $n_{\phi}=dim(\phi)$ and $\phi_{ij}$ is the $(i,j)$-th entry of $\phi:G\rightarrow GL(V_{\phi})$. In other words all irreducible unitary representations (say $\phi$ is one of them) of $G$ occur in $L^2(G)$ with multiplicities $n_{\phi}$. The direct sum here should be understood in the sense of Schauder basis with respect to the topology induced by $\langle\ ,\ \rangle$. Note that $\lbrace\sqrt{n_{\phi}}\phi_{i,j}\rbrace$ gives an orthonormal basis of $L^2(G)$.

Now here is a set of natural questions:

(1) Do all the irreducible unitary representations of a semi-semiple (reductive) algebraic group over $\mathbf{R}$ occur in $L^2(G)$?

(2) On the other side of the spectrum, what about algebraic solvable groups?

(3) What is the minimal example of a locally compact topological group $G$ (with an non artificial tailor made topology, in particular $G$ has to be infinite) for which one can find an irreducible unitary representation which does not occur in $L^2(G)$?

2 added 111 characters in body

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of groups but for the sake of convenience let us make the following definition:

Let $(V,\langle\ ,\ \rangle)$ be an Hilbert space over $\mathbf{C}$ of countable (finite or infinite) orthonormal Schauder basis. Let $\rho:G\rightarrow GL_{cont}(V)$ be a group homomorphism where $GL_{cont}(V)$ stands for the set of bounded (with respect to the operator norm) linear operators on $V$. We may view $GL_{cont}(V)$ as a topological group via the discrete topology. Now let $M$ be a $\mathbf{C}$ vector space with a linear $G$-action. We will say that $M$ is a unitary irreducible representation of $G$ if there exists an abstract isomorphism of $\mathbf{C}$ vector spaces $f:M\rightarrow V$ (where $V$ is chosen as above) such that the natural map $GL(M)\rightarrow GL(V)$

(1) factors through $GL_{cont}(V)$

(2) $V$ is irreducible as a $G$-module

(3) For all $g\in G$ and all $v,w\in V$ one has that $\langle\rho(g)v,\rho(g)w\rangle=\langle v,w\rangle$.

Now let us consider the space $L^2(G)$ of all functions $f:G\rightarrow\mathbf{C}$ where $f$ is measurable and square integrable with respect to the Haar measure. Note that this space has a natural structure of a $G$-module through left action.

Now in the special case where $G$ is a compact Lie group ($G$ is not necessarily connected so in particular this covers all finite groups) then all irreducible representation are unitary (the average trick) and finite dimensional (this I think is non-trivial and follows from Peter-Weyl, actually I never looked at the proof of this result). Moreover, if $\widehat{G}$ denote a complete set irreducible $\mathbf{C}$ representations of $G$ (up to isomorphisms as (unitary) $G$-modules) then one has that

$L^2(G)=\bigoplus_{\phi\in\widehat{G}} n_{\phi}\phi_{ij}$ L^2(G)=\bigoplus_{\phi\in\widehat{G}}\oplus_{i,j}\sqrt{n_{\phi}}\phi_{ij}$where$n_{\phi}=dim(\phi)$and$\phi_{ij}$is the$(i,j)$-th entry of$\phi:G\rightarrow GL(V_{\phi})$. In other words all irreducible unitary representations (say$\phi$is one of them) of$G$occur in$L^2(G)$with multiplicities$n_{\phi}$. The direct sum here should be understood in the sense of Schauder basis with respect to the topology induced by$\langle\ ,\ \rangle$. Note that$\lbrace\sqrt{n_{\phi}}\phi_{i,j}\rbrace$gives an orthonormal basis of$L^2(G)$. Now here is a set of natural questions: (1) Do all the irreducible unitary representations of a semi-semiple (reductive) algebraic group over$\mathbf{R}$occur in$L^2(G)$? (2) On the other side of the spectrum, what about algebraic solvable groups? (3) What is the minimal example of a locally compact topological group$G$(with an non artificial tailor made topology, in particular$G$has to be infinite) for which one can find an irreducible unitary representation which does not occur in$L^2(G)\$?

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