A Generalized Gelfand Pair Attached to a 3-Step Nilpotent Lie Group
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(2020) 26:62
A Generalized Gelfand Pair Attached to a 3-Step Nilpotent Lie Group Andrea L. Gallo1 · Linda V. Saal1 Received: 18 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let N be a nilpotent Lie group and K a compact subgroup of the automorphism group Aut(N ) of N . It is well-known that if (K N , K ) is a Gelfand pair then N is at most 2-step nilpotent Lie group. The notion of Gelfand pair was generalized when K is a non-compact group. In this work, we give an example of a 3-step nilpotent Lie group and a non-compact subgroup K of Aut(N ) such that (K N , N ) is a generalized Gelfand pair. Keywords Generalizd Gelfand pairs · Nilpotent Lie group Mathematics Subject Classification 43A80 · 22E25
1 Introduction Let G be a Lie group and K a compact subgroup of G. We denote by D(G/K ) the space of C ∞ -functions on G/K with compact support and by D K (G) the subspace of D(G) of functions on G which are right K -invariant. Both spaces are identified by mapping f ∈ D(G/K ) to f 0 := f ◦ P, where P : G → G/K is the natural projection. It follows from the Schwartz’s kernel Theorem, that every linear operator which maps continuously D(G/K ) in D (G/K ) with respect to the standard topologies
Communicated by Fulvio Ricci. The authors are partially supported by CONICET and SECYT-UNC.
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Andrea L. Gallo [email protected] Linda V. Saal [email protected]
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FaMAF Universidad Nacional de Córdoba CIEM (CONICET), 5000 Córdoba, Argentina 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:62
and commuting with the action of G is a convolution operator with a K -bi-invariant distribution in D (G). In particular, we consider the subalgebra of convolution operators which kernels are K -bi-invariant integrable functions on G. When this algebra is commutative, we can expect a kind of simultaneous “diagonalization” of all these operators. This motivated, in part, the study of Gelfand pairs and the corresponding spherical analysis. In this sense, we begin by introducing the concept of Gelfand pair. The following statements are equivalent: (i) The convolution algebra of K -bi-invariant integrable functions on G is commutative. (ii) For any irreducible unitary representation (π, H) of G, the subspace H K of vectors fixed by K is at most one dimensional. When any of the above holds, we say that (G, K ) is a Gelfand pair. Very well studied examples of Gelfand pairs are provided by symmetric pairs of compact or non-compact types. More recent works have put attention on Gelfand pairs of the form (K N , K ) where N is a nilpotent Lie group and K is a subgroup of the automorphism group Aut(N ) of N (see [1–4,6,7,12], among others). One of the first results, proved in [2], stated that if (K N , K ) (in short (K , N )) is a Gelfand pair then N is abelian or a 2-step nilpotent group. The notion of Gelfand pair was extended to the case when K is non-compact. Observe that, in this case, the space of K -invariant integrable functions on G/
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