REPRESENTATION THEORY.WEEK 8VERA SERGANOVA1. Representations of SL2(R)In this se ctionG = SL2(R) = {g ∈ GL2(R) | det g = 1} .Let K be the subgroup of matricesgθ=cos θ sin θ− sin θ cos θ.The group K is a maximal compact subgroup of G, clearly K is isomorphic to S1. Ifρ: G → GL (V ) is a unitary representation of G in a H ilbert space then then ResKρsplits into the sum of 1-dimensional representations of V . In particular, one can findv ∈ V such that ρgθ(v) = einθv. Define the matrix coefficient function f : G → Cgiven byf (g) = hv, ρgvi .Then f satisfies the conditionf (ggθ) = einθf (g) .Thus, one can consider f as a section of a linear bundle on the space G/K (if n = 0,then f is a function). Thus, it is clear that the space G/K is an important geometricobject, where the representations of G are “realized”.Consider the Lobachevsky planeH = {z ∈ C | Im z > 0}with metric defined by the formuladx2+dy2y2and the volume formdxdyy2. Then Gcoinci des with the group of rigid motions of H preserving orientation. The action ofG on H is given by the formulaz 7→az + bcz + d.One c an check easily that G acts transitively on H, preserves the m etric and volume.Moreover, the stabilizer of i ∈ H coincides with K. Thus, we ide ntify H with G/K.The first series of representations we describe is called the representations of dis-crete series. Those are the representations with matrix coefficients in L2(G). LetDate: November 7, 2005.12 VERA SERGANOVAH+nbe the space of holomorphic densities on H, the e xpressions ϕ (z) (dz)n/2, whereϕ (z) is a holomorphic function on H satisfying the condition thatZ|ϕ|2yn−2dzd¯zis finite. Define the representation of G in H+nbyρgϕ (z) (dz)n/2= ϕaz + bcz + d1(cz + d)n(dz)n/2,and Hermitian product on Hnthe formula(1.1)Dϕ (dz)n/2, ψ (dz)n/2E=Z¯ϕψyn−2dzd¯z,for n > 1. For n = 1 the product is defined by(1.2)Dϕ (dz)n/2, ψ (dz)n/2E=Z∞−∞¯ϕψdx,in this case H+1consists of all de nsities which converge to L2-functions on the bound-ary (real line). Check that this Hermitian product is invariant.Let us show that Hnis irreducible. It is convenient to consider Poincar´e model ofLobachevsky plane using the conformal mapw =z − iz + i,that maps H to a unit disk |w| < 1. Then the group G acts on the uni t disk bylinear-fractional maps w →aw+b¯bw+¯afor all complex a, bsatisfying |a|2− |b|2= 1, and K is defined by the condition b = 0. If a = eiθ, thenρgθ(w) = e2iθw. The invariant volume form isdwd ¯w1− ¯ww.It is clear that wk(dw)n/2for all k ≥ 0 form an orthogonal basis in H+n, each vectorwk(dw)n/2is an eigen vector with respect to K, namelyρgθwk(dw)n/2= e(2k+n)iθwk(dw)n/2.It is easy to check now that H+nis irreducible. Indeed, every invariant closed subspaceV has a topological basis consisting of eigenvectors of K, in other words wk(dw)n/2forsome positive k must form a topological basis of V . Without loss of generality assumethat V contains (dw)n/2, then by applying ρgone can get that1(bw+a)n(dw)n/2, andin Taylor series for1(bw+a)nall elements of the basis appear with non-zero coefficients.That implies wk(dw)n/2∈ V for all k ≥ 0, hence V = H+n.One can construct another series of representations H−nby considering holomorphicdensities in the lower half-plane Im z < 0.Principal series. These representations are parameteriz ed by a continuous pa-rameter s ∈ Ri (s 6= 0). Consider now the action of G on a real line by linear fractionalREPRESENTATION THEORY. WEEK 8 3transformations x 7→ax+bcx+d. Let P+sdenotes the space of densities ϕ (x) (dx)1+s2withG-action given byρgϕ (x) (dx)1+s2= ϕax + bcx + d|cx + d|−s−1(dx)1+s2.The He r mitian product gi ven by(1.3) hϕ, ψi =Z∞−∞¯ϕψdxis invari ant. The property of invariance justif y the choice of weight for the density as(dx)1+s2(dx)1+¯s2= dx, thus the integration is invariant. To check that the represen-tation is irreducible one can m ove the real line to the unit circle as in the exampleof di screte series and then use eikθ(dθ)1+s2as an orthonormal basis in P+s. Note thatthe eigen values of ρgθin this case are e2kiθfor all integer k.The se cond principal se r ies P−scan be obtained if instead of densi t ies we considerthe pseudo densities which are transformed by the lawρgϕ (x) (dx)1+s2= ϕax + bcx + d|cx + d|−s−1sgn (cx + d) dx1+s2.Complementary series. Those are representations which do not appear in theregular repre se ntation L2(G). They can be r ealized as the representations in Csofall densities ϕ (x) (dx)1+s2for real 0 < s < 1. An invariant Hermitian product is(1.4) hϕ, ψi =Z∞−∞Z∞−∞¯ϕ (x) ψ (y) |x − y|s−1dxdy.2. Semisimple modules and density theoremWe assume that R is a unital ring. Recall that an R-module is semi-simple if forany submodule N ⊂ M there exists a submodule N′such that M = N ⊕ N′andR-module M is simple if any submodule of M is either M or 0.Lemma 2.1. Every submodule and every quotie nt of a semisimple mo dule is semisim-ple.Proof. If Let N be a submodule of a semisimpl e modul e M, and let P be a submoduleof N. Sinc e M = P ⊕ P′, then there exists an R-invariant projector p : M → P . Therestriction of p to N de fines the projector N → P . Lemma 2.2. Any semisimple R-module contains a simple submodule.Proof. Let M be semisimple , m ∈ M. Let N be a maximal submodule in Rm whichdoes not c ontain m (exists by Zorn’s lemma, take all submodules which do not containm ). Then Rm is semisimple and Rm = N ⊕N′. We claim that N′is simple. Inde ed,if N′is not simple, then it contains a prop er submodule P . But m /∈ P ⊕ N, sinceP ⊕ N 6= Rm. That contradicts maximality of N. 4 VERA SERGANOVALemma 2.3. The following conditions on a module M are equivalent(1) M is semisimple;(2) M =Pi∈IMifor some simpl e submodules Mi;(3) M = ⊕j∈ JMjfor some simple submodules Mj.Proof. (1) ⇒ (2) Let {Mi}i∈Ibe the collection of all simple submodules. Let N =Pi∈IMi, assume that N 6= M, then M = N ⊕ N′and N′contains a simple submod-ule. Contradiction.To prove (2) ⇒ (3) let J ⊂ I be minimal such that M =Pj∈ JMj(check that itexists by Zorn’s lemma). By minimality of J for any k ∈ J, Mkdoes not belong toPj∈ J−kMj. Therefore M = ⊕j∈ JMj.Finally, let us prove (3) ⇒ (1). Let N ⊂ M be a submodule and S ⊂ J be amaximal subset such that N ∩ ⊕j∈ SMj= 0. Let M′= N ⊕ (⊕j∈ SMj). We claim