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On the Fourier transform of a compact semisimple Lie group. (English) Zbl 0842.22015

The difficulties of the standard Fourier transform theory in the nonabelian case are well known and have led to numerous attempts of reformulating the theory in a way which avoids problems caused by the “operator-valuedness” of the conventional Fourier transform. The paper describes such an alternative framework for the study of the Fourier transform for a compact connected semisimple Lie group, denoted throughout by \(G\). The key idea is to replace for each irreducible representation \((V,\rho)\) of \(G\) the space of endomorphisms of \(V\) by a distinguished algebra of functions on an appropriate space, the composition of endomorphisms corresponding to a certain multiplication in this algebra. The construction is quite elaborate and relies on a detailed study of orbits of the coadjoint action of \(G\) in the dual \({\mathfrak g}^*\) of its Lie algebra \(\mathfrak g\) and their association to irreducible representations \((V,\rho)\) of \(G\). A class of \(G\)-orbits in the representation spaces \(V\) of \(G\) is singled out under the name of effective orbits and the symbol map assigning to endomorphisms of \(V\) certain functions on such an orbit is introduced and studied in detail. By using the moment map the author constructs a canonical \(S^1\) bundle associated to an irreducible representation \((V,\rho)\) of \(G\). It is of the form \(\Phi : {\mathcal M}_\rho \to {\mathcal O}_\rho\), where \({\mathcal M}_\rho \subset V\) is an effective \(G\)-orbit, \({\mathcal O}_\rho \subset {\mathfrak g}^*\) is a uniquely defined, extremal in an appropriate sense, co-adjoint orbit and \(\Phi\) is the moment map. Via this map the symbols of operators on \(V\) are then interpreted as functions on the orbit \({\mathcal O}_\rho\) and form a distinguished finite dimensional space of functions on \({\mathcal O}_\rho\), denoted \(A_\rho\), which obtains an algebra structure with respect to a \(*\)-product, which is the product of operators on \(V\) transfered to \(A_\rho\). The set of extremal orbits coincides with the set \({\mathfrak g}^*_{\text{INT}}\) of integral coadjoint orbits in the sense of geometric quantization. The formula \[ e(g,f) = \langle g^{-1} \cdot \Phi^{-1} (f), \Phi^{-1}(f)\rangle \] defines the Fourier kernel \(e(g,f)\) of \(G\) as a function on \(G \times {\mathfrak g}^*_{\text{INT}} = T^* G_{\text{INT}} \subset T^* G\). The Fourier transform \(F : {\mathcal F}(G) \to {\mathcal F}({\mathfrak g}^*_{\text{INT}})\) is now defined for suitable functions on \(G\) by the formula \[ F \phi(f) = \int_G \phi(g) e(g,f) dg. \] It is connected with the standard Fourier transform by the fact that the restriction of \(F\phi\) to the extremal orbit \({\mathcal O}_\rho\) corresponding to a given irreducible representation \((V, \rho)\) is the previously defined symbol of the operator \(\rho (\phi)\) for each \(\phi \in L^1(G)\). One of the main results concerning the so defined Fourier transform is that it maps the space \(U_\rho\) of matrix coefficients of \((V,\rho)\) isomorphically onto \(A_\rho\) and the standard convolution of functions on \(G\) corresponds on each orbit \({\mathcal O}_\rho\) to the \(*\)-product of their Fourier transforms. There is also proven the following interesting analog of the Kirillov character formula: If \(\chi_\rho\) is the character of an irreducible representation \((V, \rho)\) of \(G\) in a space of dimension \(n\), then \(F \overline {\chi}_\rho (f) = 1/n\) for all \(f \in {\mathcal O}_\rho\).
The paper concludes with a detailed illustration of the theory in the case of the \(\text{SU} (2)\) group. The construction given in this work is related to some of the earlier results, notably those elaborated for the case of the euclidean sphere by T. O. Sherman in [Trans. Am. Math. Soc. 209, 1-31 91975; Zbl 0308.43009)] and later treated in complete generality by the same author in [Acta Math. 164, 73-144 (1990; Zbl 0707.43001)] and also bears some similarity to the earlier work of the author on the Fourier transform for nilpotent Lie groups. It has also some relation to the studies of the so called \(*\)-products in the theory of geometric quantization of symplectic manifolds.

MSC:

22E46 Semisimple Lie groups and their representations
43A32 Other transforms and operators of Fourier type
53D50 Geometric quantization
43A75 Harmonic analysis on specific compact groups
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