# Finite-dimensional representation

A linear representation of a topological group in a finite-dimensional vector space. The theory of finite-dimensional representations is one of the most important and most developed parts of the representation theory of groups. An irreducible finite-dimensional representation is completely irreducible (see Schur lemma), but an operator-irreducible finite-dimensional representation can be reducible. A measurable finite-dimensional representation of a locally compact group locally coincides almost-everywhere with a continuous finite-dimensional representation. A bounded finite-dimensional representation of a locally compact group is equivalent to a unitary representation. A locally compact group having a faithful finite-dimensional representation is a Lie group [7].

A unitary finite-dimensional representation is a direct sum of irreducible unitary finite-dimensional representations. The intersection of the kernels of the continuous homomorphisms of a topological group $G$ coincides with the intersection of the kernels of the irreducible unitary finite-dimensional representations of $G$; if this set contains only the identity of $G$, then there is a continuous monomorphism from $G$ into some compact group, and $G$ is said to be imbeddable in a compact group, or to be a maximally almost-periodic group (a MAP-group). If $G$ is a MAP-group, then the family of matrix entries of irreducible finite-dimensional representations of $G$ separates points in $G$. Commutative groups and compact groups are MAP-groups; a connected locally compact group is a MAP-group if and only if it is the direct product of a connected compact group and $\mathbf R ^ {n}$( see [5]). A MAP-group can have infinite-dimensional irreducible unitary representations and need not be a group of type 1. Every continuous irreducible unitary representation of a locally compact group $G$ is finite-dimensional if and only if $G$ is a projective limit of finite extensions of groups $H$ of the form $( K \cdot D) \times V$, where $K$, $D$ and $V$ are closed subgroups of $H$ such that $V$ is isomorphic to $\mathbf R ^ {n}$, $K$ is compact and $D$ is a discrete group that is central in $H$[8]; a sufficient condition is that the quotient group of $G$ by its centre be compact. Moreover, for many locally compact groups (in particular, for non-compact simple Lie groups), the only irreducible unitary finite-dimensional representation is the trivial representation.

Non-unitary finite-dimensional representations of topological groups have been classified (up to equivalence) only for special groups; in particular, for $\mathbf R$ and $\mathbf Z$, where the problem of describing finite-dimensional representations is solved by reducing matrices to Jordan form, and — in the case of $\mathbf R$— is related to the theory of ordinary differential equations with constant coefficients, in that the solution space of such an equation is a finite-dimensional invariant subspace of the regular representation of $\mathbf R$ in the space of continuous functions on $\mathbf R$. In addition, the finite-dimensional representations of connected semi-simple Lie groups are known. More precisely, these are direct sums of irreducible finite-dimensional representations that can be described as follows. If $G$ is a semi-simple complex Lie group and $U$ is a maximal compact subgroup, then every continuous irreducible unitary representation $\pi$ of $U$ in a space $E$ can be extended: 1) to an irreducible representation $\pi ^ {G}$ of $G$ in $E$ whose matrix entries are analytic functions on $G$; and 2) to an irreducible representation $\overline \pi \; {} ^ {G}$ of $G$ whose matrix entries are complex conjugates of analytic functions on $G$; $\pi ^ {G}$ and $\overline \pi \; {} ^ {G}$ are determined uniquely by $\pi$. The tensor product $\pi _ {1} ^ {G} \otimes \overline \pi \; {} _ {2} ^ {G}$ is an irreducible finite-dimensional representation of $G$ for arbitrary irreducible unitary finite-dimensional representations $\pi _ {1}$ and $\pi _ {2}$ of $U$, and every irreducible finite-dimensional representation of $G$ is equivalent to a representation of the form $\pi _ {1} ^ {G} \otimes \overline \pi \; {} _ {2} ^ {G}$. A description of the finite-dimensional representations of a simply-connected connected complex semi-simple Lie group can also be given in terms of the exponentials of finite-dimensional representations of its Lie algebra, and also by using the Gauss decomposition $G \supset Z _ {-} D Z _ {+}$ of $G$: Let $\alpha$ be a continuous function on $G$ such that $\alpha ( z _ {-} \delta z _ {+} ) = \alpha ( \delta )$ for all $z _ {-} \in Z _ {-}$, $\delta \in D$, $z _ {+} \in Z _ {+}$, and suppose that the linear hull $\Phi _ \alpha$ of the functions $g \rightarrow \alpha ( gg _ {0} )$, $g _ {0} \in G$, is finite-dimensional; then the formula $[ T _ \alpha ( g _ {0} ) f] ( g) = f ( gg _ {0} )$, $g, g _ {0} \in G$, $f \in \Phi _ \alpha$, defines an irreducible finite-dimensional representation of $G$, and all irreducible finite-dimensional representations of $G$ can be obtained in this way. If $G$ is a real semi-simple Lie group having complex form $G ^ {\mathbf C }$, then every irreducible finite-dimensional representation of $G$ is the restriction to $G$ of some unique irreducible finite-dimensional representation of $G ^ {\mathbf C }$ whose matrix entries are analytic on $G ^ {\mathbf C }$( so that the theory of finite-dimensional representations of semi-simple connected Lie groups reduces essentially to that of compact Lie groups). A finite-dimensional representation of an arbitrary Lie group is real analytic; if $G$ is a simply-connected Lie group, then there is a one-to-one correspondence between the finite-dimensional representations of $G$ and of its Lie algebras (associating to a representation of the Lie group its differential; the inverse mapping associates to a representation of the Lie algebra its exponential). However, the classification of finite-dimensional representations of arbitrary Lie groups is far from being completely solved (1988), even for the special cases of $\mathbf R ^ {n}$, $n \geq 2$, and $\mathbf Z ^ {n}$, $n \geq 2$( see [6]). On the other hand, the irreducible finite-dimensional representations $\pi$ of a connected Lie group $G$ are known [2]: They have the form $\pi = \chi \otimes \pi _ {0}$, where $\chi$ is a one-dimensional representation of $G$( that is, essentially of its commutative quotient group by the commutator subgroup), and $\pi _ {0}$ is a finite-dimensional representation of the semi-simple quotient group of $G$ by the maximal connected solvable normal subgroup of $G$( see Levi–Mal'tsev decomposition).

#### References

 [1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) [2] D.P. Zhelobenko, "Compact Lie groups and representations" , Amer. Math. Soc. (1973) (Translated from Russian) [3] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) [4] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French) [5] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940) [6] I.M. Gel'fand, V.A. Ponomarev, "Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space" Funct. Anal. Appl. , 3 : 4 (1969) pp. 325–326 Funktsional. Anal. i Prilozhen. , 3 : 4 (1969) pp. 81–82 [7] V.M. Glushkov, "The structure of locally compact groups and Hilbert's fifth problem" Transl. Amer. Math. Soc. , 15 (1960) pp. 55–93 Uspekhi Mat. Nauk , 12 : 2 (1957) pp. 3–41 [8] A.I. Shtern, "Locally bicompact groups with finite-dimensional irreducible representations" Math. USSR Sb. , 19 : 1 (1973) pp. 85–94 Mat. Sb. , 90 : 1 (1973) pp. 86–95