# Representation theory of SL2(R)

In

mathematics , the main results concerning irreducibleof theunitary representation sLie group SL_{2}(**R**) are due to Gelfand and Naimark (1946),V. Bargmann (1947), andHarish-Chandra (1952).**Structure of the complexified Lie algebra**We choose a basis "H", "X", "Y" for the complexification of the Lie algebra of SL

_{2}(**R**)so that "iH" generates the Lie algebra of a**compact**Cartan subgroup "K" (so in particular unitary representations split as a sum of eigenspaces of "H"), and {"H","X","Y"} is an sl_{2}-triple, which means that they satisfy the relations: $[H,X]\; =2X,\; quad\; [H,Y]\; =-2Y,\; quad\; [X,Y]\; =H.$

One way of doing this is as follows:

:$H=egin\{pmatrix\}0\; -i\backslash \; i\; 0end\{pmatrix\}$ corresponding to the subgroup "K" of matrices $egin\{pmatrix\}cos(\; heta)\; -sin(\; heta)\backslash \; sin(\; heta)\; cos(\; heta)end\{pmatrix\}$:$X=\{1over\; 2\}egin\{pmatrix\}1\; i\backslash \; i\; -1end\{pmatrix\}$:$Y=\{1over\; 2\}egin\{pmatrix\}1\; -i\backslash \; -i\; -1end\{pmatrix\}$

The

Casimir operator Ω is defined to be:$Omega=\; H^2+1+2XY+2YX.$

It generates the center of the

universal enveloping algebra of the complexified Lie algebra of SL_{2}(**R**). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ^{2}. Thus in the case of the Lie algebra sl_{2}, theinfinitesimal character of an irreducible representation is specified by one complex number.The center "Z" of the group SL

_{2}(**R**) is a cyclic group {"I",-"I"} of order 2, consisting of the identity matrix and its negative. On any irreducible representation, the center either acts trivially, or by the non trivial character of "Z", which represents the matrix -"I" by multiplication by -1 in the representation space. Correspondingly, one speaks of the trivial or nontrivial "central character". The central character and the infinitesimal character of an irreducible representation of any reductive Lie group are important invariants of the representation. In the case of irreducible admissible representations of SL_{2}(**R**), it turns out that, generically, there is exactly one representation, up to an isomorphism, with the specified central and infinitesimal characters. In the exceptional cases there are two or three representations with the prescribed parameters, all of which have been determined.**Finite dimensional representations**For each nonnegative integer "n", the group SL

_{2}(**R**) has an irreducible representation of dimension "n"+1, which is unique up to an isomorphism. This representation can be constructed in the space of homogeneous polynomials of degree "n" in two variables. The case "n" = 0 corresponds to thetrivial representation . An irreducible finite dimensional representation of a noncompactsimple Lie group of dimension greater than 1 is never unitary. Thus this construction produces only one unitary representation of SL_{2}(**R**), the trivial representation.The "finite-dimensional" representation theory of the noncompact group SL

_{2}(**R**) is equivalent to therepresentation theory of SU(2) , its compact form, essentially because their Lie algebras have the same complexification and they are "algebraically simply connected". (More precisely the group SU(2) is simply connected and SL_{2}(**R**) is not, but has no non-trivial algebraic central extensions.) However, in the general "infinite-dimensional" case, there is no close correspondence between representations of a group and the representations of its Lie algebra. In fact, it follows from thePeter-Weyl theorem that all irreducible representations of the compact Lie group SU(2) are finite-dimensional and unitary. The situation with SL_{2}(**R**) is completely different: it possesses infinite-dimensional irreducible representations, some of which are unitary, and some are not.**Principal series representations**A major technique of constructing representations of a reductive Lie group is the method of

parabolic induction . In the case of the group SL_{2}(**R**), there is up to conjugacy only one properparabolic subgroup , theBorel subgroup of the upper-triangular matrices of determinant 1. The inducing parameter of an inducedis a (possibly non-unitrary) character of the multiplicative group of real numbers, which is specified by choosing ε = ± 1 and a complex number μ. The corresponding principal series representation is denoted "I"principal series representation _{ε,μ}. It turns out that ε is the central character of the induced representation and the complex number μ may be identified with theinfinitesimal character via theHarish-Chandra homomorphism .The principal series representation "I"

_{ε,μ}(or more precisely its Harish-Chandra module of "K"-finite elements) admits a basis consisting of elements "w"_{"j"}, where the index "j" runs through the even integers if ε=1 and the odd integers if ε=−1. The action of "X", "Y", and "H" is given by the formulas:$H(w\_j)\; =\; jw\_j$:$X(w\_j)\; =\; \{mu+j+1over\; 2\}w\_\{j+2\}$:$Y(w\_j)\; =\; \{mu-j+1over\; 2\}w\_\{j-2\}$**Admissible representations**Using the fact that it is an eigenvector of the Casimir operator and has an eigenvector for "H", it follows easily that any irreducible

admissible representation is a subrepresentation of a parabolically induced representation. (This also is true for more general reductive Lie groups and is known as.) Thus the irreducible admissible representations of SLCasselman's subrepresentation theorem _{2}(**R**) can be found by decomposing the principal series representations "I"_{ε,μ}into irreducible components and determining the isomorphisms. We summarize the decompositions as follows:

*"I"_{ε,μ}is reducible if and only if μ is an integer and ε=−(−1)^{μ}. If "I"_{ε,μ}is irreducible then it is isomorphic to "I"_{ε,−μ}.

*"I"_{−1, 0}splits as the direct sum "I"_{ε,0 }= "D"_{+0}+ "D"_{−0}of two irreducible representations, called limit of discrete series representations. "D"_{+0}has a basis "w"_{"j"}for "j"≥1, and "D"_{-0}has a basis "w"_{"j"}for "j"≤−1,

*If "I"_{ε,μ}is reducible with μ>0 (so ε=−(−1)^{μ}) then it has a unique irreducible quotient which has finite dimension μ, and the kernel is the sum of two discrete series representations "D"_{+μ}+ "D"_{−μ}. The representation "D"_{μ}has a basis "w"_{μ+"j"}for "j"≥1, and "D"_{-μ}has a basis "w"_{−μ−"j"}for "j"≤−1.

*If "I"_{ε,μ}is reducible with μ<0 (so ε=−(−1)^{μ}) then it has a unique irreducible subrepresentation, which has finite dimension μ, and the quotient is the sum of two discrete series representations "D"_{+μ}+ "D"_{−μ}.This gives the following list of irreducible admissible representations:

*A finite dimensional representation of dimension μ for each positive integer μ, with central character −(−1)^{μ}.

*Two limit of discrete series representations "D"_{+0}, "D"_{−0}, with μ=0 and non-trivial central character.

*Discrete series representations "D"_{μ}for μ a non-zero integer, with central character −(−1)^{μ}

*Two families of irreducible principle series representations "I"_{ε,μ}for ε≠−(−1)^{μ}(where "I"_{ε,μ}is isomorphic to "I"_{ε,−μ}).**Relation with the Langlands classification**According to the

Langlands classification , the irreducible admissible representations are parametrized by certain tempered representations of Levi subgroups "M" of parabolic subgroups "P"="MAN". This works as follows:

*The discrete series, limit of discrete series, and unitary principle series representations "I"_{ε,μ}with μ imaginary are already tempered, so in these cases the parabolic subgroup "P" is SL_{2}itself.

*The finite dimensional representations and the representations "I"_{ε,μ}for ℜμ>0, μ not an integer or ε≠−(−1)^{μ}are the irreducible quotients of the principal series representations "I"_{ε,μ}for ℜμ>0, which are induced from tempered representations of the parabolic subgroup "P"="MAN" of upper triangular matrices, with "A" the positive diagonal matrices and "M" the center of order 2. For μ a positive integer and ε=−(−1)^{μ}the principal series representation has a finite dimensional representation as its irreducible quotient, and otherwise it is already irreducible.**Unitary representations**The irreducible unitary representations can be found by checking which of the irreducible admissible representations admit an invariant positively-definite Hermitian form. This results in the following list of unitary representations of SL

_{2}(**R**):

*The trivial representation

*The twolimit of discrete series representation s "D"_{+"0"}, "D"_{−"0"}.

*Thediscrete series representation s "D"_{"k"}, indexed by non-zero integers "k". They are all distinct.

*The two families of irreducibleprincipal series representation , consisting of the spherical principal series "I"_{+,"i"μ}indexed by the real numbers μ, and the non-spherical unitary principal series "I"_{-,"i"μ}indexed by the non-zero real numbers μ. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.

*Thecomplementary series representation s "I"_{+,μ}for 0<|μ|<1. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.Of these, the two limit of discrete series representations, the discrete series representations, and the two families of principal series representations are tempered, while the finite dimensional and complementary series representations are not tempered.

**References***V. Bargmann, [

*http://links.jstor.org/sici?sici=0003-486X%28194707%292%3A48%3A3%3C568%3AIUROTL%3E2.0.CO%3B2-Z, "Irreducible Unitary Representations of the Lorentz Group"*] , The Annals of Mathematics, 2nd Ser., Vol. 48, No. 3 (Jul., 1947), pp. 568-640

* Gelfand, I.; Neumark, M. "Unitary representations of the Lorentz group." Acad. Sci. USSR. J. Phys. 10, (1946), pp. 93--94

* Harish-Chandra, "Plancherel formula for the 2×2 real unimodular group." Proc. Nat. Acad. Sci. U.S.A. 38 (1952), pp. 337--342

*Roger Howe, Eng-Chye Tan, "Nonabelian harmonic analysis. Applications of SL(2,**R**)." Universitext. Springer-Verlag, New York, 1992. ISBN 0-387-97768-6

*Knapp, Anthony W. "Representation theory of semisimple groups. An overview based on examples." Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001. xx+773 pp. ISBN 0-691-09089-0

*Kunze, R. A.; Stein, E. M. "Uniformly bounded representations and harmonic analysis of the 2×2 real unimodular group." Amer. J. Math. 82 (1960), pp. 1--62

*D. Vogan, "Representations of real reductive Lie groups", ISBN 3-7643-3037-6

*N. R. Wallach, Real reductive groups I. ISBN 0-12-732960-9

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