Principal Gelfand pairs

Yakimova, Oksana

doi:10.1007/s00031-005-1110-9

Cited by 26 publications

(42 citation statements)

References 20 publications

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“…Proof. The first part is a restatement of the well-known fact that SO(n) ⋉ R n , SO(n) , U (n) ⋉ H n , U (n) are strong Gelfand pairs [27,Thm. 3].…”

Section: Spherical Functions Of Positive Typementioning

confidence: 99%

Spherical analysis on homogeneous vector bundles

Ricci

Samanta

2018

Advances in Mathematics

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Given a Lie group G, a compact subgroup K and a representation τ ∈ K, we assume that the algebra of End(Vτ )-valued, bi-τ -equivariant, integrable functions on G is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of G-invariant differential operators on the homogeneous bundle Eτ over G/K. In particular, we observe that, under the above assumptions, (G, K) is a Gelfand pair and show that the Gelfand spectrum for the triple (G, K, τ ) admits homeomorphic embeddings in C n .In the second part, we develop in greater detail the spherical analysis for G = K ⋉ H with H nilpotent. In particular, for H = R n and K ⊂ SO(n) and for the Heisenberg group Hn and K ⊂ U (n), we characterize the representations τ ∈ K giving a commutative algebra.2010 Mathematics Subject Classification. 43A90, 43A85.

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“…Proof. The first part is a restatement of the well-known fact that SO(n) ⋉ R n , SO(n) , U (n) ⋉ H n , U (n) are strong Gelfand pairs [27,Thm. 3].…”

Section: Spherical Functions Of Positive Typementioning

confidence: 99%

Spherical analysis on homogeneous vector bundles

Ricci

Samanta

2018

Advances in Mathematics

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“…For k = 1 the statements are wrong, see [1] for a counterexample. In the global setting (i.e., when the manifold is complete), both statements were proved in [1]; the proof is quite involving, is essentially global, and in particular extensively uses the quite nontrivial results on weakly symmetric spaces from [3,4,5,6]. Our proof is much easier, shorter, and works locally.…”

Section: Introductionmentioning

confidence: 85%

Locally 2-fold symmetric manifolds are locally symmetric

Deng

Matveev

2017

Arch. Math.

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A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that for $k \ge 2$, Riemannian, pseudoriemannian and Finslerian locally $k$-fold symmetric manifolds are locally symmetric

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“…These (G n /K n ) are constructed from certain basic ones that satisfy several technical conditions (indecomposable, principal, maximal and Sp(1)-saturated). See [16,[19][20][21] and [18]. The basic such direct systems, with K n reducible on n n /z n , dim z n bounded and {K n } parabolic, are tabulated in [18, Table 9.15] as follows.…”

Section: Reducible Quotientsmentioning

confidence: 99%