The Spectrum of the Symmetric Space SP (l)/SU(l)

Gr. Tsagas and K. Kalogeridis

Abstract

The aim of the present paper is to determine the spectrum of the Laplace operator on the functions on SP(l)/SU (l).

Mathematics Subject Classification: 58G25 Key words: symmetric space, Lie algebra, Lie group, roots and Cartan subalgebra

1 Intoduction

Let (M, g) be a compact Riemannian manifold of dimension n. From this Riemannian manifold we obtain the vector spaces Λ^{q}(M, R), q = 0, 1, ..., n. If we apply the Laplace operator Δ = dδ + δd on Λ^{q}(M, R) we obtain the spectrum Sp^{q}(M, g).

One problem in the spectral theory is to determine Sp^{q}(M, g), where (M, g) is a known compact Riemannian manifold.

The aim of the present paper is to determine Sp^{0}(M, g), that means the spectrum of Δ on the functions on M, where M = SP (l)/SU(l) and g is the Riemannian metric on M, coming from the Killing-Cartan form of the Lie algebra sp(l).

2 Basic elements for symmetric spaces

Let M = G/H be a compact homogenous space. One of the problems concerning of the spectrum is to calculate

Sp^{q}(M = G/H, g), q = 0, 1, ..., dim M (2.1)

Let t and v be the Lie algebras of G and H respectively. If m is the tangent space of M at its origin, we have the relation

t = v ⊕ m (2.2)

Let ⟨, ⟩ be a positively definite inner product on the vector space m which comes from the Riemannian metric g, which is an ad(H)-invariant. Conversely, an inner product on m which is an ad(H)-invariant gives a G-invariant metric on the homogeneous space M, which is also called homogeneous manifold.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 109-114.

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We assume that the Lie group G is compact and semisimple and the Lie subgroup H of G is its closed subgroup.

Let b be the Cartan subalgebra of t. The set of the simple roots which are linear forms on the bC, that is elements of bC^{∗}, is given as follows:

Ψ = {ψ_{1}, ..., ψ_{l}} , (2.3)

where l = dim b.

We denote by Λ the set of all roots of bC,

Λ = {λ = ∑_{i=1}^{l} r_{i}ψ_{i}/r_{i} ∈ Z} (2.4)

and with Λ^{+} the set of all positive roots, so we have

Λ^{+} = {λ = ∑_{i=1}^{l} r_{i}ψ_{i}/r_{i} ≥ 0} .

For each element p ∈ b^{∗}, we take p^{∗} = \frac{2p}{< p, p >} so we can construct the following

set:

Ψ^{∗} = {ψ_{1}^{∗}, ..., ψ_{l}^{∗}} (2.5)

where:

ψ_{1}^{∗} = \frac{2ψ_{1}}{⟨ψ_{1}, ψ_{1}⟩}, ..., ψ_{l}^{∗} = \frac{2ψ_{l}}{⟨ψ_{l}, ψ_{l}⟩}

and consequently for each λ, we take λ^{∗} defined as follows:

λ^{∗} = \frac{2λ}{< λ, λ >}. (2.6)

Now, we can define the fundamental weights μ_{i} ∈ b, i = 1, ..., l, via

< μ_{i}, ψ_{j} >= δ_{ij}, i = 1, ...l, j = 1, ..., l (2.7)

We construct the set of the weights

B = {μ = ∑_{i=1}^{l} m_{i}μ_{i}/m_{i} ∈ Z,m_{i} ≥ 0} . (2.8)

If we consider the set

{I = μ ∈ b/ < μ, ψ_{i}^{∗} >∈ Z, i = 1, ..., l} , (2.9)

then B is equivalent to the set N_{1} = {μ ∈ / < μ, λ >≥ 0, ∀λ ∈ Λ} , (2.10)

or

N_{2} = {μ ∈ I/ < μ, ψ_{i} >≥ 0, i = 1, ..., l} . (2.11)

The above relations are used for the calculations of the spectrum of a symmetric space M = G/H, where G is a simply connected compact semisimple Lie group and H is a closed subgroup of G, where the Riemannian metric on M comes from the Killing-Cartan form ⟨, ⟩ on the Lie algebra t of G. If v is the Lie algebra of H, then we have

t = v ⊕ s,

where v is the eigenspace with the eigenvalue +1 and s is the eigenspace with the eigenvalue −1.

Let m be the maximal abelian subspace of s and Λ_{m}^{+} the system of positive roots on mC.

We define the subset k of v as follows

k = {x ∈ v/[x, m] = 0} , (2.12)

and we consider the Cartan subalgebra b_{1} of k.

The following relaton is valid

b = b_{1} + m,

and therefore b is a Cartan subalgebra of t. We define the Λ_{m}^{+} as follows:

Λ_{m}^{+} = {λ/m : λ ∈ Λ^{+} and λ/m ≠ 0} , (2.13)

and we construct the set

B_{m} = \{μ ∈^{∗} |\frac{< μ, λ >}{< λ, λ >} ≥ 0, ∀λ ∈ Λ_{m}^{+}\} . (2.14)

The following theorem is valid Theorem. Let M = G/H the compact symmetric space of Riemann, where G is a simply connected, compact, semisimple Lie group and H is a closed subgroup of G, where the metric g on M comes from the Killing-Cartan form of the Lie algebra t of G. The spectrum Sp(M, g) is given by

Sp(M, g) = {< μ, μ > +2 < μ, δ_{m} >, μ ∈ B_{m}} , (2.15)

where δ is the half of the sum of the positive roots and δ_{m} is the restriction of δ on m, which is the maximal abelian subspace of s (the eigenspace with eigenvalue −1). The multiplicity of the eigenvalue is given by

P (μ) = ∏_{λ∈Λ^{+}} \frac{< μ + δ, λ >}{< δ, λ >}. (2.16)

3 The spectrum Sp^{0}(M, g), where M = SP (l)/SU(l)

Let sp(l) and su(l) be the Lie algebras of SP (l) and SU(l) respectively. Then we have

sp(l) = su(l) ⊕ m = h ⊕ m,

where the direct sum is valid if the Lie algebras are considered as vector spaces and m is the tangent space of the SP (l)/SU(l) at its origin.

The Lie algebra sp(l) has the form

sp(l) = \{X ∈ M_{2l}|XK + KX^{T} = O, where K = [ \begin{matrix}O & I_{l} \\ I_{l} & O \end{matrix} ]\} .

An element of the Lie algebra sp(l) is split into an element of su(l) and an element of the tangent space m in the following way

X ∈ sp(l) ⇒ X = [ \begin{matrix}{A \atop l × l} & {B \atop l × l} \\ {C \atop l × l} & \begin{matrix}A & ^{T} \\ l × l \end{matrix} \end{matrix} ] ,

where B and C are symmetric martices of order l.

The matrix A can be written A = [ \begin{matrix}a_{11} & \begin{matrix}A & 12 \\ 1 × (l − 1) \end{matrix} \\ \begin{matrix}A & 21 \\ (l − 1) × 1 \end{matrix} & \begin{matrix}A & 22 \\ (l − 1) × (l − 1) \end{matrix} \end{matrix} ] .

Therefore the vector X takes the form

X = [ \begin{matrix}\begin{matrix}a_{11} & \begin{matrix}A & 12 \\ 1 × (l − 1) \end{matrix} \\ \begin{matrix}A & 21 \\ (l − 1) × 1 \end{matrix} & \begin{matrix}A & 22 \\ (l − 1) × (l − 1) \end{matrix} \end{matrix} & {B \atop l × l} \\ {C \atop l × l} & \begin{matrix}−a_{11} & −A_{21}^{T} \\ −A_{12}^{T} & −A_{22}^{T} \end{matrix} \end{matrix} ] =

= [ \begin{matrix}\begin{matrix}−tr(A_{22}) & A_{12} \\ A_{21} & A_{22} \end{matrix} & O \\ O & \begin{matrix}tr(A_{22}) & −A_{21}^{T} \\ −A_{12}^{T} & −A_{22}^{T} \end{matrix} \end{matrix} ] ⊕

⊕ [ \begin{matrix}\begin{matrix}tr(A_{22}) + a_{11} & O \\ O & O \end{matrix} & B \\ C & \begin{matrix}−tr(A_{22}) − a_{11} & O \\ O & O \end{matrix} \end{matrix} ]

Hence the tangent space m takes the following form

m = \{[ \begin{matrix}\begin{matrix}k & O \\ O & O \end{matrix} & B \\ C & \begin{matrix}−k & O \\ O & O \end{matrix} \end{matrix} ] , k ∈ R\} ,

and the Lie algebra h becomes

\{[ \begin{matrix}\begin{matrix}−tr(A_{22}) & A_{12} \\ A_{21} & A_{22} \end{matrix} & O \\ O & \begin{matrix}tr(A_{22}) & −A_{21}^{T} \\ −A_{12}^{T} & −A_{22}^{T} \end{matrix} \end{matrix} ]\} .

Therefore the Lie algebra sp(l) can be written sp(l) = h ⊕ m. The roots of sp(l)

are given by \{±e_{i} ± e_{k}, ±2e_{i}|i < k, i, k = 1, ..., l\} , where e_{i} = \frac{1}{4l + 4} ·(E_{i} − E_{i+l}) , i =

1, ..., l and E_{i} is a matrix of order 2l which has at the position (i, i) the number one and everywhere else the number zero.

Taking into consideration the form of the tangent space m, the quantities e_{i} take the following form:

- for i = 1, e_{i} = \frac{1}{4l + 4} · (E_{1} − E_{1+l}) = O ⊕ \frac{1}{4l + 4} · (E_{1} − E_{l+1})

- for 2 ≤ i ≤ l, e_{i} = \frac{1}{4l + 4} · (E_{i} − E_{i+l}) = \frac{1}{4l + 4} · (E_{i} − E_{i+l}) ⊕ O.

At first we construct the weights of sp(l),

w_{i} = e_{1} + e_{2} + ... + e_{i} ⇒ w_{i_{m}} = e_{1} = \frac{1}{4l + 4} · (E_{1} − E_{1+l}) , i = 1, ..., l .

Then we construct the quantity

μ = m_{1}w_{1_{m}} + ... + m_{l}w_{l_{m}} where m_{1}, ..., m_{l} ∈ Z^{+} .

Taking into consideration the form of the weights restricted to the tangent space, the quantity μ takes the form μ = (m_{1} + m_{2} + ... + m_{l}) · e_{1} = m · e_{1}.

Now, we construct the half of the sum of all positive roots δ

2δ = 2le_{1} + (2l − 2) e_{2} + ... + 4e_{l−1} + 2e_{l}.

Subsequently the restriction of the half of the sum of all positive roots to the tangent space takes the following form δ_{m} = l · e_{1}. At first we calculate the quantities which appeare in formula (17)

⟨μ, μ⟩ = ⟨m · e_{1}, m · e_{1}⟩ = m^{2}⟨e_{1}, e_{1}⟩ = \frac{m^{2}}{4l + 4};

⟨μ, δ_{m}⟩ = ⟨m · e_{1}, l · e_{1}⟩ = m · l · ⟨e_{1}, e_{1}⟩ = \frac{ml}{4l + 4}.

So, the spectrum takes the form

Sp(SP (l)/SU(l), g) = \{\frac{m^{2} + 2ml}{4l + 4} , m ∈ Z\} . (3.1)

The positive roots are i) e_{i} − e_{j}, 1 ≤ i < j ≤ l

ii) e_{i} + e_{j}, 1 ≤ i < j ≤ l
iii) 2e_{i}, 1 ≤ i ≤ l

and for

μ + δ = me_{1} + le_{1} + (l − 1)e_{2} + ... + e_{l} = (m + l)e_{1} + (l − 1)e_{2} + ... + e_{l},

we construct

P_{1} = ∏_{1≤i<j≤l} \frac{< μ + δ, e_{i} − e_{j} >}{< δ, e_{i} − e_{j} >} = ∏_{2≤j≤l} \frac{< μ + δ, e_{1} − e_{j} >}{< δ, e_{1} − e_{j} >} = ∏_{2≤j≤l} \frac{m + j − 1}{j − 1} =

( \begin{matrix}m + l − 1 \\ l − 1 \end{matrix} );

P_{2} = ∏_{1≤i<j≤l} \frac{< μ + δ, e_{i} + e_{j} >}{< δ, e_{i} + e_{j} >} = ∏_{2≤j≤l} \frac{< μ + δ, e_{1} + e_{j} >}{< δ, e_{1} + e_{j} >} = ∏_{2≤j≤l} ∏\frac{m + 2l − j + 1}{2l − j + 1};

P_{3} = ∏_{1≤i≤l} \frac{< μ + δ, 2e_{i} >}{< δ, 2e_{i} >} = \frac{< μ + δ, 2e_{1} >}{< δ, 2e_{1} >} = \frac{m + l}{l} .

Having taken into consideration the relation (18) the multiplicity of the eigenvalue is taken then

dμ^{2} = P_{1}P_{2}P_{3} = ∏_{k=1}^{2l−1} \frac{m + k}{k} = ( \begin{matrix}m + 2l − 1 \\ 2l − 1 \end{matrix} ) . (3.2)

We conclude with the following Theorem. The spectrum Sp(M, g), where M = SP (L)/SU(l), is given by the relation (19). The multiplicity of the eigenvalue is given by the relation (20).

References

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Division of Mathematics

Department of Mathematics and Physics

Aristotle University of Thessaloniki, Thessaloniki 54 250

email: kkalogerid@yahoo.gr