Maximal degenerate series for Sl(n, r) of rank two Текст научной статьи по специальности «Математика»

Maximal degenerate series for SL(n, R) of rank two

A.S.Rakityansky Orenburg State Pedagogical University 460000 Orenburg, Sovietskaya 19

We consider representations of the group SL(n, E) induced by characters of standard maximal parabolic subgroups P± corresponding to the partition n = (n — 2)+ 2. These representations can be realized on functions on the Grassmann manifolds of rank two. It explains the title. In this paper we restrict ourselves to the structure of these representations (irreducibility and reducibility). Other properties (intertwining operators, invariant Hermitian forms, unitarizability) will be considered elsewhere.

1 Maximal degenerate series of SL(n, R) of rank two

Let G = SL(n, M), n > 4. Any element g E G can be written as a block matrix

,-(■?) ».I)

according to the partition n = (n — 2) + 2 of n. Let P± denote the two maximal parabolic subgroups of G corresponding to this partition. They consist of upper and lower block matrices respectively:

l*:(t I (1.2)

0 c J ’ • V b c

For /x £ C, £ = 0,1, let us denote by t^,e the character of R* :

t^>e = sgnet

Define the character uof P± by formula:

= (det cY'E

where p has one of forms (1.2). Consider the representations

I^, = Ind(G,^,uw)

Let us describe these representations in ’’the compact picture”. Let K = SO(n), a maximal compact subgroup of G . One has the following decompositions

G=P+K = P~I< (1.3)

(Iwasawa type decompositions). For the corresponding decompositions g = pk the factors p and k are defined up to the factor in L = K D P+ = K fl P~ : pk = p\k\ with p\ = pl~l, ki = Ik,I € L. The subgroup L is S(0(n — 2) x 0(2)).

Take the Iwasawa type decomposition g = pk,p E P+,k E K, with g and p given by (1.1) and (1.2)

respectively. For the block c we have the equation cc1 = 77' +¿6' (the prime denotes matrix transposition).

We can take for c a symmetric positive definite matrix:

c = (77' + 66')lyf2 (1-4)

The connected component Le of L containing the identity element is S0(n — 2) x SO(2). The coset

space S = I\/Le is Grassmann manifold of rank two, the manifold of two-dimensional oriented planes in

Ru. We let K act on S from the right (i.e. S is the right coset space), so that S can be realized as the space of real 2 x n matrices

U\ ... uT Vi ... vr

with the condition ss' = E (so that the rows of s are unit orthogonal vectors) and the identification s ~ rs,r E SO(2). We shall denote by s the equivalence class containing s. Let us denote

o_ / 0 ... 1 0

v 0 ... 0 1

Let Ve{S) be the space of complex valued C°° functions on S of parity e :

<p(ws) = (det w)£<p(s), w E 0(2)

(we write functions on S as ip(s) keeping in mind the agreement above or thinking of tp to be left invariant with respect to S0(2)).

The group G acts on S (from the right):

s i—► s = s ■ g (1-5)

as follows. Let k be an element of K such that s°k = s. Decompose kg according to (1.3): kg = pk,p E

P+, k E K. Then s = s°k. The restriction of this action to K is the action by translations: s • k = sk.

The representation T~£ acts on V£(S) by the formula:

(3^.(iMW = v(5Q (delcT' (1-6)

where c is the c-block of the matrix p . The definitions (1.5) and (1.6) are well-defined.

Let us write (1.6) in a more detailed way. Using the choice (1.4), we obtain

c = (sgg's')1/2.

Therefore

C(s) = <P (\sgg's')~l,2sg) {det(sgg's1)}^12 The representation T+ is reduced to T~ by the outer automorphism g i—»• g -1(a Cartan involution):

TX,(9) = T-C(9~l)

The representations T^e are continuous [1], Ch.8, in the sense that the function T^e(g)<p from G x Ve(S) into Ve(S) in continuous, and indefinitely differentiable, i.e. for each E 'De(S) the function T^e(g)<p from G to VE(S) is of class C°°. We preserve the symbols of these representations for the corresponding representations of the Lie algebra g of G .

For n — 4 one has a homomorphism of G onto SOo(3,3) with the kernel {i-E1}. On the other hand, under T^€ this kernel goes to the indentity operator. So in fact are representations of S'Oo(3,3). They were studied in [4]. So from now on we shall assume n > 4. Notice that for n = 4 the subgroups P+ and P~ are conjugated by an inner automorphism - the conjugation by means of the matrix

0 E E 0

so that and E are equivalent.

2 Harmonic analysis on the Grassmann manifold

The manifold S = K/Le is a compact symmetric space, indeed, L is the fixed point subgroup of the involution a(k) = Ikl where I is the diagonal block matrix with the diagonal — 1,-1. The

Lie algebra t of K is the direct sum t=(+m of eigenspaces of a. The rank of S is equal to 2. The quasiregular representations 7r of K on T>(S) (the representations by translations) is the direct sum of the representations e = 0,1, acting on V£(S). Since S is a symmetric space, both 7r(f) decompose into the multiplicity free direct sum of irreducible representations. Each irreducible subspace contains exactly one (up to the factor) spherical function, it is constant on Le-orbits. So let us describe these orbits. Let us take in m the maximal Abelian subalgebra a consisting of matrices

X = U(Enl — Ein) + t2(En-i,2 — E2,n-i) where Eij is the standard matrix basic. Let A = exp a. The set s°v4 consists of points

_ ( 0 sin<2 0 ... 0 cos/2 0

S~ ^ sin<i 0 0 ... 0 0 cos<i

One has the Cartan decomposition K = LeALe. Hence any Le-orbit on S is completely defined by its

intersection with s°A. This intersection is obtained from a given point by the following transformations: a) t\ —► — ¿i, b) <2 —*• —hi c) (<i,<2) —► (*i + 7T,<2 + tt), d) (¿i,¿2) —* Therefore, Le-orbits can be

parameterized by the two functions x, y:

x = (l/2)(cos2<i -1- cos2<2), y = cos<icosi2, (2.1)

so that functions on S constant on Le-orbits are functions of x,y:

tp(s) = F(x,y) (2.2)

For variables x, y given by (2.1), points (x, y) fill out the domain D on the plane xOy defined by inequalities |y| < x < (1/2)(y2 + 1). Let ds be a /^-invariant measure on S. For functions tp invariant with respect to Le (i.e. of the form (2.2)) this measure gives rise to the measure

C|sin<isini2|n-4|sin2ii — sin2t2\dtidt2 =

= C( 1 - 2x + y2)(n~5V2dxdy (2.3)

Let (, ) denote the inner product correspoding to the measure (2.3) (with C = 1):

{FuF2) = J Fi(*. y)F2(x, y)( 1 - 2x + y2)("'5)/2 dx dy (2.4)

D

4

Similarly, L-orbits on S can be parameterized by the functions (they were used in [3]):

£ = sin2ii + sin2<2 = 2(1 — x), 77 = sin2<isin2i2 = 1 — 2x + y2 (2.5)

In [3], Koornwinder introduced a family of polynomials R°(£, tj) *n £>*7 highest term const ■

yyhich are obtained by orthogonalization of the sequence 1, ?7, £2, £*7, f?2, £3, • • • with respect to

the measure

7?a(l - £ + vfH2 ~ ^rj)1 d^dr]

on the domain 2y/rj < £ < 1 + 77.

Theorem 2.1. Irreducible consiituents of can be labelled by pairs z = (l,m) of integers

with I > m > 0. The spherical function in the corresponding space Vz^ C T>e{S) normalized by

is the following polynomial in x,y, see (2.1) and (2.2):

= (2-6)

with a = (n — 5)/2, (3 = e — 1/2,7 = 0 where £, r] are expressed in terms of x,y by (2.5).

Proof. For £=0 the theorem was proved in [3]. For e =1 the theorem is proved by the direct check that functions (2.6) satisfy differential equations defining spherical functions □.

Write down explicit expressions. We have

= (2.7)

where the summation is taken over (k, r) such that 0 <r<k<l,r<m,

M l2k o , n (-mf't-QWt-' - j)M(< + « + l,m,k,r (_/)W(a=2)W(«=3)W(|)Wr!

/-k + r,-l + m,-l-m-e-(n- 4)/2,1/2; \

4 \ -I+ r,-l - k - e - (n - 5)/2,l; /’

W»,r(*, y) = (1 - 2x + y2f+r)/2Pt-r( ),

>/1 - 2x + y

P;(/) is the Legendre polynomial, = a(a + 1)... (a + k — 1) . One can see that <I>/£) is a linear combination of functions (1 — x)p-?(l — 2x + y2)q ye where p < l,p + q < I + m.

3 The structure of maximal degenerate series representations

Let ( , ) be the inner product in L2(S,ds) :

(<p,rl>) = J <p(s)xp(s)ds (3.1)

5

The measure ds is transformed under (1 11) as follows:

ds = |det c]-n ds

Therefore the form (3.1) is invariant with respect to the pairs , T+-_n and ^T“e,Tj-_n £^:

(T*M<p,rp) = (3.2)

So T±e are unitarizable for Re// = — n/2 , the invariant inner product is (3.1). We shall see below, that these representations are irreducible. Their unitary completions form the continuous series of unitary irreducible representations.

The centralizer of L in g is one-dimensional, let us take for a basis the element

r 2 2 2 2

Zo = diag ----1,----1

[n n Tl Tl

The operator T^£{Zq) preserves the set of Le-invariant functions in Ve{S) , so that it gives rise to a

differential operator in variables x, y . This differential operator is ±2£M where

£„=( 2^-x-^)A + (x-1)!,A + /1g-I)

It follows from (3.2) that

(C„F1F2) = {F1,C-1I-nF2)

Let us take 4 vectors on the plane:

ei = (1,0), e, = (0,1), e3 = (0,-1), e„ = (-1,0)

I

and for /i £ C, £ = 0,1 define 4 linear functions of z — (I, m) :

Pi(/jl,£-,z) = (l/2)(/z -£)-/,

P2{H, £\ z) = (l/2)(/x + 1 - £) - m,

/?3(/i, £; z) = (1/2)(// + n - 3 + g) -I- m,

/?4(/i, g; z) = (l/2)(/i + n — 2 + g) + /.

There are relations between them:

e; z) + /?5_,-(a£, g; z) = n + (n - 2)/2

/?i(/i,g;z)+ $;_*(//, g;z) = -1

where

fX* = —/i — n

Lemma 3.1 The operator carries Wz = W/>m ¿o a linear combination ofWz, Wz+ei, Wz+e2 :

= (-*¡*11 + l+m)W,+ (%-!) Wz+'t +

+ (l-mf i(/-m)2 - |1 (‘‘f1 - m) W,+e,

The lemma is proved by means of the differentiation formulae for the Legendre polynomials, see, for example, [2] 10.10.

Lemma 3.2 The operator carries to a linear combination of and neighboring functions

$

(O

z+a •

c^{£) = 70 (a*, e\ z) + J2 7»• (£;2) A- (a*, e;*) *!+Ci

1—1

where

_ (<-m+l)(< + af2)(< + a=3 + £) (i + m + a^ + e)

7 (/ — m -(- |) (/ + m 4- + g) (2/ + ^ + g) (2/ •+■ + g)

(/ - m) (m + rL^p) (m + + g) (/ + m + + g)

72 (/ - m + i) (/ + m + 2f2 + e) (2m + + g) (2m + ^ + g)

(/ — m + 1) m (m — ^ + g) (/ + m -f + g)

(/ — m + (/ + m + n 2 3 + £) (2m + + g) (2m + n2 4 + g)

(/ - m) (/ + |) (/ + g) (/ + m + + g)

74 “ ~ (/ _ m + i) (/ + m + a=3 + c) (2/ + azd + g) (2/ + + g)

70 = (n - 4) (f + J) •

1 n - 6 /2 + m2 + (gf2 + g) / + (a=i -f g) m + \ + g) (^ + g)

“ n + ~1T ' (2/ + a=4 + g) (2/ + | H- c) (2m + ^ + g) (2m + 2=* + e)

The lemma is proved using Lemma 3.1, formulae (2.7), (3.3) and pairwise orthogonality of <$>i^ with respect to the inner product (2.4) and some formulae about 4F3 from [6]. We omit these rather

cumbersome calculations. In particular, in certain cases we resorted to the help of a computer (Maple

n

Let Z be the lattice of the pairs z = (/, m) of integers with / > m > 0 (K-types for 7Te'). Let us call the line /?,• (n,£\ z) = 0 on the z-plane a barrier if it meets the lattice Z and the intersection Zn{/3i > 0} does not coincide with Z. Then n has to be integer. If the line /?,• = 0 is a barrier, we denote by V{ (/¿,0 the subspace given by the inequality /?,• (/¿,£;z) > 0 , i.e. the sum of 7^ with z satisfying this inequality. The following lemma follows from the definition of /?,• immediately.

Lemma 3.3 For given fi,e the number of barriers is not greater than 1.

Let us point out p.,6 (p. e Z) for which /?,• = 0 is a barrier (the sign = denotes the congruence modulo

2);

i = 1 : n > e,n = e

i = 2 : fi > e — l,/i = £— 1

* = 3 : /i < 1 — n — g, /i = 1 — n — g (i.e. p.* > g — 1, n* = e — 1 )

i = 4 : fi < —n — £,// = n + g (i.e. /i* >£,ii*=£)

___________________________________________________________________________________Вестник ТГУ, т.З, вып. 1, 1998

Theorem 3.4 If the line /?» (/i,e;z) = 0 is a barrier, then the subspace Vi (/¿,e) is invariant for T^e. There is no other invariant subspaces. The representation T^e is irreducible except when \i is integer and

satisfies one of two inequalities: /i > z — 1 or ц < 1 — n — e (i.e. //* > e — \). In this case there is exactly

one invariant subspace Vi (fj.,e)(see the list after Lemma 3.3).

The proof is carried out analogously to the proofs of similar theorems in [4], [5].

References

[1] Bourbaki N. Elements de Mathématique. Livre VI: Integration, Ch. VI—VIII, Hermann, Paris, 1965.

[2] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. Higher Transcendental Functions I, II. Me Graw-Hill, New York, 1953, 1955.

[3] Koornwinder T. Harmonics and spherical functions on Grassmann manifolds of rank two and two-variable analogues of Jacobi polynomials. Lect. Notes Math., 1977, v.571, 141-154.

[4] Molchanov V.F. Representations of the pseudo-orthogonal group associated with a cone. Mat. Sb., 1970, t.81, N3, 358-375. Engl, transi.: Math. USSR-Sb. 1970, v.10, N3, 333-347.

[5] Molchanov V.F. Representations of the pseudo-unitary group associated with a cone. Functs. Anal. Ulyanovsk, 1984, N22, 55-66.

[6] Prudnikov A.P , Brychkov Yu.A. , Marichev O.I. , Integrals and Series. Complementary Chapters., Moscow, Nauka, 1986.