Pizzetti formula on the Grassmannian of 2-planes
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Pizzetti formula on the Grassmannian of 2‑planes D. Eelbode1 · Y. Homma2 Received: 11 March 2020 / Accepted: 27 July 2020 / Published online: 6 August 2020 © Springer Nature B.V. 2020
Abstract This paper is devoted to the role played by the Higgs algebra H3 in the generalisation of classical harmonic analysis from the sphere Sm−1 to the (oriented) Grassmann manifold Gro (m, 2) of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group SO(m) acting on functions on the Grassmannian. This is then used to obtain a Pizzetti formula for integration over this manifold. The resulting formulas are finally compared to formulas obtained earlier for the Pizzetti integration over Stiefel manifolds, using an argument involving symmetry reduction.
1 Introduction The theory of spherical harmonics on ℝm is a beautiful piece of mathematics, with many applications in, for instance, representation theory, physics and even engineering. It has grown out of, and is centred around, the notion of the Laplace operator on ℝm , given by
∑ 𝜕2 𝜕2 𝜕2 +⋯+ 2 = . 2 2 𝜕xm 𝜕x1 j=1 𝜕xj m
Δx =
Null solutions for Δx in the polynomial ring P(ℝm , ℂ) ∶= ℂ[x1 , … , xm ] are commonly referred to as harmonic polynomials on ℝm . Their restrictions to the sphere Sm−1 ⊂ ℝm are the so-called spherical harmonics (indexed by a positive integer k ∈ ℤ+ , which then refers to the degree of homogeneity of the harmonic polynomial it uniquely extends to). These functions realise the eigenspaces of the Laplace–Beltrami operator on the homogeneous space Sm−1 , which is closely connected to the Casimir operator of order 2 for the Lie algebra 𝔰𝔬(m) . From a purely algebraic point of view, the operator Δx arises if one wants to understand the behaviour of the space P(ℝm , ℂ) as a representation for the (special) orthogonal group, under the regular action ( ) H ∶ SO(m) → Aut P(ℝm , ℂ)
* D. Eelbode [email protected] 1
University of Antwerp, Antwerpen, Belgium
2
Waseda University, Tokyo, Japan
13
Vol.:(0123456789)
326
Annals of Global Analysis and Geometry (2020) 58:325–350
with P(x) ↦ H(g)[P(x)] ∶= P(g−1 x) , where x = (x1 , … , xm )T and with g an arbitrary element of the group SO(m). It is well known that under this action, one has that
P(ℝm , ℂ) =
∞ �
Pk (ℝm , ℂ) =
k=0
⌊k⌋
∞ 2 � �
r2j Hk−2j (ℝm , ℂ),
(1)
k=0 j=0
2 denotes the squared norm of the vector where Hk = Pk ∩ ker Δx and r2 = x12 + ⋯ + xm m x ∈ ℝ . Here, the spaces Hk of k-homogeneous harmonic polynomials define an irreducible module for SO(m) with the highest weight (k, 0, … , 0) . A crucial observation which can be made here is that the Lie algebra spanned by Δx and r2 , seen as a subalgebra of the Weyl algebra W(ℝm , ℂ) acting on the space P(ℝm , ℂ) , is given by ) ( 1 1 m , 𝔰𝔩(2) = Alg(X, Y, H) ≅ Alg r2 , − Δx , 𝔼x + (2) 2 2 2 ∑ with 𝔼x = j xj 𝜕xj the so-called Euler operator (acting as a constant k on homogeneous polynomials of degree k). This had led to the celebrated Howe duality theorem, which in this particular ca
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