CC BY
29
MSC 43A85, 22E46
Eigenvalues of the Berezin transform 1
© S. V. Tsykina
Derzhavin Tambov State University, Tambov, Russia
We consider polynomial quantization on para-Hermitian symmetric spaces G/H with the pseudo-orthogonal group G = SO0(p, q). We give explicit expressions of the Berezin transform in terms of Laplace operators. For that, we compute eigenvalues of the Berezin
G/H
Keywords: svmplectic manifolds, pseudo-orthogonal groups, polynomial quantization, Berezin transform.
We consider polynomial quantization, a variant of quantization in the spirit of Berezin, on para-Hermitian symmetric spaces G/H with pseudo-orthogonal group G = SO0(p, q), For all these spaces, the connected component He of the subgroup H containing the identity of G is the direct product SO0(p — 1,q — 1) x SO0(1,1), so that G/H is covered by G/He (with multiplicity 1, 2 or 4). The dimension of G/H is equal to 2n — 4, where n = p + q, the signature is (n — 2, n — 2). We restrict
G/H G G
In polynomial quantization, covariant and contravariant symbols are polynomials G/H
in [1]. Our spaces G/H with group G = SO0(p,q) have rank two.
One of the main formulas in polynomial quantization is the expression of the
G/H
expression one needs to know eigenvalues of the Berezin transform on irreducible finite dimensional subspaces.
Thus this paper has two goals: for our spaces G/H with group G = SO0(p, q), we first compute eigenvalues of the Berezin transform, and then we write explicit expressions of the Berezin transform in terms of Laplace operators.
Let us introduce in Rn the following bilinear form:
where Ai = 1 for i ^ p, Ai = — 1 for i > p, and x = (xi,..., xn), y = (yi,..., yn) are vectors in Rn, The group G = SO0(p,q) is the connected component of the identity
1 Supported by the Russian Foundation for Basic Research (RFBR): grant 09-01-00325-a, Sci. Progr. "Development of Scientific Potential of Higher School": project 1.1.2/9191, Fed. Object Progr. 14.740.11.0349 and Templan 1.5.07
n
i= 1
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in the group SL(n, R) preserving the form [x, y], We consider that G acts on Rn from the right: x M xg so that we write vectors x G Rn in the row form. We consider the general case p > 1, q > 1,
Let us introduce on G/H horospherical coordinates £, n where £ and n are vectors in Rn-2 as follows.
Let C be the cone [x,x] = 0 x = 0 in Rn, The group G acts on it transitively. Let us take in the cone two points: s- = (1, 0,..., 0, —1) s+ = (1, 0,..., 0,1), and consider the following sections of the cone:
r- = {xi — xn = 2}, r+ = {xi + xn = 2}.
These sections intersect almost all generating lines of the cone. Therefore the linear G
on r- and r+. We introduce coord mates £ on r- and n on r+ as follows: for points u G r- and v G r+ we set:
u = u(£) = (1 + <£,£), 2£, —1 + (£,£)),
v = v(n) = (1 + <n,n), 2n, 1 — <n,n)),
where <<£, 0) denotes the following bilinear form in Rn-2:
n- i
<^,0) = Ai^j0j.
i=2
Notice that
[u,v] = —2N (£,n^
where
N (£,n) = 1 — 2(£,n) + <£,£)<n,n).
These coordinates £, n °n G/H have to satisfy the condition N(£, n) = 0,
We refer to [1] concerning covariant and contravariant symbols. The key moment is to construct explicitly an operator A, for which both covariant and contravariant symbols are written explicitly,
G
G/H
representations with highest weights (k + l,k — l), where k,l G N k ^ /, acting on subspaces Hk,i- The weights are taken with respect to the secondary diagonal Abelian subalgebra of the Lie algebra of G, Here N = {0,1, 2,...},
Further we use the notation:
a[r] = a(a + 1)... (a + r — 1), a(r) = a(a — 1)... (a — r + 1).
Theorem 2.6 Let a be the parameter labeling co- and contravariant symbols. The
eigenvalues bk,i (a) of the Berezin transform Ba on are given by
( ) (a + n — 2)[k] (a + m + 2)[l]
k’1 a(k) (a + m)(l)
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In order to find these eigenvalues, it is sufficiently to trace some polynomial in Hk,i- As such polynomial, we take the lowest vector
F = (n,n)1 K6 - £»-i)(n,n) - (n2 - Vn-i)}k-1
N (e,n)k •
It is the eontravariant symbol of the operator
. = 1 ( d d \k-1 = 2k+1 a(k)(a + m)(0 V d& + d&-J ° ’
where
n-1 d 2 A=£ Ak ^
k_2 'dS
In turn, this operator A has the covariant symbol just the polynomial
Fi = bk,i (a) ■ F.
Now we can write an expression of the Berezin transform Ba in terms of Laplace operators, see also [2], For our space G/H, we have two Laplace operators A2 and A4, they are differential operators of the second and of the fourth order, respectively.
Theorem 2.7 Denote m = (n — 4)/2. We have
= r(a + n — 2 + k)r(a + 1 — k)
a r(a + n — 2)r(a + 1)
x
r(a + m + 2 + 1)r(a + m +1 — l) r(a + m + 2)r(a + m +1)
where k, I some variables. In fact, the right hand side depends on A2 and A4 only, where A2 = 2(a1 +a2^ A2 = 16(a1a2 — ma1+m2a2) and a1 = k(k+n—3), a2 = 1(1 + 1). A2 A4 A2 A4
References
1, V, F, Molchanov, N, B, Volotova, Polynomial quantization on rank one para-Hermitian symmetric spaces, Acta Appl, Math,, 2004, vol. 81, Nos, 1-3, 215-232,
2, S, V, Tsykina. Polynomial quantization on para-Hermitian symmetric spaces with pseudo-orthogonal group of translations. International workshop "Idempotent and tropical mathematics and problems of mathematical physics Moscow, Aug, 25-30, 2007, vol. II, 63—71.
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