Quantization and Projective Representations of Solvable Lie Groups

Moscovici, Henri; Verona, Andrei

doi:10.2307/1997970

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Corollary 12 With the Assumptions And Notations As In1

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2022

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“…A particular motivation for obtaining Theorem 1.3 was to investigate the optimality of the density conditions in Corollary 1.2 for the existence of frames and Riesz sequences for general exponential Lie groups, i.e., Lie groups for which the exponential map is a diffeomorphism. For a description of the projective discrete series of an exponential Lie group in terms of the Kirillov correspondence, see [12,20]; in particular, cf. [20,Proposition 4].…”

Section: Corollary 12 With the Assumptions And Notations As Inmentioning

confidence: 99%

On sufficient density conditions for lattice orbits of relative discrete series

Enstad

Velthoven

2022

Arch. Math.

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This note provides new criteria on a unimodular group G and a discrete series representation $$(\pi , \mathcal {H}_{\pi })$$ ( π , H π ) of formal degree $$d_{\pi } > 0$$ d π > 0 under which any lattice $$\Gamma \le G$$ Γ ≤ G with $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi } \le 1$$ vol ( G / Γ ) d π ≤ 1 (resp. $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi } \ge 1$$ vol ( G / Γ ) d π ≥ 1 ) admits $$g \in \mathcal {H}_{\pi }$$ g ∈ H π such that $$\pi (\Gamma ) g$$ π ( Γ ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

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Section: Corollary 12 With the Assumptions And Notations As Inmentioning

confidence: 99%