Representations of SL(2, R)

Mateusz S. Szczygiel

Faculty of Physics University of Warsaw

June 8, 2018 Plan of the presentation

1. Lie algebra sl(2, C) and sl(2, R). 2. Complex representations of sl(2, C). 3. Representations of sl(2, R). 4. Lie group SL(2, R). 5. Unitary representations of SL(2, R).

First we will start from the complex representations of sl(2, C), because it is a complexiﬁcation of sl(2, R). Also we ﬁrst want to outline the theory on the level of Lie algebras in order to obtain the Lie group representations by the exponential map. Lie algebra sl(2, C)

It consists of 2 by 2 traceless matrices equipped with the scalar product hX|Y i = Tr(XY ),X,Y ∈ sl(2, C). (1) We have the orthogonal basis of sl(2, C) consisting of Pauli matrices: 0 1 0 −i 1 0 σ = , σ = , σ = , (2) 1 1 0 2 i 0 3 0 −1 and the commutation relation hiσ iσ i iσ i , j = − k . (3) 2 2 ijk 2 Lie algebra sl(2, C)

It is convenient to introduce a triplet of operators A−,A+,N, which we will call the standard triplet, satisfying commutation relations

[N,A±] = ±A±, [A+,A−] = 2N. (4)

Example of such a triplet:

1 0 1 0 0 2 0 A+ = ,A− = ,N = 1 . (5) 0 0 1 0 0 − 2

It can be constructed as another basis of sl(2, C) by transformation 1 1 A = (σ ± iσ ),N = σ (6) ± 2 1 2 2 3 Representations of sl(2, C) Let us introduce the Casimir operator 1 C = (σ2 + σ2 + σ2), (7) 4 1 2 3 in terms of the triplet A−,A+,N 1 C = (A A + A A ) + N 2. (8) 2 + − − + The Casimir operator commutes with the representation. Let us deﬁne a speciﬁc representation

l −1 l 2 l A− = ∂w + lw ,A+ = −w ∂w + lw, N = w∂w, (9) where l ∈ C. It satisﬁes the sl(2, C) commutation relations. The Casimir operator reads

Cl = l(l + 1). (10) Representations of sl(2, C)

Let us deﬁne the space of monomials

η k W := {w | k ∈ Z + η}. (11)

We have the representation

l,η η sl(2, C) 3 X 7→ X ∈ L(W ), (12)

l,η parametrized by l, η ∈ C. The action of X :

l k k−1 A−w = (k + l)w , l k k+1 A+w = (−k + l)w , (13) N lwk = kwk.

l,η l,η+n We have X = X for n ∈ Z. Representations of sl(2, C)

1. The representation Xl,η is irreducible if and only if l, −l∈ / Z + η. 2. Consider Xl,−l. Then it has an invariant subspace {wk | k = −l, −l + 1,...}. Let us define Xl,lw as Xl,−l restricted to this subspace. It is 1 irreducible if l 6= 0, 2 , 1,... 3. Consider Xl,l. Then it has an invariant subspace {wk | k = . . . , l − 1, l}. Let us define Xl,hw as Xl,l restricted to this subspace. It is 1 irreducible if l 6= 0, 2 , 1,... 1 l,−l l,l 4. If l = 0, 2 , 1,..., then X = X and we have an invariant subspace {wk | k = −l, . . . , l}. Let us define Xfin as Xl,−l = Xl,l restricted to this subspace. Representations of sl(2, C)

l, −l∈ / Z + η l − 2 l − 1 l l + 1 l + 2

lowest weight −l − 2 −l − 1 −l −l + 1 −l + 2

highest weight l − 2 l − 1 l l + 1 l + 2

ﬁnite −l − 1 −l l l + 1 Equivalence of representations sl(2, C)

For l, −l∈ / Z + η let us introduce diagonal linear map l,η l,η Wl : X → X . Γ(−l + k) W wk := wk. (14) l Γ(l + 1 + k) By direct calculation we can check that

−1 l −1−l Wl X Wl = X ,X ∈ sl(2, C). (15) This shows the equivalence of representations Xl,η and X−l−1,η for l, −l∈ / Z + η. This condition is given by the behavior of the Γ function. Consistently the Casimir operator is invariant with respect to transformation l 7→ −1 − l:

Cl = l(l + 1) = C−1−l. (16) Alternative forms of representation

Let us compute Xl,− := wlXlw−l. We obtain

l,− A− = ∂w, l,− N = w∂w − l, (17) l,− 2 A+ = −w ∂w + 2lw.

With this representation in the case of Xl,lw, the representation space is {wk|k = 0, 1,...}. For the representation Xl,ﬁn, the representation space is {wk|k = 0, 1,..., 2l}. Alternative forms of representation

Another form of representations is Xl,+ := w−lXlwl. We obtain

l,+ A− = ∂w + 2l, l,+ N = w∂w + l, (18) l,+ 2 A+ = −w ∂w.

With this representation in the case of Xl,lw, the representation space is {wk|k = 0, −1,...}. For the representation Xl,ﬁn, the representation space is {wk|k = 0, −1,..., −2l}. SL(2, R) group

It consists of 2 by 2 matrices over R whose determinant is equal ∼ to 1. Its center is {1, −1} = Z2. We have the group PSL(2, R) := SL(2, R)/Z2 with the neutral element as a center. This group is isomorphic to homographies preserving C+ := {z ∈ C : Im z > 0}. The important decomposition (valid also for SL(2, C)) h h 1 0 1 t 1 0 11 12 = , (19) h21 h22 s2 1 0 1 s1 1 where h11 − 1 h22 − 1 t = h12, s1 = , s2 = . (20) h12 h12

If h12 = 0 the decomposition is invalid. We will treat this decomposition as a tool to heuristically construct group representations. Representations of SL(2, R) Alternative form of decomposition

1 0 1 t 1 0 = es2A− etA+ es1A− . (21) s2 1 0 1 s1 1

To ﬁnd representations of the group we need to exponentiate l l l A+,N ,A−, obtaining

tAl −1 l e − f(w) = (1 + tw ) f(w + t), l etN f(w) = f(etw), (22) tAl l w e + f(w) = (1 + tw) f . 1 + wt Thus we have the representation by homographies

l −1 l l h11w + h12 h f(w) = (h11 + h12w ) (h21w + h22) f (23) h12w + h22 Representations of SL(2, R)

l,− l,− l,− By using operators A+ ,N ,A− we have

tAl,− e − f(w) = f(w + t), l,− etN f(w) = e−lt f(etw), (24) tAl,− l w e + f(w) = (1 + tw) f . 1 + wt The representation

l,− 2l h11w + h12 h f(w) = (h21w + h22) f (25) h12w + h22 These formulas can suﬀer from possible multivaluedness of power functions. This is solved by going to the representations of the universal cover SLf (2, R) or by other tricks showed in the following slides. Lie algebra sl(2, R)

Lie algebra sl(2, R) consists of matrices of the form X X X = 11 12 . (26) X21 −X11

It has a scalar product

Tr(XY ) = 2X11Y11 + X12Y21 + X21Y12. (27)

• Let us consider representation X of sl(2, R). It can be extended by complex linearity to representation of sl(2, C). Therefore representations of sl(2, R) are the same as complex representations of sl(2, C). Representations of sl(2, R) The main parameter describing representations of sl(2, C) was l and η. This parametrization is standard for the theory of su(2). In the case of sl(2, R) the popular in the literature parametrization is given by parameter m, deﬁned by m + 1 m = −2l − 1, l = − . (28) 2 The representation Xl with l expressed by m, will be denoted Xm. In our new notation we have m + 1 Amwk = k − wk−1, − 2 N mwk = kwk, (29) m + 1 Amwk = − k − wk+1. + 2

m m2 1 The Casimir operator is given by C = 4 − 4 . Representations of sl(2, R)

1. The representation preserving Wη is dented Xm,η. It is irreducible if and only if m, −m∈ / 2Z + 1 + 2η. m, m+1 2. Consider X 2 . Then it has an invariant k m+1 m+3 subspace {w | k = 2 , 2 ,...}. m,lw m, m+1 Let us define X as X 2 restricted to this subspace. It is irreducible if m 6= −1, −2, −3,.... m,− m+1 3. Consider X 2 . Then it has an invariant subspace k m+3 m+1 {w | k = ..., − 2 , − 2 }. m,hw m,− m+1 Let us define X as X 2 restricted to this subspace. It is irreducible if m 6= −1, −2, −3,.... m, m+1 m,− m+1 4. If m = −1, −2, −3,..., then X 2 = X 2 and we k m+1 m+1 have invariant subspace {w | k = 2 ,..., − 2 }. m,lw m,hw X and X give representations of PSL(2, R) for m ∈ 2Z + 1 and of SL(2, R) for m ∈ Z. Unitarity of representations of SL(2, R) Let η ∈ R. 1. Suppose that we equip the space Wη with the sesquilinear scalar product (f|g) where f, g ∈ W, defined on the canonical basis by

k k0 (w |w ) = δk,k0 . (30)

Then (X−m,ηf|g) + (f|Xm,ηg) = 0. (31)

2. Suppose we have another sesquilinear product, (f|g)m, where f, g ∈ W, deﬁned on the canonical basis by

m 1 k k0 Γ( 2 + 2 + k) (w |w )m = δk,k0 m 1 . (32) Γ(− 2 + 2 + k) Then m,η m,η (X f|g)m + (f|X g)m = 0. (33) Unitary representations of SL(2, R) We have positive scalar product in the following cases: 1. The principal series: m = iµ ∈ iR, 0 ≤ η < 1. Then Xiµ,η is unitary in the canonical scalar product (wk|wk) = 1. 2. The complementary series: −1 < m < 1, 1−|m| 1−|m| m,η − 2 < η < 2 . Then X is unitary in the scalar product m 1 k k Γ( 2 + 2 + k) (w |w )m = m 1 . (34) Γ(− 2 + 2 + k) 3. The holomorphic or lowest weight series: m > −1 , Xm,lw. It is unitary in the scalar product k! (wk|wk) = . (35) m Γ(k + m + 1)

4. The antiholomorphic or highest weight series: m > −1 , Xm,hw. Unitary representations of SL(2, R)

principal series Im m complementary series holomorphic and antiholomorphic series

Re m −1 1

For general m > −1 the holomorphic and anthiholomorphic series gives the representations of SLf (2, R). Representations by homographies

The group SL(2, R) acts on the R. Let us integrate the ﬂows m,− m,− generated by A− and N

tAm,− e − f(x) = f(x + t), (36) tN m,− m+1 t e f(x) = e 2 f(e w).

m,− 1 The ﬂow generated by A+ for η = 0 and η = 2

tAm,− −m−1 x e + f(x) = |1 + tx| f , η = 0, 1 + xt tAm,− −m−1 x 1 e + f(x) = sgn(1 + tx)|1 + tx| f , η = . 1 + xt 2 (37) Representations by homographies This yields representation of PSL(2, R) for η = 0 and of 1 SL(2, R) for η = 2

m,0 −m−1 h11x + h21 h f(x) = |h12x + h22| f , h12x + h22 m, 1 −m−1 h11x + h21 h 2 f(x) = sgn(h12x + h22)|h12x + h22| f . h12x + h22 (38)

For arbitrary η we obtain representations of universal cover SLf (2, R). The center of SLf (2, R) is {1n| n ∈ Z}. For h˜ ∈ SLf (2, R) corresponding to matrix h ∈ SL(2, R) we can ﬁnd n such that

˜ 1 h ∈ Ell 1 ∪ Hypn ∪ Parn ∪ { n} ∪ Ell 1 . (39) n− 2 n+ 2

This means that h˜ can be accessed from 1n by one-parameter path. Representations of SLf (2, R)

For h˜ belonging to nth sector (as in previous slide) we have h˜m,η = ein2πη ×

 −m−1 h11x+h21 n |h22 + h12x| f , (−1) (h22 + h12x) > 0;  h12x+h22  −i2πη −m−1 h11x+h21 n e |h22 + h12x| f , (−1) (h22 + h12x) < 0,  h12x+h22 n (−1) h12 < 0;  i2πη −m−1 h11x+h21 n e |h22 + h12x| f , (−1) (h22 + h12x) < 0,  h12x+h22  n  (−1) h12 > 0; Principal series representations

2 2 Consider spaces L (R) and L (S) and a unitary map 2 2 U : L (R) → L (S)

1 2 2 u − 1 Uf(u) = f i , (40) |u + 1| u + 1

2 1 i + x U −1p(x) = 2 p . (41) x2 + 1 i − x 2 k k0 0 Orthogonal basis on L (S): (u |u ) = δk,k0 2π, k, k ∈ Z + η. 2 Corresponding basis on L (R)

2 1 i + xk U −1uk = 2 . (42) x2 + 1 i − x

(f|g)m = 1 ZZ − f(x) − f(y)|x − y|m−1g(x) − g(y)dx dy, 2Γ(m) − 1 < m < 0.

We have the equivalence of representations for m and −m. By similar procedure as in the previous slide, we arrive at the basis in (·|·)m

π − 1 2 1+m i + xk wk = 2π cos m 2 2 . (43) 2 x2 + 1 i − x Holomorphic series representations Consider the upper complex half-plane: C+ := {z = x + iy ∈ C : y ≥ 0}, where x = Re z and y = Im z. For F , G analytic on C+, we deﬁne the scalar product 1 Z (F |G) = ym−1F (z)G(z)d2z, 0 < m; m Γ(m) C+ Z (F |G)0 = F (x)G(x)dx, R 1 Z (F |G) = ym−1F (z)G(z) − F (x)G(x)d2z, m Γ(m) C+ − 1 < m < 0.

Similarly as in preceding frames we obtain basis in (·|·)m k k − 1 m (i + z) w = π 2 2 . (44) (i − z)k+m+1 Antiholomorphic series representations are equivalent. They are obtained by taking antiholomorphic functions on C+. Thank you for your attention !