A fundamental problem, motivated by the applications to harmonic analysis and number theory, is the classification of the unitary representations.

A unitary representation of a group $G$ is a continuous homomorphism $\pi$ from $G$ to the group of unitary operators on a complex Hilbert space. Many questions about a representation $\pi$ reduce to its invariant subspaces and one defines the irreducible unitary representations to be those with no proper nontrivial invariant subspaces. The classification problem consists of determining (explicitly) the unitary dual for $G$, i.e. the set of all irreducible unitary representations of $G$.

This problem for an algebraic group $G$ over a local field $\backslash BbbF$ can be reduced to the case when $G$ is reductive (locally isomorphic to a product of simple algebraic groups) and further to simple algebraic groups. The simple algebraic groups belong to four infinite families of classical groups (types $A-D$, e.g.: the special linear, symplectic and orthogonal groups) and five finite families of exceptional groups indexed $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$. By the classification of local fields, $\backslash BbbF$ can be archimedean ( $\backslash BbbC$ or $\backslash BbbR$) or nonarchimedean (finite extension of the $p$-adic numbers or the fractions field of the ring of formal series with coefficients in a finite field). The unitary dual for type $A$ was determined by Vogan (archimedean case) and Tadic (nonarchimedean). Barbasch classified the unitary dual of classical groups when $\backslash BbbF=\backslash BbbC$. If $G$ is split connected, Barbasch also determined the spherical unitary dual for real groups and (with Moy) for $p$-adic groups. There are some other particular cases in which the unitary dual is known, but in general, this is very much an open problem.

A basic example from harmonic analysis which motivates the study of unitary representations is the following. Assume the group $G$ acts as symmetries of a space $X$ and it preserves a measure on $X$. Consider $L^2(X)$, the Hilbert space of square-integrable functions on $X$. The group $G$ then also acts on $L^2(X)$ by left translations: $(g·f)(x):=f(g^{-1}·x)$ and this representation decomposes into irreducible unitary representations.

Of great importance for number theory is the case $G=\mathrm{SL}(2,\backslash BbbR)$ and $X=G/\Gamma$, where $\Gamma$ is a congruence subgroup of $G$. Work by Kim and Shahidi concerning the analytic properties of $L$-functions used the explicit classification of the unitary dual for groups of exceptional type $G_2$ as obtained by Vogan (real case) and Mui\' c ( $p$-adic case).

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