CC BY
24
MSG 22E46, 22E43, 22E60
On normalization of representations 1
© R. S. Ismagilov
N. E. Bauman Moscow State Technical University, Moscow, Russia
It is well known that principal series representations Ts of the group SL(2, R) satisfy the
condition Ts ~ T-s for s = ±1, ±2,____ In I960 Kunze and Stein constructed a family
of representations V(s), holomorphic in the strip |Re s| < 1 and satisfying the condition: V(s) = V(—s), V(s) ~ Ts, |Re s| < 1; this family was called a normalized family. In the paper we discuss different modifications of this subject and apply normalized families to construct representations of rather general kind
Keywords: Lie groups and Lie algebras, representations, Laplacian, spectrum, normalized families of representations
§ 1. Example
We begin with the following example in order to explain a notion "normalization". Let a group G act on a connect complex manifold X bv holomorphic transformations x M xg. Let x M f (x), x E X, be a holomorphic function with values in n x n matrices. Suppose that on some open subset X0 C X the matrix f (x) has simple eigenvalues and f (xg) ~ f (x) for any x E X0 (here ~ stands for matrix similarity). We are going to construct another holomorphic function x M fi(x), x E X, with values in n x n matrices, such that f (x) ~ fi(x) on some open dence subset X1 C X and f1(xg) = f1(x) x E Xg E G, To do it, we consider the characteristic polynomial of the matrix f (x):
det(AE — f (x)) = A” + dn-1(x) A”-1 + ... + d1(x) A + d0(x)
and put
0 ... 0 —d0(x) ^
0 ... 0 —d1(x)
0 ... 1 —d”_1(x)
1 Supported by the Russian Foundation for Basic Research (RFBR): grants 11-01-00790-a, 09-01-00325-a
f1(x)
0 1
0
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This function satisfies our requirement. It is called a normalization of the initial function.
§ 2. Normalization of representations
The first example of normalization of representations was constructed by Kunze and Stein in [1] for the group SL(2, R),
It is convenient to replace this group bv the isomorphic group G = SU(1,1), Denote by r the circle {u E C, |u| = 1}. The representations Ts, s E C, of the group G act on the space L2(r) by
(T,(g)/)(u) = |<m + SI'-1 f (^) , g = ( a a ) . (1)
There is a symmetry relation: T-s ~ Ts if s = ±1, ±2,....
Kunze and Stein have constructed a holomorphie family of representations V(s) for the strip {s E C : |Re s| < 1}, such that V(s) ~ T^d V(s) = V(—s) for all s in this strip. Later the author constructed a normalization V(s, a), where a > 1 is an arbitrary number, for the strip {s E C : |Res| < a}, see [2]. This strip contains "singular points" ±1, ±2,..., so that V (s,a) ~ Ts, V (s,a) = V (—s,a) for all s in s = ±1, ±2, . . .
The similar prolem can be also posed for other groups. In [3] the group Mot(R”) R”
holomorphie on the whole complex plane. Recently the authors of [3] have considered the group of hyperbolic motions.
§ 3. Applications: construction of "arbitrary" spherical representations
Consider the group G = SU(1,1) and its representation T by bounded operators in a Hilbert space H, Assume that this representation is spherical with respect to the rotation subgroup K (diagonal matrices). It means that the representation T is unitary on K and the subspace H0 = {h : T(k)h = h, k E K} generates H, The question is: how to "describe all" such representations?
We do not try to set the problem in strict terms. Instead we consider the following (rather natural) approach to this problem.
Consider the Laplaeian A on G, its image T(A) under T and the restriction A of T(A) to H0, Thus the spherical representation T results in a pair (H0, A), It is easy to
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prove that the spectrum of A lies in the domain Da = {z = x+iy,x < —(y/2a)2+a2} with some a and for points z E Da we have ||(A — z/)-1|| < C |Re (^z)|,
T
(H0, A)? This is an "inverse representation problem" for our group G,
Before discussing this problem, let us begin with a very simple construction of spherical representations. Consider the Hilbert space L2(r,H0) of veetor-funetions, an operator S in H0 and define a representation TS bv the formula (1) where the number s is replaced by this operator S. The operator S is assumed to generate an one-parameter subgroup t M exp(tS), One would think that we obtain a suffieentlv big family of representations. However such an impression is false. The reason is
A
representation (in the sense just explained) has the form A = (S/2)2 — (1/4)/, so that the operator 4A + / admits a square root. It is known however that there exist operators even bounded which have no square root. So we see that the operators TS do not solve our "inverse problem",
V(a, s)
with sufficiently big parameter a. Since the function s M V(s, a) is even (and hence is a holomorphie function on s2), we can substitute V4A + / instead of s (although the operator may 4A + / have no square roots). This leads to a solution of our "inverse problem", (Of course this substitution is possible under some restrictions;
A Da
discussed in [2].)
The "inverse"problem for the group Mot(R”) was investigated in [3],
The authors of [3] are going to discuss the further development of this subject in a forthcoming paper.
References
1, E, A, Kunze, E, M, Stein, Uniformly bounded representations and harmonic 2x 2
SL(2, R)
Sbornik, 1967, vol. 74, No. 4, 495-515.
3. E. S. Ismagilov, Sh. Sh. Sultanov. A normalized family of representations of the group of motions of Euclidean space and the inverse problem of representation theory for this group, Matem, Sbornik, 2005, vol. 195, No. 12, 47-56.
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