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On Gelfand pairs associated with solvable Lie groups. (English) Zbl 0704.22006

Let \(G\) be a locally compact group and let \(K\) be a compact subgroup of the group of automorphisms of \(G\). Then \(K\) naturally acts on the convolution algebra \(L^ 1(G)\) and we denote by \(L^ 1_ K(G)\) the subalgebra of all invariant elements. The pair \((K,G)\) is called a Gelfand pair if \(L^ 1_ K(G)\) is commutative.
In this paper, the authors characterize Gelfand pairs \((K,S)\), \(K\) being connected, for a connected, simply connected solvable Lie group \(S\) and study the associated \(K\)-spherical functions.
They first show that, for a Lie group \(G\), \((K,G)\) is a Gelfand pair if and only if \((K\cdot x)(K\cdot y)=(K\cdot y)(K\cdot x)\) for all \(x,y\in G\). This criterion serves repeatedly to characterize a Gelfand pair. For example, \(N\) being a connected and simply connected nilpotent Lie group, we see: if \((K,N)\) is a Gelfand pair, then \(N\) is at most two step. Hence the classification of Gelfand pairs \((K,N)\) reduces to that of Gelfand pairs \((K,H_ n)\), where \(H_ n\) is the \(2n+1\)-dimensional Heisenberg group. Now \(K\) acts on the unitary dual \(\hat H_ n\) of \(H_ n\) and one constructs the intertwining representation \(W_{\pi}\) of the isotropy subgroup \(K_{\pi}\) of \(\pi\in \hat H_ n\). To determine if \((K,H_ n)\) is a Gelfand pair, they apply to \(W_{\pi}\) a result due to G. Carcano [Boll. Unione Mat. Ital., VII. Ser. B 1, 1091–1105 (1987; Zbl 0632.22005)].
Let \((K,G)\) be a Gelfand pair. A continuous, complex valued function \(\phi\) on \(G\) is called a \(K\)-spherical function if \(\phi\) satisfies \(\phi (e)=1\) and \(\int_{K}\phi (xk\cdot y)\,dk=\phi (x)\phi (y)\). Their main result on spherical functions is the following. Let \(K, N\) be as before such that \((K,N)\) is a Gelfand pair. Then \(\phi\) is a bounded \(K\)-spherical function if and only if there is a \(\pi\in N\) and a \(\xi\in V\subseteq {\mathcal H}_{\pi}\), \(\| \xi \| =1\), such that for each \(x\in N\), \(\phi (x)=\phi_{\pi,\xi}(x):=\int_{K}\langle\pi (k\cdot x)\xi,\xi \rangle\,dk\). Here \(V\) is an irreducible subspace for the intertwining representation \(W_{\pi}\). Furthermore, \(\phi_{\pi,\xi}=\phi_{\pi ',\xi '}\) if and only if \(\pi '=\pi_ k\), \(\pi_ k(x)=\pi (k\cdot x)\) (x\(\in N)\), for some \(k\in K\) and \(\xi,\xi '\) belong to the same \(V\).
Reviewer: H. Fujiwara

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E30 Analysis on real and complex Lie groups
22E25 Nilpotent and solvable Lie groups
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A90 Harmonic analysis and spherical functions
22D15 Group algebras of locally compact groups

Citations:

Zbl 0632.22005
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