Asymptotic properties of unitary representations

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If ρ{variant} is a unitary representation of a locally compact group G, one is very often interested in the asymptotic behavior of the operators ρ{variant}(g) as g tends to ∞ in G. Put another way one is interested in what operators not in ρ{variant}(G) can be (weak) limits of the ρ{variant}(g) as g tends to ∞. The first part of the paper deals with the question of what kinds of unitary operators can be limits of ρ{variant}(g), and we show that for connected G the possibilities are very limited. The second part of the paper deals with the question of when one can conclude that essentially the only operator of any kind not in ρ{variant}(G) obtained as a limit of the ρ{variant}(g) is 0. This amounts to showing that the matrix coefficients of ρ{variant} "vanish at ∞," and we establish affirmative results for irreducible representations of connected algebraic groups over local fields (archimedian and non-archimedian). Such results turn out to have applications to the theory of automorphic forms and to ergodic theory. © 1979.




Howe, R. E., & Moore, C. C. (1979). Asymptotic properties of unitary representations. Journal of Functional Analysis, 32(1), 72–96.

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