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REPRESENTATIONS OF THE GENERALIZED LORENTZ GROUP ON DIFFERENTIAL FORMS ON A CONE1
A. V. Opimakh
G. R. Derzhavin Tambov State University, Russia
Let G be the pseudo-orthogonal group SOo (l,n — 1) (also called the generalized Lorentz group) acting linearly on the space K". Let Xq be the cone —x\ + x\ ... + x\ — 0, x\ > 0, in Rn. The group G acts on this cone transitively.
Representations Ta, a 6 C, of the group G in homogeneous functions on the cone were investigated in [5].
In this paper we study representations of the group G in homogeneous differential forms on the cone, of degree one. The representation is the tensor product Ta Q p\ of the representation Ta and the tautological representation p\. It decomposes into the direct sum of two representations, one of them is isomorphic to Ta, the other, denoted by Ra, is subject to study. We give a description of the structure of the representations Ra and determine intertwining operators (in the matrix form) for this series. We use results of our papers [2], [3],
[4]. To avoid various exceptions and particular cases, we consider a general case: n > 5.
Let us introduce some notation and agreements.
By N we denote {0,1,2,...}.
For a manifold M, let V(M) denote the Schwartz space of compactly supported infinitely differentiable C-valued functions on M, with a usual topology. Let A 1(M) denote the space of differential forms on M of degree one with coefficients of class C°°.
For a differentiable representation of the Lie group G, we retain the same symbol for the corresponding representations of the Lie algebra g of G and of the universal enveloping algebra. We use the notation for a ’’generalized power”:
afml = a(a + 1)... (a + m — 1)
(instead of the Pochhammer symbol (a)m).
1 The generalized Lorentz group
Let us equip the space Mn, n > 5, with the following bilinear form
[x, y] = -Xiyi + ... + xnyn
Tl
i=l
where
Ai = —1, A2 = ... = A„ = 1,
and x = (x\,..., xn), y = (yi,... ,yn) are vectors in Rn.
Supported by the Russian Foundation for Basic Research (grant No. 05-01-00074a), the Scientific Programs ’’Universities of Russia” (grant No. ur.04.01.465) and’’Devel. Sci. Potent. High. School”, (Templan, No. 1.2.02).
Let G denote the group SOo (1, n — 1), the connected component of the identity in the group of linear transformations with determinant 1 of the space R", preserving the bilinear form [x,y\. The latter means that g E G satisfies the condition
g'ig = i, (i.i)
where the prime denotes matrix transposition and I is the matrix of the bilinear form [x,y], i.e.
I = diag {—1,1,..., 1}.
We assume that G acts linearly on Rn from the right: x \-¥x = xg. In accordance with that, we write vectors in the row form.
Let K denote the subgroup of G consisting of orthogonal matrces, i.e. K = G fl SO(n). By (1.1) we obtain that K consists of matrices k G G commuting with I, i.e. kl = Ik. Therefore, K is the subgroup of fixed points of the following involution r of the group G\
r{g) = Igl (= g'~l).
Let us write matrices g G G in the block form corresponding to the partition n = 1 + (n — 1). The subgroup K consists of block diagonal matrices
k=(l °
\0 »
where v G SO(n — 1). This subgroup is a maximal compact subgroup of G.
The Lie algebra g of G consists of real matrices X of order n satisfying the condition X'l + IX = 0. Dimension of g and G is equal to n(n — 1)/2.
The involution r of G induces an involution of g, which we denote by the same letter. The algebra g splits into the direct sum of +1, — 1-eigenspaces of r: g = t + p. The space t is the Lie algebra of the subgroup K. This decomposition is orthogonal with respect to the Killing form. The Lie algebra 6 and the space p consist of matrices respectively:
° 0\ / 0 v
0 u / ’ P ' V ip' 0
Here u is a matrix in the Lie algebra of SOo(n — 1), i.e. a real skew-symmetric matrix of order n — l, and ip is a row in IRn_1.
Here are commutation relations:
[fi,p] Cp, [C,t] C e, [p,p] c t
Therefore, the subgroup K preserves the subspace p under the adjoint action (recall we consider the right actions of groups):
Z k~lZk, k G K.
For a basis of the algebra g, we take matrices Lj,j = Eij — XiEji, i < j , where Eij is the matrix unit (1 stands at the place (i,j) and 0 at other places). In particular, matrices L\j = E\j + Ejj, j = 2,... ,n, form a basis of p, and matrices Lij — Eij — Eji, 1 < i < j ^ n, form a basis of t. The space p has dimension n — l, the dimension of 6 is equal to (n — 1 )(n — 2)/2.
Rank of algebra t is equal to [(n — l)/2] ([a] denotes the entier part of a number a G R). For a Cartan subalgebra of 6 we take the subalgebra a with the basis £2fc,2fc+i5 k = 1,..., [(n — 1)/2].
Let n and n denote the positive and negative root subalgebras of Ec. For n odd, a basis in n is formed by elements
Ek,m = 2^2k>2m + *-^2fc,2m+l ± *(-^2/c+l,2m + *-^2fc+l,2m+l) 15 where 1 ^ k < m ^ [(n — l)/2]. For n even, to these elements one has to add elements
Ek — -t'2A;,n HJ2k+l,m 1 ^ k [(u 2)/2].
To obtain a basis in n, we have to take elements of the basis in n with complex conjugation.
Let 7t(°) denote the representation of K by rotations on the space A1(5') of differential forms of degree one on the sphere S. It is obtained when we restrict the representation to the sphere S. Since on the sphere S we have Q = 1 hence dQ = 0, the series of representations 7rm acting on HmdQ disappears. We obtain
Theorem 2.2 The representation of the group K by rotations on the space A1 (S') decomposes into the direct multiplicity free sum of irreducible representations 7rm, and nmtx, m ^ 1.
Dominant vectors of representations nm are d{(x2 + ix3)m}, dominant vectors of representations 7rTOii are given by (2.2).
3 Representations associated with a cone
The group G = SOo(l,n — 1) preserves manifolds [x, x] = c, c € R, in Mn. Let Xq be the cone \x,x] — 0, x\ > 0, in K71. The group G acts on this cone transitively. For coordinates on Xq, we can take variables X2,... ,xn.
In this Section we consider representations of G by translations on spaces of homogeneous functions and homogeneous differential forms on the cone Xq.
First we consider representations of G on homogeneous functions. They were investigated in [5]. Let us recall results from [5].
Let a E C. Let Va(X0) denote the space of functions in C°°(Xq) of homogeneity a:
f(tx) = taf(x), x E Xq, t > 0.
The group G preserves this subspace. Denote by Ta the corresponding representation of G.
The section of the cone Xq by the hyperplane x\ = 1 is just the sphere S introduced above.
Functions / in Va(Xo) are completely defined by their restrictions ip to S. These restrictions give the whole space T>(S). In this realization the representation Ta acts as follows:
{Ta{g)ip){s) = K(^)(s0)i’
where s E S and (sg)i is the first coordinate of sg. The restriction of the representation Ta to the subgroup K is the representation 7r^°\ see Section 2. It is the direct sum of representations 7rm, m ^ 0, acting on spaces Hm(S). The space p in the Lie algebra g links i^-types as follows: <
Ta(p)Hm(S) = {a - m)Hm+i(S) + (a + n - 3 + m)Hm-i{S).
These barrier functions a — m and a + n — 3 + moim give possibility to describe the structure of the representations Ta. Namely,
Theorem 3.1 (see [5]). For all a, with the exception of a E N and a E 2 — n — N, the representations Ta are irreducible. If a E N, then T>(S) has an invariant irreducible subspace H~, it is finite-dimensional and is the sum of Hm(S) with m ^ a. The factor-space is irreducible.
If cr E 2 —n —N, then the pictire is dual: T>(S) contains an invariant irreducible subspace H+, it is the sum of Hm(S) with m ^ 3 — n — a, the factor-space is irreducible and finite-dimensional.
Now consider representations of G on homogeneous differential forms of degree one.
The group G acts on the space of differential forms A.1{Xq) by translations: an element g E G carries a differential form u — Y^^k(x)dxk to the differential form ui =
Let A1 (Xq), a E C, be a subspace of A1(A’o) consisting of forms of homogeneous degree a:
xi-^tx ~ t~UJ, t ]> 0.
Let denote the representation of K by rotations on the space A1 (5') of differential forms of degree one on the sphere S'. It is obtained when we restrict the representation to the sphere S. Since on the sphere S we have Q = 1 hence dQ = 0, the series of representations 7rm acting on HmdQ disappears. We obtain
Theorem 2.2 The representation tt^1) of the group K by rotations on the space A1 (S') decomposes into the direct multiplicity free sum of irreducible representations 7rm, and m ^ 1.
Dominant vectors of representations 7rTO are d{(x2 + ix^)m}, dominant vectors of representations ■Km, 1 are given by (2.2).
3 Representations associated with a cone
The group G = SOo(l,n — 1) preserves manifolds [x,x\ = c, c G R, in R”. Let Xq be the cone
[x,x] = 0, x\ > 0, in Rn. The group G acts on this cone transitively. For coordinates on Xq, we
can take variables x'2- ..., xn.
In this Section we consider representations of G by translations on spaces of homogeneous functions and homogeneous differential forms on the cone Xq.
First we consider representations of G on homogeneous functions. They were investigated in
[5]. Let us recall results from [5].
Let a G C. Let T>a(Xo) denote the space of functions in C°°(X0) of homogeneity a:
f{tx) = taf(x), x G X0, t > 0. f.'
The group G preserves this subspace. Denote by Ta the corresponding representation of G.
The section of the cone Xq by the hyperplane X\ = 1 is just the sphere S introduced above.
Functions / in Va(X0) are completely defined by their restrictions ip to S. These restrictions give the whole space V(S). In this realization the representation Ta acts as follows:
(T„(9)V)M = v(^-)(ss)f,
where s 6 S and (sg)i is the first coordinate of sg. The restriction of the representation Ta to i
the subgroup K is the representation ir(°\ see Section 2. It is the direct sum of representations flVnj m ^ 0, acting on spaces Hm(S). The space p in the Lie algebra g links if-types as follows:
Ta(p)Hm(S) - (a - m)Hm+1(S) + (a + n - 3 + m)Hm^i(S).
These barrier functions a — m and a + n — 3 + mofm give possibility to describe the structure of the representations Ta. Namely,
Theorem 3.1 (see [5]). For all a, with the exception of cr £ N and a G 2 — n — N, the representations Ta are irreducible. If a £ N, then T>{S) has an invariant irreducible subspace H~, it is finite-dimensional and is the sum of Hm(S) with m ^ a. The factor-space is irreducible.
If a G 2 —n —N, then the pictire is dual: V(S) contains an invariant irreducible subspace , it
is the sum of Hm(S) with m ^ 3 — n — a, the factor-space is irreducible and finite-dimensional. j
Now consider representations of G on homogeneous differential forms of degree one.
The group G acts on the space of differential forms A1(A'o) by translations: an element g G G carries a differential form u = Y]u>k(x)dxk to the differential form u> = Y\uk(x)dxk
Let A1 (Xq), cr G C, be a subspace of A1(A’o) consisting of forms of homogeneous degree a:
428
The group G preserves this subspace. Denote by the corresponding representation of G.
Introduce on the cone Xq polar coordinates x = rs, where r = x\ > 0, s € 5. Any form u; in A1 (Xq) decomposes into the following sum:
u = df + r°(, (3.1)
where / = ra(f, ip E T>(S), ( E A1 (S'). On functions / = raip, and hence on their differentials, the group G acts by means of the representation Tai see above. Therefore, formula (3.1) gives the decomposition of into the sum of two representations:
=Ta + Ra. (3.2)
The representation Ra acts on A1 (S’) in the following way. Let us take for local coordinates on S variables S2, ■ ■ ■, sn omitting someone of them. Then for a form £ = ^ (j(s)dsj in A1 (S') we have
Ra(g)C = (ss)i
where s = {sg)/{sg)i. It remains to study Ra.
The restriction of Ra to K is the representation of K by rotations on A1 (5) considered in Section 2. It does not depend on a. Recall that it decomposes into the direct multiplicity free sum of the representations 7rm and 7rm>i, m ^ 1.
Let Vm and Wm, m ^ 1, denote subspaces of A1(S'), where representations 7rm and 7rm>i act, respectively. Denote
V = Y,Vm, W = J2wm, rn>l-
Theorem 3.2 Operators Ra(X), X E p, link above-mentioned K-types as follows:
Ra(p)Wm = [a+ n-4 + m)Wm-i + [a+ n - 4)Vm +(a - m-l)Wm+i,
Ra(p)Vm = (cr + n - 4 + m)Fm_i + aWm + (<? - m - l)Vm+1.
These barrier functions a + n — 4 + m, a + n — 4, cr, a — m— 1 give a full information concerning the structure of Ra. Namely, we have
Theorem 3.3 The representation Ra is irreducible for all a with the exception of a E N and
a E 4 — n — N. If a = 1, 2,..then the space A1(S<) has an invariant irreducible subspace
+ Wm), m ^ a — 1, it is finite-dimensional. The factor-space is irreducible. In the case a — 3 — n, 2 — n,... the situation is dual: A1 (S') contains an invariant irreducible subspace YKVm + Wm), m ^ 4—n—a, the factor-space is irreducible and finite-dimensional. If a = 0, then
V is invariant and irreducible, the factor-space is isomorphic W and irreducible. If u = 4 — n, then the picture is dual: W is invariant and irreducible, the factor-space is isomorphic V and irreducible.
4 Intertwining operators
In this Section we write intertwining operators for both series: Ta and in the matrix form.
For the first series the answer was obtained in [5]. It consists in the following. Let A be an operator in T>(S) intertwining Ta with T/;, i.e.
Tll(g)A = ATa(g), gEG.
Such an operator A exists only for u = a and u, = 2 — n — a. If u = a, then A is a scalar operator. Let fi = 2 — n — a, then an operator A on every Hm(S) is multiplication by a number
am(a). In irreducible case, i.e. for all cr, with the exception of a G N and a G 2 — n — N, provided a0(a) = 1 these numbers are expressed as follows:
, N T(a + n-2 + m)T(-a)
am(v) = -p7-------\t-!/ .------TyT I4'1)
1 (m — a)L [cr + n — 2)
(a + n — 2)tml
~~ (-cr)H
If cr G N, then A vanishes on H~ and
(a + n + m — 3)!
(4.2)
«rn(ff) = C
(m — a — 1)!
If cr G 2 — n — N, then A vanishes on H+ and coefficients am(a) with m ^ 2 — n — a are expressed by the same formulas (4.1), (4.2).
Now consider the second series. Let C be an operator on A1^), intertwining with t[ 1 -1. Accordingly to the decomposition = Ta + Ra, see (3.2), C is the sum of two operators A and B which intertwine Ta with and Ra with respectively. As stated above, the operator A intertwines either Ta with Ta (then it is a scalar operator) or Ta with T2_n_(T. For the second operator B we have the following theorem.
Theorem 4.1 An operator B on A1 (5) intertwining Ra with Ru exists only for v = a and v = 4 — n — a. In both cases it is unique up to a factor, for v = a it is a scalar operator. If v = 4 — n — a and Ra is irreducible, then the operator B vanishes on the irreducible invariant subspace for Ra, and its image is the irreducible invariant subspace for R^n~a.
The theorem follows from exploit formulas for eigenvalues vm(a) and wm(a) of the operator B on K-types Vrn and Wrn respectively. These formulas are proved by means of Theorem 3.2. For v = cr these eigenvalues are equal to each other. For v = 4 — n — a. except the irreduble case cr G N and a G 2 — n — N, we have (up to a factor)
VmACT =
wm{a) =
If cr = 1,2,..., then
Vm{cr) =
r(cr + n — 3 + m)F(l — cr) T(cr + n — 3)T(1 — cr + m) {a + n- 3)H (1 - tj )[ml ’
V(a n — 3 4- m.'irf—a)
\ ' " ■ • • - V -/
F(cr + n — 4)T(1 — a + m) (cr + n — 4)tm+1l (_cr)[m+1]
(a + n- 3)H (m - cr)!
1 (cr + n — 4)[m+1l ^m(cr) =
If cr = 4 — n — k, k = 1,2,..., then
(*-1)!
< \ ( i \m
= \-l)
(k — m — 1)! (k + n — 3)[ml ’
430 Wm(a) = (-l)m+1-----------------------
(vx
(k — m — 1)! (k + n — 4)tm+1]
Finally, if a = 0, then
(n — 4 + m)\
vm(a) = 0, wm(a) =-------------j-----,
m\
and if a = 4 — n, then
Tfl)
vm(&) = 7--------------x7> Wm(o-) = 0.
(n — 4 + my.
5 Example: n = 7
In this Section we show explicitly how an element L G p links if-types in the representation Ra for n — 7. It is sufficient to consider only one element in p. For such an element L we take Lyi-It is convenient to introduce new variables instead of £2,..., £7:
U = X2 + ix3, V = £4 + iX5, W = X6 + 1X7,
U = X2 — 2X3, V = £4 — ix$, W = Xq — 1X7.
For brevity, denote
£ = du, rj = dv, £ = dw, £ = du, rj = dv, ( = dw.
Here are operators corresponding to elements of the negative root subalgebra n in the representation :
r, - ^ ^ 7 ^
= ~V^~ + UTT ~ V at +
cm aw at, or)
„ d _ d d-d F2 = -V— + U-Z -r)— +
du dv dt, or)
_d _d -d -d
d _ d d-d
rr — d -7 9 _9
#1 = ^ a----Co- +
dv dw dr) oQ
u _ <9 ^ _ <9 f d ^_d
±±2 ~ ~Wdv + VftB ~sdr)+r,l%’
Dominant vectors in Vrn and Wm are
em = fm = um~lvi - umT),
respectively. We have
Ra(Li2)em = (a -m- l)Xem+i + oY 1 + {a + m + 3) cem-i,
where
X m! +
2 1 (m + l)(m + 2)
(m + 1 )(m + 2) 1
l)(m + 3 (m — l)(m + 4)
2(m + 2)(m + 3) ’
V —________________________________( Q. _L W. J
(m + l)(m + 3) v 2 7
and
Ra{Li2)fm — (a ~ m — l)Xfm+1 + (a + 3)yem + (a + m + 3) c/m_i,
where
- (m2 + 5m + 9)FiF2 +
2m(m + 2)2(rn + 4) I
— m(m + 5) [3HiS2jF2 + H2S1F2 + SiS2]},
1 F m(m + 4) 2’
(m2 — l)(m + 4)
2 m(m + 2)
REFERENCES
1. D. Levine. Systems of singular integrals on spheres. Trans. Amer. Math. Soc., 1969, vol. 144, 493-522.
2. A. V. Opimakh. Harmonic analysis on the space of differential forms of degree one on a sphere. Vestnik Tambov Univ., 2004, tom 9, issue 1, 96-97.
3. A. V. Opimakh. Representations of the generalized Lorentz group on differential forms of degree one on a cone. Vestnik Tambov Univ., 2004, tom 9, issue 1, 98-100.
4. A. V. Opimakh. Intertwining operators for representations of the generalized Lorentz group on differential forms on a cone. Vestnik Tambov Univ., 2005, tom 10, issue 1, 51-53.
5. N. Ya. Vilenkin. Special Functions and the Theory of Group Representations, Nauka, Moscow, 1965. Engl, transl.: Transl. Math. Monographs 22, Amer. Math. Soc., Providence R.I. 1968.
Y =
